Advertisement

Nonlocal Problems

Part of the Birkhäuser Advanced Texts / Basler Lehrbücher book series (BAT)

Abstract

In this chapter, we study various problems with nonlocal nonlinearities. The equations that we consider involve nonlocal terms taking the form of an integral in space, or in time. These terms may also be combined with local ones, either in an additive or in a multiplicative way.

Keywords

Maximum Principle Dirichlet Boundary Condition Global Existence Comparison Principle Nonlocal Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    M. Abramowitz and I. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables,, Wiley, New York, 1972.MATHGoogle Scholar
  2. [2]
    N. Ackermann and T. Bartsch, Superstable manifolds of semilinear parabolic problems, J. Dynam. Differential Equations 17 (2005), 115–173.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    N. Ackermann, T. Bartsch, P. Kaplický and P. Quittner, A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems, Trans. Amer. Math. Soc. (to appear).Google Scholar
  4. [4]
    S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand, Princeton, N.J., 1965.Google Scholar
  5. [5]
    J. Aguirre and M. Escobedo, On the blow up of solutions for a convective reaction diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 449–470.MathSciNetGoogle Scholar
  6. [6]
    N. Alaa, Solutions faibles d’équations paraboliques quasi-linéaires avec données initiales mesures, Ann. Math. Blaise Pascal 3 (1996), 1–15.MATHMathSciNetGoogle Scholar
  7. [7]
    S. Alama and M. Del Pino, Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 95–115.MATHGoogle Scholar
  8. [8]
    S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993), 439–475.MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    F. Alessio, P. Caldiroli and P. Montecchiari, Infinitely many solutions for a class of semilinear elliptic equations inN, Boll. Unione Mat. Ital. (8) 4-B (2001), 311–318.MathSciNetGoogle Scholar
  10. [10]
    L. Alfonsi and F.B. Weissler, Blow up in ℝ n for a parabolic equation with a damping nonlinear gradient term, Progress in nonlinear differential equations (N.G. Lloyd et al., eds.), Birkhäuser, Basel, 1992.Google Scholar
  11. [11]
    N.D. Alikakos, L p bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979), 827–868.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    N.D. Alikakos, An application of the invariance principle to reaction diffusion equations, J. Differential Equations 33 (1979), 201–225.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), 593–676.MATHMathSciNetGoogle Scholar
  14. [14]
    H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations 72 (1988), 201–269.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis (H.J. Schmeisser and H. Triebel, eds.), Teubner, Stuttgart, Leipzig, 1993, pp. 9–126.Google Scholar
  16. [16]
    H. Amann, Linear and quasilinear parabolic problems, Volume I: Abstract linear theory, Birkhäuser, Basel, 1995.Google Scholar
  17. [17]
    H. Amann, Linear and quasilinear parabolic problems, Volume II, in preparation.Google Scholar
  18. [18]
    H. Amann, M. Hieber and G. Simonett, Bounded H -calculus for elliptic operators, Differential Integral Equations 7 (1994), 613–653.MATHMathSciNetGoogle Scholar
  19. [19]
    H. Amann and J. Lopez-Gomez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations 146 (1998), 336–374.MATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    H. Amann and P. Quittner, Elliptic boundary value problems involving measures: existence, regularity, and multiplicity, Adv. Differential Equations 3 (1998), 753–813.MATHMathSciNetGoogle Scholar
  21. [21]
    H. Amann and P. Quittner, Semilinear parabolic equations involving measures and low regularity data, Trans. Amer. Math. Soc. 356 (2004), 1045–1119.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    H. Amann and P. Quittner, Optimal control problems with final observation governed by explosive parabolic equations, SIAM J. Control Optim. 44 (2005), 1215–1238.MATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.MATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    A. Ambrosetti and P.N. Srikanth, Superlinear elliptic problems and the dual principle in critical point theory, J. Math. Phys. Sci. 18 (1984), 441–451.MATHMathSciNetGoogle Scholar
  25. [25]
    A. Ambrosetti and M. Struwe, A note on the problemΔu = λu + u¦u¦ 2*-2, Manuscripta Math. 54 (1986), 373–379.MATHMathSciNetCrossRefGoogle Scholar
  26. [26]
    L. Amour and M. Ben-Artzi, Global existence and decay for viscous Hamilton-Jacobi equations, Nonlinear Anal. 31 (1998), 621–628.MATHMathSciNetCrossRefGoogle Scholar
  27. [27]
    D. Andreucci, Degenerate parabolic equations with initial data measures, Trans. Amer. Math. Soc. 349 (1997), 3911–3923.MATHMathSciNetCrossRefGoogle Scholar
  28. [28]
    D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), 363–441.MATHMathSciNetGoogle Scholar
  29. [29]
    D. Andreucci, M.A. Herrero and J.J.L. Velázquez, Liouville theorems and blow up behaviour in semilinear reaction diffusion systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 1–53.MATHCrossRefGoogle Scholar
  30. [30]
    S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79–96.MATHMathSciNetGoogle Scholar
  31. [31]
    S. Angenent and M. Fila, Interior gradient blow-up in a semilinear parabolic equation, Differential Integral Equations 9 (1996), 865–877.MATHMathSciNetGoogle Scholar
  32. [32]
    S. Angenent and R. van der Vorst, A priori bounds and renormalized Morse indices of solutions of an elliptic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), 277–306.MATHCrossRefGoogle Scholar
  33. [33]
    S.N. Antontsev and M. Chipot, The thermistor problem: existence, smoothness, uniqueness, blow-up, SIAM J. Math. Anal. 25 (1994), 1128–1156.MATHMathSciNetCrossRefGoogle Scholar
  34. [34]
    G. Arioli, F. Gazzola, H.-Ch. Grünau and E. Sassone, The second bifurcation branch for radial solutions of the Brezis-Nirenberg problem in dimension four, NoDEA Nonlinear Differential Equations Appl. (to appear). ][35]_D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math. 30 (1978), 33–76.Google Scholar
  35. [36]
    J.M. Arrieta and A. Rodriguez-Bernal, Non well posedness of parabolic equations with supercritical nonlinearities, Comm. Contemp. Math. 6 (2004), 733–764.MATHMathSciNetCrossRefGoogle Scholar
  36. [37]
    J.M. Arrieta, A. Rodriguez-Bernal, J.W. Cholewa and T. Dlotko, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci. 14 (2004), 253–293.MATHMathSciNetCrossRefGoogle Scholar
  37. [38]
    J.M. Arrieta, A. Rodriguez-Bernal and Ph. Souplet, Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3 (2004), 1–15.MATHMathSciNetGoogle Scholar
  38. [39]
    F.V. Atkinson, H. Brezis and L.A. Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents, J. Differential Equations 85 (1990), 151–170.MATHMathSciNetCrossRefGoogle Scholar
  39. [40]
    F.V. Atkinson and L.A. Peletier, Emden-Fowler equations involving critical exponents, Nonlinear Anal. 10 (1986), 755–776.MATHMathSciNetCrossRefGoogle Scholar
  40. [41]
    F.V. Atkinson and L.A. Peletier, Large solutions of elliptic equations involving critical exponents, Asymptotic Anal. 1 (1988), 139–160.MATHMathSciNetGoogle Scholar
  41. [42]
    F.V. Atkinson and L.A. Peletier, Oscillations of solutions of perturbed autonomous equations with an application to nonlinear elliptic eigenvalue problems involving critical Sobolev exponents, Differential Integral Equations 3 (1990), 401–433.MATHMathSciNetGoogle Scholar
  42. [43]
    T. Aubin, Problèmes isopérimétriques de Sobolev, J. Differential Geometry 11 (1976), 573–598.MATHMathSciNetGoogle Scholar
  43. [44]
    P. Aviles, On isolated singularities in some nonlinear partial differential equations, Indiana Univ. Math. J. 32 (1983), 773–791.MATHMathSciNetCrossRefGoogle Scholar
  44. [45]
    A. Bahri, Topological results on a certain class of functionals and application, J. Funct. Anal. 41 (1981), 397–427.MATHMathSciNetCrossRefGoogle Scholar
  45. [46]
    A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), 1–32.MATHMathSciNetCrossRefGoogle Scholar
  46. [47]
    A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253–294.MATHMathSciNetCrossRefGoogle Scholar
  47. [48]
    A. Bahri and P.-L. Lions, Morse index of some min-max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988), 1027–1037.MATHMathSciNetCrossRefGoogle Scholar
  48. [49]
    A. Bahri and P.-L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992), 1205–1215.MATHMathSciNetCrossRefGoogle Scholar
  49. [50]
    A. Bahri and P.-L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 365–413.MATHMathSciNetCrossRefGoogle Scholar
  50. [51]
    J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. (2) 28 (1977), 473–486.MATHMathSciNetCrossRefGoogle Scholar
  51. [52]
    A. Barabanova, On the global existence of solutions of a reaction-diffusion equation with exponential nonlinearity, Proc. Amer. Math. Soc. 122 (1994), 827–831.MATHMathSciNetCrossRefGoogle Scholar
  52. [53]
    P. Baras, Non unicité des solutions d’une équation d’évolution non linéaire, Ann. Fac. Sci. Toulouse Math. (5) 5 (1983), 287–302.MATHMathSciNetGoogle Scholar
  53. [54]
    P. Baras and L. Cohen, Complete blow-up after T max for the solution of a semilinear heat equation, J. Funct. Anal. 71 (1987), 142–174.MATHMathSciNetCrossRefGoogle Scholar
  54. [55]
    P. Baras and R. Kersner, Local and global solvability of a class of semilinear parabolic equations, J. Differential Equations 68 (1987), 238–252.MATHMathSciNetCrossRefGoogle Scholar
  55. [56]
    P. Baras and M. Pierre, Critère d’existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 185–212.MATHMathSciNetGoogle Scholar
  56. [57]
    G. Barles and F. Da Lio, On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations, J. Math. Pures Appl. 83 (2004), 53–75.MATHMathSciNetGoogle Scholar
  57. [58]
    J.-Ph. Bartier, Global behavior of solutions of a reaction diffusion equation with gradient absorption in unbounded domains, Asympt. Anal. 46 (2006), 325–347.MATHMathSciNetGoogle Scholar
  58. [59]
    J.-Ph. Bartier and Ph. Souplet, Gradient bounds for solutions of semilinear parabolic equations without Bernstein’s quadratic condition, C. R. Acad. Sci. Paris Sér. I Math. 338 (2004), 533–538.MATHMathSciNetGoogle Scholar
  59. [60]
    J. Bebernes, A. Bressan and A.A. Lacey, Total blow-up versus single-point blow-up, J. Differential Equations 73 (1988), 30–44.MATHMathSciNetCrossRefGoogle Scholar
  60. [61]
    J. Bebernes and S. Bricher, Final time blowup profiles for semilinear parabolic equations via center manifold theory, SIAM J. Math. Anal. 23 (1992), 852–869.MATHMathSciNetCrossRefGoogle Scholar
  61. [62]
    J. Bebernes and D. Eberly, A description of self-similar blow-up for dimensions n ≥ 3, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), 1–21.MATHMathSciNetGoogle Scholar
  62. [63]
    J. Bebernes and D. Eberly, Mathematical problems from combustion theory, Springer, New York, 1989.MATHGoogle Scholar
  63. [64]
    J. Bebernes and A.A. Lacey, Finite-time blowup for a particular parabolic system, SIAM J. Math. Anal. 21 (1990), 1415–1425.MATHMathSciNetCrossRefGoogle Scholar
  64. [65]
    J. Bebernes and A.A. Lacey, Finite time blowup for semilinear reactivediffusive systems, J. Differential Equations 95 (1992), 105–129.MATHMathSciNetCrossRefGoogle Scholar
  65. [66]
    J. Bebernes and A.A. Lacey, Global existence and finite-time blow-up for a class of nonlocal parabolic problems, Adv. Differential Equations 2 (1996), 927–953.MathSciNetGoogle Scholar
  66. [67]
    H. Bellout, Blow-up of solutions of parabolic equations with nonlinear memory, J. Differential Equations 70 (1987), 42–68.MATHMathSciNetCrossRefGoogle Scholar
  67. [68]
    V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal. 114 (1991), 79–93.MATHMathSciNetCrossRefGoogle Scholar
  68. [69]
    S. Benachour, S. Dăbuleanu-Hapca and Ph. Laurençot, Decay estimates for a viscous Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions, Asympt. Anal. 51 (2007), 209–229.MATHGoogle Scholar
  69. [70]
    S. Benachour, G. Karch and Ph. Laurençot, Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations, J. Math. Pures Appl. 83 (2004), 1275–1308.MATHMathSciNetGoogle Scholar
  70. [71]
    S. Benachour and Ph. Laurençot, Global solutions to viscous Hamilton Jacobi equations with irregular data, Comm. Partial Differential Equations 24 (1999), 1999–2021.MATHMathSciNetCrossRefGoogle Scholar
  71. [72]
    M. Ben-Artzi, Ph. Souplet and F.B. Weissler, The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces, J. Math. Pures Appl. 81 (2002), 343–378.MATHMathSciNetCrossRefGoogle Scholar
  72. [73]
    R.D. Benguria, J. Dolbeault and M.J. Esteban, Classification of the solutions of semilinear elliptic problems in a ball, J. Differential Equations 167 (2000), 438–466.MATHMathSciNetCrossRefGoogle Scholar
  73. [74]
    H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994), 59–78.MATHMathSciNetGoogle Scholar
  74. [75]
    H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl. 2 (1995), 553–572.MATHMathSciNetCrossRefGoogle Scholar
  75. [76]
    H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313–345.MATHMathSciNetGoogle Scholar
  76. [77]
    J. Bergh and J. Löfström, Interpolation spaces. An introduction, Springer, Berlin-Heidelberg New York, 1976.MATHGoogle Scholar
  77. [78]
    G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on ℝ n or R+n through the method of moving planes, Comm. Partial Differential Equations 22 (1997), 1671–1690.MATHMathSciNetCrossRefGoogle Scholar
  78. [79]
    M.-F. Bidaut-Véron, Initial blow-up for the solutions of a semilinear parabolic equation with source term, Equations aux dérivées partielles et applications, articles dédiés à Jacques-Louis Lions, Gauthier-Villars, Paris, 1998, pp. 189–198.Google Scholar
  79. [80]
    M.-F. Bidaut-Véron, Local behaviour of the solutions of a class of nonlinear elliptic systems, Adv. Differential Equations 5 (2000), 147–192.MATHMathSciNetGoogle Scholar
  80. [81]
    M.-F. Bidaut-Véron, A.C. Ponce and L. Véron, Boundary singularities of positive solutions of some nonlinear elliptic equations, C. R. Math. Acad. Sci. Paris 344 (2007), 83–88.MATHMathSciNetGoogle Scholar
  81. [82]
    M.-F. Bidaut-Véron and Th. Raoux, Asymptotics of solutions of some nonlinear elliptic systems, Comm. Partial Differential Equations 21 (1996), 1035–1086.MATHMathSciNetCrossRefGoogle Scholar
  82. [83]
    M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), 489–539.MATHMathSciNetCrossRefGoogle Scholar
  83. [84]
    M.-F. Bidaut-Véron and L. Vivier, An elliptic semilinear equation with source term involving boundary measures: the subcritical case, Rev. Mat. Iberoamericana 16 (2000), 477–513.MATHMathSciNetGoogle Scholar
  84. [85]
    M.-F. Bidaut-Véron and C. Yarur, Semilinear elliptic equations and systems with measure data: existence and a priori estimates, Adv. Differential Equations 7 (2002), 257–296.MATHMathSciNetGoogle Scholar
  85. [86]
    P. Biler, M. Guedda and G. Karch, Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation, J. Evol. Equ. 4 (2004), 75–97.MATHMathSciNetCrossRefGoogle Scholar
  86. [87]
    I. Birindelli and E. Mitidieri, Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1217–1247.MATHMathSciNetGoogle Scholar
  87. [88]
    L. Boccardo, F. Murat and J.-P. Puel, Existence results for some quasilinear parabolic equations, Nonlinear Anal. 13 (1989), 373–392.MATHMathSciNetCrossRefGoogle Scholar
  88. [89]
    M. Bouhar and L. Véron, Integral representation of solutions of semilinear elliptic equations in cylinders and applications, Nonlinear Anal. 23 (1994), 275–296.MATHMathSciNetCrossRefGoogle Scholar
  89. [90]
    H. Brezis, Analyse fonctionnelle, Masson, Paris, 1983.Google Scholar
  90. [91]
    H. Brezis, unpublished manuscript.Google Scholar
  91. [92]
    H. Brezis and X. Cabré, Some simple nonlinear PDE’s without solutions, Boll. Unione Mat. Ital. (8) 1-B (1999), 223–262.Google Scholar
  92. [93]
    H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math. 68 (1996), 277–304.MATHMathSciNetGoogle Scholar
  93. [94]
    H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for u t Δu = g(u) revisited, Adv. Differential Equations 1 (1996), 73–90.MATHMathSciNetGoogle Scholar
  94. [95]
    H. Brezis and A. Friedman, A nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), 73–97.MATHMathSciNetGoogle Scholar
  95. [96]
    H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. 58 (1979), 137–151.MATHMathSciNetGoogle Scholar
  96. [97]
    H. Brezis and P.-L. Lions, A note on isolated singularities for linear elliptic equations, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., 7a“ Academic Press, New York-London, 1981, pp. 263–266.Google Scholar
  97. [98]
    H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math. 36 (1983), 437–477.MATHMathSciNetCrossRefGoogle Scholar
  98. [99]
    H. Brezis and R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), 601–614.MATHMathSciNetCrossRefGoogle Scholar
  99. [100]
    J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity 7 (1994), 539–575.MATHMathSciNetCrossRefGoogle Scholar
  100. [101]
    P. Brunovský and B. Fiedler, Number of zeros on invariant manifolds in reaction-diffusion equations, Nonlinear Anal. 10 (1986), 179–193.MATHMathSciNetCrossRefGoogle Scholar
  101. [102]
    P. Brunovský, P. Poláčik and B. Sandstede, Convergence in general periodic parabolic equations in one space dimension, Nonlinear Anal. 18 (1992), 209–215.MATHMathSciNetCrossRefGoogle Scholar
  102. [103]
    C. Budd, B. Dold and A. Stuart, Blowup in a partial differential equation with conserved first integral, SIAM J. Appl. Math. 53 (1993), 718–742.MATHMathSciNetCrossRefGoogle Scholar
  103. [104]
    C. Budd and J. Norbury, Semilinear elliptic equations and supercritical growth, J. Differential Equations 68 (1987), 169–197.MATHMathSciNetCrossRefGoogle Scholar
  104. [105]
    C. Budd and Y.-W. Qi, The existence of bounded solutions of a semilinear elliptic equation, J. Differential Equations 82 (1989), 207–218.MATHMathSciNetCrossRefGoogle Scholar
  105. [106]
    J. Busca and R. Manasevich, A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J. 51 (2002), 37–51.MATHMathSciNetGoogle Scholar
  106. [107]
    X. Cabré and Y. Martel, Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal. 156 (1998), 30–56.MATHMathSciNetCrossRefGoogle Scholar
  107. [108]
    L.A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271–297.MATHMathSciNetCrossRefGoogle Scholar
  108. [109]
    G. Cai, On the heat flow for the two-dimensional Gelfand equation, Nonlinear Anal. (to appear).Google Scholar
  109. [110]
    G. Caristi and E. Mitidieri, Blow-up estimates of positive solutions of a parabolic system, J. Differential Equations 113 (1994), 265–271.MATHMathSciNetCrossRefGoogle Scholar
  110. [111]
    F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001), 229–258.MATHMathSciNetCrossRefGoogle Scholar
  111. [112]
    T. Cazenave, F. Dickstein and F.B. Weissler, An equation whose Fujita critical exponent is not given by scaling, Nonlinear Anal. (to appear).Google Scholar
  112. [113]
    T. Cazenave and A. Haraux, Introduction aux problèmes d’évolution semilinéaires, Ellipses, Paris, 1990, English translation: The Clarendon Press, Oxford University Press, New York, 1998.Google Scholar
  113. [114]
    T. Cazenave and P.-L. Lions, Solutions globales d’équations de la chaleur semi linéaires, Comm. Partial Differential Equations 9 (1984), 955–978.MATHMathSciNetCrossRefGoogle Scholar
  114. [115]
    T. Cazenave and F.B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z. 228 (1998), 83–120.MATHMathSciNetCrossRefGoogle Scholar
  115. [116]
    C. Celik and Z. Zhou, No local L 1 solution for a nonlinear heat equation, Comm. Partial Differential Equations 28 (2003), 1807–1831.MATHMathSciNetCrossRefGoogle Scholar
  116. [117]
    G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal. 69 (1986), 289–306.Google Scholar
  117. [118]
    K. Cerqueti, A uniqueness result for a semilinear elliptic equation involving the critical Sobolev exponent in symmetric domains, Asymptot. Anal. 21 (1999), 99–115.MATHMathSciNetGoogle Scholar
  118. [119]
    J.M. Chadam, A. Peirce and H.-M. Yin, The blow-up property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl. 169 (1992), 313–328.MATHMathSciNetCrossRefGoogle Scholar
  119. [120]
    C. Y. Chan, Recent advances in quenching phenomena, Proceedings of Dynamic Systems and Applications, Vol. 2, Dynamic, Atlanta, GA, 1996, pp. 107–113.Google Scholar
  120. [121]
    K.-C. Chang and M.-Y. Jiang, Dirichlet problem with indefinite nonlinearities, Calc. Var. Partial Differential Equations 20 (2004), 257–282.MATHMathSciNetCrossRefGoogle Scholar
  121. [122]
    H. Chen, Positive steady-state solutions of a non-linear reaction-diffusion system, Math. Methods Appl. Sci. 20 (1997), 625–634.MATHMathSciNetCrossRefGoogle Scholar
  122. [123]
    W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615–622.MATHMathSciNetCrossRefGoogle Scholar
  123. [124]
    W. Chen and C. Li, Indefinite elliptic problems in a domain, Discrete Contin. Dyn. Syst. 3 (1997), 333–340.MATHGoogle Scholar
  124. [125]
    X. Chen, M. Fila and J.-S. Guo, Boundedness of global solutions of a supercritical parabolic equation, Nonlinear Anal. (to appear).Google Scholar
  125. [126]
    X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finitepoint blow-up in one-dimensional semilinear heat equations, J. Differential Equations 78 (1989), 160–190.MATHMathSciNetCrossRefGoogle Scholar
  126. [127]
    X.-Y. Chen and P. Poláčik, Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball, J. Reine Angew. Math. 472 (1996), 17–51.MATHMathSciNetGoogle Scholar
  127. [128]
    M. Chipot and P. Quittner, Equilibria, connecting orbits and a priori bounds for semilinear parabolic equations with nonlinear boundary conditions, J. Dynam. Differential Equations 16 (2004), 91–138.MATHMathSciNetCrossRefGoogle Scholar
  128. [129]
    M. Chipot and F.B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal. 20 (1989), 886–907.MATHMathSciNetCrossRefGoogle Scholar
  129. [130]
    M. Chlebik and M. Fila, From critical exponents to blowup rates for parabolic problems, Rend. Mat. Appl. (7) 19 (1999), 449–470.MATHMathSciNetGoogle Scholar
  130. [131]
    M. Chlebik, M. Fila and P. Quittner, Blow-up of positive solutions of a semilinear parabolic equation with a gradient term, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10 (2003), 525–537.MATHMathSciNetGoogle Scholar
  131. [132]
    K.-S. Chou, S.-Z. Du and G.-F. Zheng, On partial regularity of the borderline solution of semilinear parabolic problems, Calc. Var. Partial Differential Equations (to appear).Google Scholar
  132. [133]
    V. Churbanov, An example of a reaction system with diffusion in which the diffusion terms lead to blowup, Dokl. Akad. Nauk SSSR 310 (1990), 1308–1309, English translation in: Soviet Math. Dokl. 41 (1990), 191-192.MathSciNetGoogle Scholar
  133. [134]
    Ph. Clément, D.G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations 17 (1992), 923–940.MathSciNetCrossRefMATHGoogle Scholar
  134. [135]
    Ph. Clément, D.G. de Figueiredo and E. Mitidieri, A priori estimates for positive solutions of semilinear elliptic systems via Hardy-Sobolev inequalities, Nonlinear partial differential equations, Pitman Res. Notes Math. Ser. 343 (A. Benkirane et al., eds.), Harlow: Longman, 1996, pp. 73–91.Google Scholar
  135. [136]
    Ph. Clément, J. Fleckinger, E. Mitidieri and F. de Thélin, Existence of positive solutions for nonvariational quasilinear system, J. Differential Equations 166 (2000), 455–477.Google Scholar
  136. [137]
    Ph. Clément and R.C.A.M. van der Vorst, On a semilinear elliptic system, Differential Integral Equations 8 (1995), 1317–1329..MATHMathSciNetGoogle Scholar
  137. [138]
    G. Conner and C. Grant, Asymptotics of blowup for a convection-diffusion equation with conservation, Differential Integral Equations 9 (1996), 719–728.MATHMathSciNetGoogle Scholar
  138. [139]
    M. Conti, L. Merizzi and S. Terracini, Radial solutions of superlinear equations onN. Part I: A global variational approach, Arch. Rational Mech. Anal. 153 (2000), 291–316.MATHMathSciNetGoogle Scholar
  139. [140]
    M. Conti and S. Terracini, Radial solutions of superlinear equations on ℝ N. Part II: The forced case, Arch. Rational Mech. Anal. 153 (2000), 317–339.MATHMathSciNetGoogle Scholar
  140. [141]
    C. Cosner, Positive solutions for a superlinear elliptic systems without variational structure, Nonlinear Anal. 8 (1984), 1427–1436.MATHMathSciNetCrossRefGoogle Scholar
  141. [142]
    M.G. Crandall and P.H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal. 58 (1975), 207–218.MATHMathSciNetCrossRefGoogle Scholar
  142. [143]
    M.G. Crandall, P.H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193–222.MATHMathSciNetCrossRefGoogle Scholar
  143. [144]
    M. Cuesta, D.G. de Figueiredo and P.N. Srikanth, On a resonant-superlinear elliptic problem, Calc. Var. Partial Differential Equations 17 (2003), 221–233.MATHMathSciNetGoogle Scholar
  144. [145]
    S. Cui, Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Anal. 43 (2001), 293–323.MATHMathSciNetCrossRefGoogle Scholar
  145. [146]
    S. Dabuleanu, Problèmes aux limites pour les équations de Hamilton-Jacobi avec viscosité et données initiales peu régulières, Doctoral Thesis, University of Nancy 1, 2003.Google Scholar
  146. [147]
    L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), 631–652.MATHMathSciNetCrossRefGoogle Scholar
  147. [148]
    E.N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations 74 (1988), 120–156.MATHMathSciNetCrossRefGoogle Scholar
  148. [149]
    E.N. Dancer, A note on an equation with critical exponent, Bull. London Math. Soc. 20 (1988), 600–602.MATHMathSciNetCrossRefGoogle Scholar
  149. [150]
    E.N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc. 46 (1992), 425–434.MathSciNetGoogle Scholar
  150. [151]
    E.N. Dancer, Superlinear problems on domains with holes of asymptotic shape and exterior problems, Math. Z. 229 (1998), 475–491.MATHMathSciNetCrossRefGoogle Scholar
  151. [152]
    D. Daners and P. Koch Medina, Abstract evolution equations, periodic problems and applications, Longman, Harlow, 1992.MATHGoogle Scholar
  152. [153]
    E.B. Davies, The equivalence of certain heat kernel and Green function bounds, J. Funct. Anal. 71 (1987), 88–103.MATHMathSciNetCrossRefGoogle Scholar
  153. [154]
    E.B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge, 1989.MATHGoogle Scholar
  154. [155]
    M. Del Pino, M. Musso and F. Pacard, Boundary singularities for weak solutions of semilinear elliptic problems, J. Funct. Anal, (to appear).Google Scholar
  155. [156]
    K. Deng, Stabilization of solutions of a nonlinear parabolic equation with a gradient term, Math. Z. 216 (1994), 147–155.MATHMathSciNetCrossRefGoogle Scholar
  156. [157]
    K. Deng, Nonlocal nonlinearity versus global blow-up, Math. Applicata 8 (1995), 124–129.Google Scholar
  157. [158]
    K. Deng, Blow-up rates for parabolic systems, Z. Angew. Math. Phys. 47 (1996), 132–143.MATHMathSciNetCrossRefGoogle Scholar
  158. [159]
    K. Deng and H.A. Levine, The role of critical exponents in blow-up theorems: The sequel, J. Math. Anal. Appl. 243 (2000), 85–126.MATHMathSciNetCrossRefGoogle Scholar
  159. [160]
    W. Deng, Y. Li and C.-H. Xie, Semilinear reaction-diffusion systems with nonlocal sources, Math. Comput. Modelling 37 (2003), 937–943.MATHMathSciNetCrossRefGoogle Scholar
  160. [161]
    R. Denk, M. Hieber and J. Prüss, -boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 788, 2003.Google Scholar
  161. [162]
    G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations 7 (2002), 1257–1280.MATHMathSciNetGoogle Scholar
  162. [163]
    C. Dohmen and M. Hirose, Structure of positive radial solutions to the Haraux-Weissler equation, Nonlinear Anal. 33 (1998), 51–69.MATHMathSciNetCrossRefGoogle Scholar
  163. [164]
    B. Dold, V.A. Galaktionov, A.A. Lacey and J.L. Vázquez, Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), 663–687.MATHMathSciNetGoogle Scholar
  164. [165]
    G.C. Dong, Nonlinear partial differential equations of second order, Amer. Math. Soc., Transi. Math. Monographs 95, Providence, RI, 1991.Google Scholar
  165. [166]
    G. Dore and A. Venni, H functional calculus for an elliptic operator on a half-space with general boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 1 (2002), 487–543.MATHMathSciNetGoogle Scholar
  166. [167]
    Y. Du and S. Li, Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations, Adv. Differential Equations 10 (2005), 841–860.MATHMathSciNetGoogle Scholar
  167. [168]
    J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 693–698.MATHMathSciNetGoogle Scholar
  168. [169]
    X.T. Duong and G. Simonett, H -calculas for elliptic operators with nonsmooth coefficients, Differential Integral Equations 10 (1997), 201–217.MATHMathSciNetGoogle Scholar
  169. [170]
    L. Dupaigne and A. Ponce, Singularities of positive supersolutions in elliptic PDEs, Selecta Math. (N.S.) 10 (2004), 341–358.MATHMathSciNetCrossRefGoogle Scholar
  170. [171]
    M. Escobedo and M.A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations 89 (1991), 176–202.MATHMathSciNetCrossRefGoogle Scholar
  171. [172]
    M. Escobedo and M.A. Herrero, A uniqueness result for a semilinear reaction diffusion system, Proc. Amer. Math. Soc. 112 (1991), 175–185.MATHMathSciNetCrossRefGoogle Scholar
  172. [173]
    M. Escobedo and M.A. Herrero, A semilinear parabolic system in a bounded domain, Ann. Mat. Pura Appl. (4) 165 (1993), 315–336.MATHMathSciNetCrossRefGoogle Scholar
  173. [174]
    M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal. 11 (1987), 1103–1133.MATHMathSciNetCrossRefGoogle Scholar
  174. [175]
    M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations inN, J. Funct. Anal. 10 (1991), 119–161.MathSciNetCrossRefGoogle Scholar
  175. [176]
    M.J. Esteban, On periodic solutions of superlinear parabolic problems, Trans. Amer. Math. Soc. 293 (1986), 171–189.MATHMathSciNetCrossRefGoogle Scholar
  176. [177]
    M.J. Esteban, A remark on the existence of positive periodic solutions of superlinear parabolic problems, Proc. Amer. Math. Soc. 102 (1988), 131–136.MATHMathSciNetCrossRefGoogle Scholar
  177. [178]
    A. Farina, Liouville-type results for solutions of Δu = ¦u¦ p-1u on unbounded domains of ℝN, C. R. Math. Acad. Sci. Paris 341 (2005), 415–418.MATHMathSciNetGoogle Scholar
  178. [179]
    P.C. Fife, Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics 28, Springer-Verlag, Berlin New York, 1979.MATHGoogle Scholar
  179. [180]
    D.G. de Figueiredo, Semilinear elliptic systems, Nonlinear Functional Analysis and Applications to Differential Equations, World Sci. Publishing, River Edge, N.J., 1998, pp. 122–152.Google Scholar
  180. [181]
    D.G. de Figueiredo and P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), 387–397.MATHMathSciNetGoogle Scholar
  181. [182]
    D.G. de Figueiredo and P. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soe. 343 (1994), 99–116.MATHCrossRefGoogle Scholar
  182. [183]
    D.G. de Figueiredo, P.-L. Lions and R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1982), 41–63.MATHMathSciNetGoogle Scholar
  183. [184]
    D.G. de Figueiredo and J. Yang, A priori bounds for positive solutions of a non-variational elliptic system, Comm. Partial Differential Equations 26 (2001), 2305–2321.MATHMathSciNetCrossRefGoogle Scholar
  184. [185]
    M. Fila, Remarks on blow up for a nonlinear parabolic equation with a gradient term, Proc. Amer. Math. Soc. 111 (1991), 795–801.MATHMathSciNetCrossRefGoogle Scholar
  185. [186]
    M. Fila, Boundedness of global solutions of nonlinear diffusion equations, J. Differential Equations 98 (1992), 226–240.MATHMathSciNetCrossRefGoogle Scholar
  186. [187]
    M. Fila, Boundedness of global solutions of nonlocal parabolic equations, Nonlinear Anal. 30 (1997), 877–885.MATHMathSciNetCrossRefGoogle Scholar
  187. [188]
    M. Fila, Blow-up of solutions of supercritical parabolic equations, Handbook of Differential Equations, Evolutionary equations, Vol. II (C.M. Dafermos et al., eds.), Elsevier/North-Holland, Amsterdam, 2005, pp. 105–158.Google Scholar
  188. [189]
    M. Fila, J.R. King, M. Winkler and E. Yanagida, Optimal lower bound of the grow-up rate for a supercritical parabolic equation, J. Differential Equations 228 (2006), 339–356.MATHMathSciNetCrossRefGoogle Scholar
  189. [190]
    M. Fila, J.R. King, M. Winkler and E. Yanagida, Grow-up rate of solutions of a semilinear parabolic equation with a critical exponent, Adv. Differential Equations 12 (2007), 1–26.MathSciNetMATHGoogle Scholar
  190. [191]
    M. Fila, J.R. King, M. Winkler and E. Yanagida, Linear behaviour of solutions of a superlinear heat equation (2006), Preprint.Google Scholar
  191. [192]
    M. Fila, H.A. Levine and Y. Uda, Fujita-type global existence-global nonexistence theorem for a system of reaction diffusion equations with differing diffusivities, Math. Methods Appl. Sci. 17 (1994), 807–835.MATHMathSciNetCrossRefGoogle Scholar
  192. [193]
    M. Fila and G. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations 7 (1994), 811–821.MATHMathSciNetGoogle Scholar
  193. [194]
    M. Fila, H. Matano and P. Poláčik, Immediate régularization after blow-up, SIAM J. Math. Anal. 37 (2005), 752–776.MATHMathSciNetCrossRefGoogle Scholar
  194. [195]
    M. Fila and N. Mizoguchi, Multiple continuation beyond blow-up, Differential Integral Equations 20 (2007), 671–680.MathSciNetGoogle Scholar
  195. [196]
    M. Fila and H. Ninomiya, Reaction versus diffusion: blow-up induced and inhibited by diffusivity, Russian Math. Surveys 60 (2005), 1217–1235.MathSciNetCrossRefMATHGoogle Scholar
  196. [197]
    M. Fila, H. Ninomiya and J.L. Vázquez, Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems, Discrete Contin. Dyn. Syst. 14 (2006), 63–74.MATHMathSciNetGoogle Scholar
  197. [198]
    M. Fila and P. Poláčik, Global solutions of a semilinear parabolic equation, Adv. Differential Equations 4 (1999), 163–196.MATHMathSciNetGoogle Scholar
  198. [199]
    M. Fila and Ph. Souplet, The blow-up rate for semilinear parabolic problems on general domains, NoDEA Nonlinear Differential Equations Appl. 8 (2001), 473–480.MATHMathSciNetCrossRefGoogle Scholar
  199. [200]
    M. Fila, Ph. Souplet and F.B. Weissler, Linear and nonlinear heat equations in L δp spaces and universal bounds for global solutions, Math. Ann. 320 (2001), 87–113.MATHMathSciNetCrossRefGoogle Scholar
  200. [201]
    M. Fila, J. Taskinen and M. Winkler, Convergence to a singular steady-state of a parabolic equation with gradient blow-up, Appl. Math. Letters 20 (2007), 578–582.MATHMathSciNetCrossRefGoogle Scholar
  201. [202]
    M. Fila and M. Winkler, Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposed, J. Eur. Math. Soc. (to appear).Google Scholar
  202. [203]
    M. Fila, M. Winkler and E. Yanagida, Grow-up rate of solutions for a supercritical semilinear diffusion equation, J. Differential Equations 205 (2004), 365–389.MATHMathSciNetCrossRefGoogle Scholar
  203. [204]
    M. Fila, M. Winkler and E. Yanagida, Slow convergence to zero for a parabolic equation with supercritical nonlinearity, Math. Ann. (to appear).Google Scholar
  204. [205]
    M. Fila, M. Winkler and E. Yanagida, Convergence to self-similar solutions in a semilinear parabolic equation (2007), Preprint.Google Scholar
  205. [206]
    S. Filippas, M.A. Herrero and J.J.L. Velázquez, Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000), 2957–2982.MATHMathSciNetGoogle Scholar
  206. [207]
    S. Filippas and R. Kohn, Refined asymptotics for the blowup of u t-Δu = up, Comm. Pure Appl. Math. 45 (1992), 821–869.MATHMathSciNetCrossRefGoogle Scholar
  207. [208]
    S. Filippas and W.-X. Liu, On the blowup of multidimensional semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), 313–344.MATHMathSciNetGoogle Scholar
  208. [209]
    S. Filippas and F. Merle, Compactness and single-point blowup of positive solutions on bounded domains, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 47–65.MATHMathSciNetGoogle Scholar
  209. [210]
    A. Filippov, Conditions for the existence of a solution of a quasi-linear parabolic equation (Russian), Dokl. Akad. Nauk SSSR 141 (1961), 568–570.MathSciNetGoogle Scholar
  210. [211]
    R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), 335–369.Google Scholar
  211. [212]
    D. Fortunato and E. Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains, Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 205–213.MathSciNetMATHGoogle Scholar
  212. [213]
    R.H. Fowler, Further studies of Emden’s and similar differential equations, Quart. J. Math. 2 (1931), 259–288.CrossRefGoogle Scholar
  213. [214]
    A. Friedman, Partial differential equations of parabolic type, Prentice Hall, 1964.Google Scholar
  214. [215]
    A. Friedman, Blow up of solutions of nonlinear parabolic equations, Nonlinear diffusion equations and their equilibrium states, I (M. Niet al., eds.), Springer, 1988, pp. 301–318.Google Scholar
  215. [216]
    A. Friedman and Y. Giga, A single point blow-up for solutions of semilinear parabolic systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), 65–79.MATHMathSciNetGoogle Scholar
  216. [217]
    A. Friedman and M. Herrero, Extinction properties of semilinear heat equations with strong absorption, J. Math. Anal. Appl. 124 (1987), 530–546.MATHMathSciNetCrossRefGoogle Scholar
  217. [218]
    A. Friedman and A.A. Lacey, Blowup of solutions of semilinear parabolic equations, J. Math. Anal. Appl. 132 (1988), 171–186.MATHMathSciNetCrossRefGoogle Scholar
  218. [219]
    A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425–447.MATHMathSciNetCrossRefGoogle Scholar
  219. [220]
    H. Fujita, On the blowing up of solutions of the Cauchy problem for u t = Δu + u1+α, J. Fac. Sci. Univ. Tokyo Sec. IA Math. 13 (1966), 109–124.MATHGoogle Scholar
  220. [221]
    I. Fukuda and R. Suzuki, Blow-up behavior for a nonlinear heat equation with a localized source in a ball, J. Differential Equations 218 (2005), 273–291.MATHMathSciNetCrossRefGoogle Scholar
  221. [222]
    V.A. Galaktionov, Geometric Sturmian theory of nonlinear parabolic equations and applications, Applied Mathematics and Nonlinear Science Series, 3, pman & Hall/CRC, Boca Raton, FL 2004.Google Scholar
  222. [223]
    V.A. Galaktionov and J.R. King, Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents, J. Differential Equations 189 (2003), 199–233.MATHMathSciNetCrossRefGoogle Scholar
  223. [224]
    V.A. Galaktionov, S.P. Kurdyumov and A.A. Samarskii, A parabolic system of quasilinear equations, I, Differential Equations 19 (1983), 2133–2143.MathSciNetGoogle Scholar
  224. [225]
    V.A. Galaktionov, S.P. Kurdyumov and A.A. Samarskii, Asymptotic stability of invariant solutions of nonlinear heat-conduction equation with sources, Differentsial’nye Uravneniya 20 (1984), 614–632, (English translation Differential Equations 20 (1984), 461-476).MathSciNetGoogle Scholar
  225. [226]
    V.A. Galaktionov, S.P. Kurdyumov and A.A. Samarskii, A parabolic system of quasilinear equations, II, Differential Equations 21 (1985), 1544–1559.MathSciNetGoogle Scholar
  226. [227]
    V.A. Galaktionov and H.A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal. 34 (1998), 1005–1027.MATHMathSciNetCrossRefGoogle Scholar
  227. [228]
    V.A. Galaktionov and S. Posashkov, The equation u t = uxx + uβ. Localization, asymptotic, Akad. Nauk SSSR Inst. Prikl. Mat. Preprint no. 97 (1985).Google Scholar
  228. [229]
    V.A. Galaktionov and S. Posashkov, Application of new comparison theorems to the investigation of unbounded solutions of nonlinear parabolic equations (Russian), Differentsial’nye Uravneniya 22 (1986), 1165–1173, (English translation: Differential Equations 22 (1986), 809-815).MathSciNetGoogle Scholar
  229. [230]
    V.A. Galaktionov and J.L. Vázquez, Regional blow up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal. 24 (1993), 1254–1276.MATHMathSciNetCrossRefGoogle Scholar
  230. [231]
    V.A. Galaktionov and J.L. Vázquez, Blowup for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations, J. Differential Equations 127 (1996), 1–40.MATHMathSciNetCrossRefGoogle Scholar
  231. [232]
    V.A. Galaktionov and J.L. Vázquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), 1–67.MATHMathSciNetCrossRefGoogle Scholar
  232. [233]
    Th. Gallouët, F. Mignot and J.-P. Puel, Quelques résultats sur le problème-Δu = λeu, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 289–292.MATHGoogle Scholar
  233. [234]
    F. Gazzola and H.-Ch. Grünau, On the role of space dimension n = 2 + 2√2 in the semilinear Brezis-Nirenberg eigenvalue problem, Analysis 20 (2000), 395–399.MATHGoogle Scholar
  234. [235]
    F. Gazzola, J. Serrin and M. Tang, Existence of ground states and free boundary problems for quasilinear elliptic operators, Adv. Differential Equations 5 (2000), 1–30.MATHMathSciNetGoogle Scholar
  235. [236]
    F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations 18 (2005), 961–990.MathSciNetMATHGoogle Scholar
  236. [237]
    J. Giacomoni, J. Prajapat and M. Ramaswamy, Positive solution branch for elliptic problems with critical indefinite nonlinearity, Differential Integral Equations 18 (2005), 721–764.MathSciNetMATHGoogle Scholar
  237. [238]
    B. Gidas, Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations, Nonlinear partial differential equations in engineering and applied science, Lecture Notes in Pure and Appl. Math. 54, ker, New York 1980, pp. 255–273.Google Scholar
  238. [239]
    B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.MATHMathSciNetCrossRefGoogle Scholar
  239. [240]
    B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525–598.MATHMathSciNetCrossRefGoogle Scholar
  240. [241]
    B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), 883–901.MATHMathSciNetCrossRefGoogle Scholar
  241. [242]
    A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972), 30–39.CrossRefGoogle Scholar
  242. [243]
    Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys. 103 (1986), 415–421.MATHMathSciNetCrossRefGoogle Scholar
  243. [244]
    Y. Giga and R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297–319.MATHMathSciNetCrossRefGoogle Scholar
  244. [245]
    Y. Giga and R.V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1–40.MATHMathSciNetCrossRefGoogle Scholar
  245. [246]
    Y. Giga and R.V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 845–884.MATHMathSciNetCrossRefGoogle Scholar
  246. [247]
    Y. Giga, S. Matsui and S. Sasayama, Blow up rate for semilinear heat equation with subcritical nonlinearity, Indiana Univ. Math. J. 53 (2004), 483–514.MATHMathSciNetCrossRefGoogle Scholar
  247. [248]
    Y. Giga, S. Matsui and S. Sasayama, On blow-up rate for sign-changing solutions in a convex domain, Math. Methods Appl. Sc. 27 (2004), 1771–1782.MATHMathSciNetCrossRefGoogle Scholar
  248. [249]
    Y. Giga and N. Umeda, On blow-up at space infinity for semilinear heat equations, J. Math. Anal. Appl. 316 (2006), 538–555.MATHMathSciNetCrossRefGoogle Scholar
  249. [250]
    D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, inger, Berlin 1998.Google Scholar
  250. [251]
    B. Gilding, The Cauchy problem for u t = Δ u + ¦Δ u¦q, large-time behaviour, J. Math. Pures Appl. 84 (2005), 753–785.MATHMathSciNetGoogle Scholar
  251. [252]
    B. Gilding, M. Guedda and R. Kersner, The Cauchy problem for u t = Δu+ ¦Δ u¦q, J. Math. Anal. Appl. 284 (2003), 733–755.MATHMathSciNetCrossRefGoogle Scholar
  252. [253]
    J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg Landau equation. II. Contraction methods, Comm. Math. Phys. 187 (1997), 45–79.MATHMathSciNetCrossRefGoogle Scholar
  253. [254]
    P. Groisman and J. Rossi, Dependence of the blow-up time with respect to parameters and numerical approximations for a parabolic problem, Asympt. Anal. 37 (2004), 79–91.MATHMathSciNetGoogle Scholar
  254. [255]
    P. Groisman, J. Rossi and H. Zaag, On the dependence of the blow-up time with respect to the initial data in a semilinear parabolic problem, Comm. Partial Differential Equations 28 (2003), 737–744.MATHMathSciNetCrossRefGoogle Scholar
  255. [256]
    M. Grossi, A uniqueness result for a semilinear elliptic equation in symmetric domains, Adv. Differential Equations 5 (2000), 193–212.MATHMathSciNetGoogle Scholar
  256. [257]
    M. Grossi, P. Magrone and M. Matzeu, Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth, Discrete Contin. Dyn. Syst. 7 (2001), 703–718.MATHMathSciNetGoogle Scholar
  257. [258]
    Y. Gu and M.-X. Wang, Existence of positive stationary solutions and threshold results for a reaction-diffusion system, J. Differential Equations 130 (1996), 177–291.MathSciNetCrossRefGoogle Scholar
  258. [259]
    M. Guedda and M. Kirane, Diffusion terms in systems of reaction diffusion equations can lead to blow up, J. Math. Anal. Appl. 218 (1998), 325–327.MATHMathSciNetCrossRefGoogle Scholar
  259. [260]
    C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation inn, Comm. Pure Appl. Math. 45 (1992), 1153–1181.MATHMathSciNetCrossRefGoogle Scholar
  260. [261]
    C. Gui, W.-M. Ni and X. Wang, Further study on a nonlinear heat equation, J. Differential Equations 169 (2001), 588–613.MATHMathSciNetCrossRefGoogle Scholar
  261. [262]
    C. Gui and X. Wang, Life span of solutions of the Cauchy problem for a semilinear heat equation, J. Differential Equations 115 (1995), 166–172.MATHMathSciNetCrossRefGoogle Scholar
  262. [263]
    J.-S. Guo and B. Hu, Blowup rate estimates for the heat equation with a nonlinear gradient source term, Discrete Contin. Dyn. Syst. (to appear).Google Scholar
  263. [264]
    J.-S. Guo and Ph. Souplet, Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up, Math. Ann. 331 (2005), 651–667.MATHMathSciNetCrossRefGoogle Scholar
  264. [265]
    J.K. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDF, Z. Angew. Math. Phys. 43 (1992), 63–124.MATHMathSciNetCrossRefGoogle Scholar
  265. [266]
    F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation inN, Arch. Ration. Mech. Anal. 157 (2001), 91–163.MATHMathSciNetCrossRefGoogle Scholar
  266. [267]
    A. Haraux, Systèmes dynamiques dissipatifs et applications, Recherches en Mathématiques Appliquées 17, Masson, Paris, 1991.Google Scholar
  267. [268]
    A. Haraux and P. Poláčik, Convergence to a positive equilibrium for some nonlinear evolution equations in a ball, Acta Math. Univ. Comenian. (N.S.) 61 (1992), 129–141.MATHMathSciNetGoogle Scholar
  268. [269]
    A. Haraux and F.B. Weissler, Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982), 167–189.MATHMathSciNetCrossRefGoogle Scholar
  269. [270]
    A. Haraux and A. Youkana, On a result of K. Masuda concerning reactiondiffusion equations, Tohoku Math. J. (2) 40 (1988), 159–163.MATHMathSciNetGoogle Scholar
  270. [271]
    K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad. 49 (1973), 503–505.MATHMathSciNetCrossRefGoogle Scholar
  271. [272]
    E. Heinz, Über die Eindeutigkeit beim Cauchyschen Anfangswertproblem einer elliptischen Differentialgleichung zweiter Ordnung, Nachr. Akad. Wiss. Göttingen IIa (1955), 1–12.MathSciNetGoogle Scholar
  272. [273]
    D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840, Springer, Berlin-Heidelberg-New York, 1981.Google Scholar
  273. [274]
    M.A. Herrero, A.A. Lacey and J.J.L. Velázquez, Global existence for reaction-diffusion systems modelling ignition, Arch. Rational Mech. Anal. 142 (1998), 219–251.MATHMathSciNetCrossRefGoogle Scholar
  274. [275]
    M.A. Herrero and J.J.L. Velázquez, Blow-up profiles in one-dimensional, semilinear parabolic problems, Comm. Partial Differential Equations 17 (1992), 205–219.MathSciNetCrossRefMATHGoogle Scholar
  275. [276]
    M.A. Herrero and J.J.L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), 131–189.MATHGoogle Scholar
  276. [277]
    M.A. Herrero and J.J.L. Velázquez, Explosion de solutions d’équations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 141–145.MATHGoogle Scholar
  277. [278]
    M.A. Herrero and J.J.L. Velazquez, A blow up result for semilinear heat equations in the supercritical case (1994), Preprint.Google Scholar
  278. [279]
    M. Hesaaraki and A. Moameni, Blow-up of positive solutions for a family of nonlinear parabolic equations in general domain inN, Michigan Math. J. 52 (2004), 375–389.MATHMathSciNetCrossRefGoogle Scholar
  279. [280]
    N. Hirano and N. Mizoguchi, Positive unstable periodic solutions for superlinear parabolic equations, Proc. Amer. Math. Soc. 123 (1995), 1487–1495.MATHMathSciNetCrossRefGoogle Scholar
  280. [281]
    S. Hollis, R. Martin and M. Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal. 18 (1987), 744–761.MATHMathSciNetCrossRefGoogle Scholar
  281. [282]
    S. Hollis and J. Morgan, Interior estimates for a class of reaction-diffusion systems from L 1 a priori estimates, J. Differential Equations 98 (1992), 260–276.MATHMathSciNetCrossRefGoogle Scholar
  282. [283]
    B. Hu, Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition, Differential Integral Equations 9 (1996), 891–901.MATHMathSciNetGoogle Scholar
  283. [284]
    B. Hu and H.-M. Yin, Semilinear parabolic equations with prescribed energy, Rend. Circ. Mat. Palermo (2) 44 (1995), 479–505.MATHMathSciNetGoogle Scholar
  284. [285]
    J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure, J. Functional Analysis 114 (1993), 32–58.MATHCrossRefGoogle Scholar
  285. [286]
    J. Húska, Periodic solutions in superlinear parabolic problems, Acta Math. Univ. Comenian. (N.S.) 71 (2002), 19–26.MATHMathSciNetGoogle Scholar
  286. [287]
    R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J. 26 (1996), 475–491.MATHMathSciNetGoogle Scholar
  287. [288]
    K. Ishige and T. Kawakami, Asymptotic behavior of solutions for some semilinear heat equations inN (2006), Preprint.Google Scholar
  288. [289]
    K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion, J. Differential Equations 212 (2005), 114–128.MATHMathSciNetCrossRefGoogle Scholar
  289. [290]
    H. Ishii, Asymptotic stability and blowing up of solutions of some nonlinear equations, J. Differential Equations 26 (1977), 291–319.MATHMathSciNetCrossRefGoogle Scholar
  290. [291]
    M.A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal. 153 (1998), 187–202.MATHMathSciNetCrossRefGoogle Scholar
  291. [292]
    H. Jiang, Global existence of solutions on an activator-inhibitor model, Discrete Contin. Dyn. Syst. 14 (2006), 737–751.MATHMathSciNetGoogle Scholar
  292. [293]
    D.D. Joseph and T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1973), 241–269.MATHMathSciNetGoogle Scholar
  293. [294]
    I. Kanel and M. Kirane, Global solutions of reaction-diffusion systems with a balance law and nonlinearities of exponential growth, J. Differential Equations 165 (2000), 24–41.MATHMathSciNetCrossRefGoogle Scholar
  294. [295]
    I. Kanel and M. Kirane, Global existence and large time behavior of positive solutions to a reaction diffusion system, Differential Integral Equations 13 (2000), 255–264.MATHMathSciNetGoogle Scholar
  295. [296]
    S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math. 16 (1963), 305–330.MATHMathSciNetCrossRefGoogle Scholar
  296. [297]
    T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), 181–205.MATHMathSciNetCrossRefGoogle Scholar
  297. [298]
    N. Kavallaris, A.A. Lacey and D. Tzanetis, Global existence and divergence of critical solutions of a non-local parabolic problem in Ohmic heating process, Nonlinear Anal. 58 (2004), 787–812.MATHMathSciNetCrossRefGoogle Scholar
  298. [299]
    O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Analyse Non Linéaire 4 (1987), 423–452.MATHMathSciNetGoogle Scholar
  299. [300]
    O. Kavian, Introduction à la théorie des points critiques, Springer, Paris, 1993.MATHGoogle Scholar
  300. [301]
    T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity, Ann. Inst. H. Poincaré Anal. non linéaire 13 (1996), 1–15.MATHMathSciNetGoogle Scholar
  301. [302]
    T. Kawanago, Existence and behavior of solutions for u t = Δ(u m)+ul, Adv. Math. Sci. Appl. 7 (1997), 367–400.MATHMathSciNetGoogle Scholar
  302. [303]
    H. Kawarada, On solutions of initial-boundary problem for u t = uxx + 1/(1-u), Publ. Res. Inst. Math. Sci. 10 (1974/75), 729–736.Google Scholar
  303. [304]
    B. Kawohl and L. Peletier, Observations on blow up and dead cores for nonlinear parabolic equations, Math. Z. 202 (1989), 207–217.MATHMathSciNetCrossRefGoogle Scholar
  304. [305]
    J.P. Keener and H.B. Keller, Positive solutions of convex nonlinear eigenvalue problems, J. Differential Equations 16 (1974), 103–125.MATHMathSciNetCrossRefGoogle Scholar
  305. [306]
    J.R. King, Personal communication 2006.Google Scholar
  306. [307]
    K. Kobayashi, T. Sirao and H. Tanaka, On the blowing up problem for semilinear heat equations, J. Math. Soc. Japan 29 (1977), 407–424.MATHMathSciNetGoogle Scholar
  307. [308]
    A.N. Kolmogorov, I.G. Petrovsky and N.S. Piskunov, Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université d’Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), Série Internationale, Section A 1 (1937), 1–26.Google Scholar
  308. [309]
    S. Kouachi, Existence of global solutions to reaction-diffusion systems via a Lyapunov functional, Electron. J. Differential Equations 2001,88 (2001), 1–13.MathSciNetGoogle Scholar
  309. [310]
    M.K. Kwong, Uniqueness of positive solutions of Δ-u + u p = 0 inn, Arch. Rational Mech. Anal. 105 (1989), 243–266.MATHMathSciNetCrossRefGoogle Scholar
  310. [311]
    A.A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math. 43 (1983), 1350–1366.MATHMathSciNetCrossRefGoogle Scholar
  311. [312]
    A.A. Lacey, The form of blow-up for nonlinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), 183–202.MathSciNetMATHGoogle Scholar
  312. [313]
    A.A. Lacey, Global blow-up of a nonlinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), 161–167.MathSciNetMATHGoogle Scholar
  313. [314]
    A.A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating. I. Model derivation and some special cases, European J. Appl. Math. 6 (1995), 127–144.MATHMathSciNetGoogle Scholar
  314. [315]
    A.A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating. II. General proof of blow-up and asymptotics of runaway, European J. Appl. Math. 6 (1995), 201–224.MATHMathSciNetGoogle Scholar
  315. [316]
    A.A. Lacey and D. Tzanetis, Global existence and convergence to a singular steady state for a semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 289–305.MathSciNetMATHGoogle Scholar
  316. [317]
    A.A. Lacey and D. Tzanetis, Complete blow-up for a semilinear diffusion equation with a sufficiently large initial condition, IMA J. Appl. Math. 41 (1988), 207–215.MATHMathSciNetCrossRefGoogle Scholar
  317. [318]
    A.A. Lacey and D. Tzanetis, Global, unbounded solutions to a parabolic equation, J. Differential Equations 101 (1993), 80–102.MATHMathSciNetCrossRefGoogle Scholar
  318. [319]
    O.A. Ladyzenskaja, Solution of the first boundary problem in the large for quasi-linear parabolic equations, Trudy Moskov. Mat. Obsc. 7 (1958), 149–177.MathSciNetGoogle Scholar
  319. [320]
    O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc., Transi. Math. Monographs, Providence, R.I., 1968.Google Scholar
  320. [321]
    Ph. Laurençot, Convergence to steady states for a one-dimensional viscous Hamilton-Jacobi equation with Dirichlet boundary conditions, Pacific J. Math. 230 (2007), 347–364.MathSciNetMATHGoogle Scholar
  321. [322]
    Ph. Laurençot and Ph. Souplet, On the growth of mass for a viscous Hamilton-Jacobi equation, J. Anal. Math. 89 (2003), 367–383.MATHMathSciNetGoogle Scholar
  322. [323]
    M. Lazzo, Solutions positives multiples pour une équation elliptique non linéaire avec l’exposant critique de Sobolev, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), 61–64.MATHMathSciNetGoogle Scholar
  323. [324]
    T. Lee and W.-M. Ni, Global existence, large time behaviour and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc. 333 (1992), 365–378.MATHMathSciNetCrossRefGoogle Scholar
  324. [325]
    L.A. Lepin, Countable spectrum of eigenfunctions of a nonlinear heat-conduction equation with distributed parameters, Differentsial’nye Uravneniya 24 (1988), 1226–1234, (English translation: Differential Equations 24 (1988), 799-805).MathSciNetGoogle Scholar
  325. [326]
    L.A. Lepin, Self-similar solutions of a semilinear heat equation, Mat. Model. 2 (1990), 63–74, (in Russian).MATHMathSciNetGoogle Scholar
  326. [327]
    H.A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu t =-Au + F(u), Arch. Rational Mech. Anal. 51 (1973), 371–386.MATHMathSciNetCrossRefGoogle Scholar
  327. [328]
    H.A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), 262–288.MATHMathSciNetCrossRefGoogle Scholar
  328. [329]
    H.A. Levine, Quenching and beyond: a survey of recent results, Nonlinear mathematical problems in industry, II, GAKUTO Internat. Ser. Math. Sci. Appl., 2, Gakkōtosho, Tokyo, 1993, pp. 501–512.Google Scholar
  329. [330]
    H.A. Levine, P. Sacks, B. Straughan and L. Payne, Analysis of a convective reaction-diffusion equation (II), SIAM J. Math. Anal. 20 (1989), 133–147.MATHMathSciNetCrossRefGoogle Scholar
  330. [331]
    F. Li, S.-H. Huang and C.-H. Xie, Global existence and blow-up of solutions to a nonlocal reaction-diffusion system, Discrete Contin. Dyn. Syst. 9 (2003), 1519–1532.MATHMathSciNetGoogle Scholar
  331. [332]
    F. Li and M.-X. Wang, Properties of blow-up solutions to a parabolic system with nonlinear localized terms, Discrete Contin. Dyn. Syst. 13 (2005), 683–700.MathSciNetGoogle Scholar
  332. [333]
    M. Li, S. Chen and Y. Qin, Boundedness and blow up for the general activator-inhibitor model, Acta Math. Appl. Sinica (English Ser.) 11 (1995), 59–68.MATHMathSciNetCrossRefGoogle Scholar
  333. [334]
    Y. Li, Asymptotic behavior of positive solutions of equation Δu+ K(x)up = 0 in ℝn, J. Differential Equations 95 (1992), 304–330.MATHMathSciNetCrossRefGoogle Scholar
  334. [335]
    Y. Li and C.-H. Xie, Blow-up for semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys. 55 (2004), 15–27.MATHMathSciNetGoogle Scholar
  335. [336]
    E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983), 441–448.MATHMathSciNetCrossRefGoogle Scholar
  336. [337]
    G.M. Lieberman, The first initial-boundary value problem for quasilinear second order parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), 347–387.MATHMathSciNetGoogle Scholar
  337. [338]
    G.M. Lieberman, Second order parabolic differential equations, World Scientific, Singapore, 1996.MATHGoogle Scholar
  338. [339]
    Z. Lin, Blowup estimates for a mutualistic model in ecology, Electron. J. Qual. Theory Differential Equ. 2002,8 (2002), 1–14.Google Scholar
  339. [340]
    P.-L. Lions, Isolated singularities in semilinear problems, J. Differential Equations 38 (1980), 441–450.MATHMathSciNetCrossRefGoogle Scholar
  340. [341]
    P.-L. Lions, Résolution de problèmes elliptiques quasilinéaires, Arch. Rational Mech. Anal. 74 (1980), 335–353.MATHMathSciNetCrossRefGoogle Scholar
  341. [342]
    P.-L. Lions, Asymptotic behavior of some nonlinear heat equations, Phys. D 5 (1982), 293–306.MathSciNetCrossRefGoogle Scholar
  342. [343]
    Y. Lou, Necessary and sufficient condition for the existence of positive solutions of certain cooperative system, Nonlinear Anal. 26 (1996), 1079–1095.MATHMathSciNetCrossRefGoogle Scholar
  343. [344]
    A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications 16, Birkhäuser, Basel, 1995.Google Scholar
  344. [345]
    Y. Martel, Complete blow up and global behaviour of solutions of u t-Δu = g(u), Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 687–723.MATHMathSciNetCrossRefGoogle Scholar
  345. [346]
    Y. Martel and Ph. Souplet, Estimations optimales en temps petit et près de la frontière pour les solutions de l’équation de la chaleur avec données non-compatibles, C. R. Acad. Sci. Paris Sér. I Math. 79 (1998), 575–580.MathSciNetGoogle Scholar
  346. [347]
    Y. Martel and Ph. Souplet, Small time boundary behavior for parabolic equations with noncompatible data, J. Math. Pures Appl. 79 (2000), 603–632.MATHMathSciNetCrossRefGoogle Scholar
  347. [348]
    R. Martin and M. Pierre, Nonlinear reaction-diffusion systems, Nonlinear equations in the applied sciences, Math. Sci. Engrg., 185, Academic Press, Boston, MA, 1992, pp. 363–398.Google Scholar
  348. [349]
    R. Martin and M. Pierre, Influence of mixed boundary conditions in some reaction-diffusion systems, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 1053–1066.MATHMathSciNetGoogle Scholar
  349. [350]
    K. Masuda, On the global existence and asymptotic behavior of solutions of reaction-diffusion equations, Hokkaido Math. J. 12 (1983), 360–370.MATHMathSciNetGoogle Scholar
  350. [351]
    K. Masuda, Analytic solutions of some nonlinear diffusion equations, Math. Z. 187 (1984), 61–73.MATHMathSciNetCrossRefGoogle Scholar
  351. [352]
    K. Masuda and K. Takahashi, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation, Japan J. Appl. Math. 4 (1987), 47–58.MATHMathSciNetCrossRefGoogle Scholar
  352. [353]
    H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. 18 (1978), 221–227.MATHMathSciNetGoogle Scholar
  353. [354]
    H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo, Sect. 1A Math. 30 (1983), 645–673.MathSciNetGoogle Scholar
  354. [355]
    H. Matano and F. Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. 57 (2004), 1494–1541.MATHMathSciNetCrossRefGoogle Scholar
  355. [356]
    H. Matano and F. Merle, in preparation.Google Scholar
  356. [357]
    J. Matos, Blow up of critical and subcritical norms in semilinear heat equations, Adv. Differential Equations 3 (1998), 497–532.MATHMathSciNetGoogle Scholar
  357. [358]
    J. Matos, Convergence of blow-up solutions of nonlinear heat equations in the supercritical case, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 1197–1227.MATHMathSciNetGoogle Scholar
  358. [359]
    J. Matos, Unfocused blow up solutions of semilinear parabolic equations, Discrete Contin. Dyn. Syst. 5 (1999), 905–928.MATHMathSciNetGoogle Scholar
  359. [360]
    J. Matos, Self-similar blow up patterns in supercritical semilinear heat equations, Comm. Appl. Anal. 5 (2001), 455–483.MATHMathSciNetGoogle Scholar
  360. [361]
    J. Matos and Ph. Souplet, Universal blow-up rates for a semilinear heat equation and applications, Adv. Differential Equations 8 (2003), 615–639.MATHMathSciNetGoogle Scholar
  361. [362]
    P.J. McKenna and W. Reichel, A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains, J. Funct. Anal. 244 (2007), 220–246.MATHMathSciNetCrossRefGoogle Scholar
  362. [363]
    P. Meier, On the critical exponent for reaction-diffusion equations, Arch. Rational Mech. Anal. 109 (1990), 63–72.MATHMathSciNetCrossRefGoogle Scholar
  363. [364]
    F. Merle, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math. 45 (1992), 263–300.MATHMathSciNetCrossRefGoogle Scholar
  364. [365]
    F. Merle and L.A. Peletier, Positive solutions of elliptic equations involving supercritical growth, Proc. Roy. Soc. Edinburgh Sect. A 118 (1991), 49–62.MathSciNetMATHGoogle Scholar
  365. [366]
    F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type u t = Δu+ ¦u¦p-1u, Duke Math. J. 86 (1997), 143–195.MATHMathSciNetCrossRefGoogle Scholar
  366. [367]
    F. Merle and H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998), 139–196.MathSciNetCrossRefGoogle Scholar
  367. [368]
    F. Merle and H. Zaag, Refined uniform estimates at blow-up and applications for nonlinear heat equations, Geom. Funct. Anal. 8 (1998), 1043–1085.MATHMathSciNetCrossRefGoogle Scholar
  368. [369]
    F. Mignot and J.-P. Puel, Sur une classe de problèmes non linéaires avec une non-linéarité positive, croissante, convexe, Comm. Partial Differential Equations 5 (1980), 791–836.MATHMathSciNetCrossRefGoogle Scholar
  369. [370]
    E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1993), 125–151.MATHMathSciNetCrossRefGoogle Scholar
  370. [371]
    E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ℝ n, Differential Integral Equations 9 (1996), 465–479.MATHMathSciNetGoogle Scholar
  371. [372]
    E. Mitidieri and S.I. Pohozaev, A priori estimates and blow-up of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math. 234 (2001), 1–362.MathSciNetGoogle Scholar
  372. [373]
    E. Mitidieri and S.I. Pohozaev, Fujita-type theorems for quasilinear parabolic inequalities with a nonlinear gradient, Dokl. Akad. Nauk 386 (2002), 160–164.MathSciNetGoogle Scholar
  373. [374]
    N. Mizoguchi, On the behavior of solutions for a semilinear parabolic equation with supercritical nonlmearity, Math. Z. 239 (2002), 215–229.MATHMathSciNetCrossRefGoogle Scholar
  374. [375]
    N. Mizoguchi, Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition, J. Differential Equations 193 (2003), 212–238.MATHMathSciNetCrossRefGoogle Scholar
  375. [376]
    N. Mizoguchi, Blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity, J. Differential Equations 205 (2004), 298–328.MATHMathSciNetCrossRefGoogle Scholar
  376. [377]
    N. Mizoguchi, Type-II blowup for a semilinear heat equation, Adv. Differential Equations 9 (2004), 1279–1316.MATHMathSciNetGoogle Scholar
  377. [378]
    N. Mizoguchi, Boundedness of global solutions for a supercritical semilinear heat equation and its application, Indiana Univ. Math. J. 54 (2005), 1047–1059.MATHMathSciNetCrossRefGoogle Scholar
  378. [379]
    N. Mizoguchi, Various behaviors of solutions for a semilinear heat equation after blowup, J. Funct. Anal. 220 (2005), 214–227.MATHMathSciNetCrossRefGoogle Scholar
  379. [380]
    N. Mizoguchi, Multiple blowup of solutions for a semilinear heat equation, Math. Ann. 331 (2005), 461–473.MATHMathSciNetCrossRefGoogle Scholar
  380. [381]
    N. Mizoguchi, Growup of solutions for a semilinear heat equation with supercritical nonlinearity, J. Differential Equations 227 (2006), 652–669.MATHMathSciNetCrossRefGoogle Scholar
  381. [382]
    N. Mizoguchi, Multiple blowup of solutions for a semilinear heat equation II, J. Differential Equations 231 (2006), 182–194.MATHMathSciNetCrossRefGoogle Scholar
  382. [383]
    N. Mizoguchi, H. Ninomiya and E. Yanagida, Diffusion induced blow-up in a nonlinear parabolic system, J. Dynam. Differential Equations 10 (1998), 619–638.MATHMathSciNetCrossRefGoogle Scholar
  383. [384]
    N. Mizoguchi and E. Yanagida, Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation, Math. Ann. 307 (1997), 663–675.MATHMathSciNetCrossRefGoogle Scholar
  384. [385]
    N. Mizoguchi and E. Yanagida, Critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation. II, J. Differential Equations 145 (1998), 295–331.MATHMathSciNetCrossRefGoogle Scholar
  385. [386]
    N. Mizoguchi and E. Yanagida, Blow-up and life span of solutions for a semilinear parabolic equation, SIAM J. Math. Anal. 29 (1998), 1434–1446.MATHMathSciNetCrossRefGoogle Scholar
  386. [387]
    J. Morgan, On a question of blow-up for semilinear parabolic systems, Differential Integral Equations 3 (1990), 973–978.MATHMathSciNetGoogle Scholar
  387. [388]
    C.E. Mueller and F.B. Weissler, Single point blow-up for general semilinear heat equation, Indiana Univ. Math. J. 34 (1985), 881–913.MATHMathSciNetCrossRefGoogle Scholar
  388. [389]
    Y. Naito, Non-uniqueness of solutions to the Cauchy problem for semilinear heat equations with singular initial data, Math. Ann. 329 (2004), 161–196.MATHMathSciNetCrossRefGoogle Scholar
  389. [390]
    Y. Naito, An ODE approach to the multiplicity of self-similar solutions for semi-linear heat equations, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 807–835.MATHMathSciNetCrossRefGoogle Scholar
  390. [391]
    W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), 9–18.MATHMathSciNetGoogle Scholar
  391. [392]
    W.-M. Ni, Qualitative properties of solutions to elliptic problems, Handbook of Differential Equations, Stationary partial differential equations. Vol. I (M. Chipot et al., eds.), Elsevier/North-Holland, Amsterdam, 2004, 157–233.CrossRefGoogle Scholar
  392. [393]
    W.-M. Ni and R. Nussbaum, Uniqueness and, nonuniqueness for positive radial solutions of Δu + f(u, r) = 0, Comm. Pure Appl. Math. 38 (1985), 67–108.MATHMathSciNetCrossRefGoogle Scholar
  393. [394]
    W.-M. Ni and P. Sacks, Singular behavior in nonlinear parabolic equations, Trans. Amer. Math. Soc. 287 (1985), 657–671.MATHMathSciNetCrossRefGoogle Scholar
  394. [395]
    W.-M. Ni and P. Sacks, The number of peaks of positive solutions of semilinear parabolic equations, SIAM J. Math. Anal. 16 (1985), 460–471.MATHMathSciNetCrossRefGoogle Scholar
  395. [396]
    W.-M. Ni, P. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984), 97–120.MATHMathSciNetCrossRefGoogle Scholar
  396. [397]
    W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case, Atti Accad. Naz. Lincei 77 (1986), 231–257.Google Scholar
  397. [398]
    W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activatorinhibitor system, J. Differential Equations 229 (2006), 4260–4465.MathSciNetCrossRefGoogle Scholar
  398. [399]
    S.M. Nikolskii, Approximation of functions of several variables and imbedding theorems, Springer, Berlin-Heidelberg-New York, 1975.Google Scholar
  399. [400]
    R.D. Nussbaum, Positive solutions of nonlinear elliptic boundary value problems, J. Math. Anal. Appl. 51 (1975), 461–482.MATHMathSciNetCrossRefGoogle Scholar
  400. [401]
    A. Okada and I. Fukuda, Total versus single point blow-up of solutions of a semilinear parabolic equation with localized reaction, J. Math. Anal. Appl. 281 (2003), 485–500.Google Scholar
  401. [402]
    O. Oleinik and S. Kruzkov, Quasi-linear parabolic second order equations with several independent variables, Uspehi Mat. Nauk 16 (1961), 115–155.MathSciNetGoogle Scholar
  402. [403]
    M. Otani, Existence and asymptotic stability of strong solutions of nonlinear evolution equations with a difference term of subdifferentials, Colloq. Math. Soc. Janos Bolyai 30, North-Holland, Amsterdam-New York, 1981, pp. 795–809.Google Scholar
  403. [404]
    F. Pacard, Existence and convergence of positive weak solutions of Δu = u n/n-2 bounded domains of ℝn, n ≥3, Calc. Var. Partial Differential Equations 1 (1993), 243–265.MATHMathSciNetCrossRefGoogle Scholar
  404. [405]
    C.-V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.MATHGoogle Scholar
  405. [406]
    D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal. 114 (1993), 97–105.MATHMathSciNetCrossRefGoogle Scholar
  406. [407]
    D. Passaseo, New nonexistence results for elliptic equations with supercritical nonlinearity, Differential Integral Equations 8 (1995), 577–586.MATHMathSciNetGoogle Scholar
  407. [408]
    D. Passaseo, Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains, Manuscripta Math. 65 (1989), 147–165.MATHMathSciNetCrossRefGoogle Scholar
  408. [409]
    L.E. Payne and D.H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), 273–303.MathSciNetCrossRefGoogle Scholar
  409. [410]
    A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, 1983.MATHGoogle Scholar
  410. [411]
    M. Pedersen and Z. Lin, Coupled diffusion systems with localized nonlinear reactions, Comput. Math. Appl. 42 (2001), 807–816.MATHMathSciNetCrossRefGoogle Scholar
  411. [412]
    L.A. Peletier, D. Terman and F.B. Weissler, On the equation Δu + x · Δu + f(u) =0, Arch. Rational Mech. Anal. 94 (1986), 83–99.MATHMathSciNetCrossRefGoogle Scholar
  412. [413]
    L.A. Peletier and R. van der Vorst, Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation, Differential Integral Equations 5 (1992), 747–767.MATHMathSciNetGoogle Scholar
  413. [414]
    M. Pierre (2003), Personal communication.Google Scholar
  414. [415]
    M. Pierre, Weak solutions and supersolutions in L1 for reaction-diffusion systems, J. Evol. Equ. 3 (2003), 153–168.MATHMathSciNetCrossRefGoogle Scholar
  415. [416]
    M. Pierre and D. Schmitt, Global existence for a reaction-diffusion system with a balance law, Semigroups of linear and nonlinear operations and applications Math. Sci. Engrg., 185, Kluwer Acad. Publ., Dordrecht, 1993, pp. 251–258.Google Scholar
  416. [417]
    M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM J. Math. Anal. 28 (1997), 259–269.MATHMathSciNetCrossRefGoogle Scholar
  417. [418]
    P. Plecháč and V. Šverák, Singular and regular solutions of a nonlinear parabolic system, Nonlinearity 16 (2003), 2083–2097.MathSciNetCrossRefMATHGoogle Scholar
  418. [419]
    S.I. Pohozaev, Eigenfunctions of the equation Δu + δf(u) = 0, Soviet Math. Dokl. 5 (1965), 1408–1411.Google Scholar
  419. [420]
    P. Polśčik, Morse indices and bifurcations of positive solutions of Δu+f(u) = 0 onN, Indiana Univ. Math. J. 50 (2001), 1407–1432.Google Scholar
  420. [421]
    P. Poláčik, Parabolic equations: Asymptotic behavior and dynamics on invariant manifolds, Handbook of dynamical systems, Vol. 2 (B. Fiedler, ed.), Elsevier, Amsterdam, 2002, pp. 835–883.Google Scholar
  421. [422]
    P. Poláčik and P. Quittner, A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation, Nonlinear Anal. 64 (2006), 1679–1689.MathSciNetCrossRefMATHGoogle Scholar
  422. [424]
    P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: elliptic equations and systems, Duke Math. J. 139 (2007) (to appear).Google Scholar
  423. [425]
    P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: parabolic equations, Indiana Univ. Math. J. 56 (2007), 879–908.MathSciNetCrossRefMATHGoogle Scholar
  424. [426]
    P. Poláčik and K.P. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations, J. Differential Equations 124 (1996), 472–494.MathSciNetCrossRefMATHGoogle Scholar
  425. [427]
    P. Poláčik and F. Simondon, Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, J. Differential Equations 186 (2002), 586–610.MathSciNetCrossRefMATHGoogle Scholar
  426. [428]
    P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann. 327 (2003), 745–771.MathSciNetCrossRefMATHGoogle Scholar
  427. [429]
    M. Protter and H. Weinberger, Maximum principles in differential equations, Prentice Hall, Englewood Cliffs, N.J., 1967.Google Scholar
  428. [430]
    J. Prüss and H. Sohr, Imaginary powers of elliptic second order differential operators in L p-spaces, Hiroshima Math. J. 23 (1993), 161–192.MATHMathSciNetGoogle Scholar
  429. [431]
    P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), 681–703.MATHMathSciNetCrossRefGoogle Scholar
  430. [432]
    F. Quiros and J. Rossi, Non-simultaneous blow-up in a semilinear parabolic system, Z. Angew. Math. Phys. 52 (2001), 342–346.MATHMathSciNetCrossRefGoogle Scholar
  431. [433]
    P. Quittner, Blow-up for semilinear parabolic equations with a gradient term, Math. Methods Appl. Sci. 14 (1991), 413–417.MATHMathSciNetCrossRefGoogle Scholar
  432. [434]
    P. Quittner, On global existence and stationary solutions for two classes of semilinear parabolic equations, Comment. Math. Univ. Carolin. 34 (1993), 105–124.MATHMathSciNetGoogle Scholar
  433. [435]
    P. Quittner, Global existence of solutions of parabolic problems with nonlinear boundary conditions, Banach Center Publ. 33 (1996), 309–314.MathSciNetGoogle Scholar
  434. [436]
    P. Quittner, Signed solutions for a semilinear elliptic problem, Differential Integral Equations 11 (1998), 551–559.MATHMathSciNetGoogle Scholar
  435. [437]
    P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenian. (N.S.) 68 (1999), 195–203.MATHMathSciNetGoogle Scholar
  436. [438]
    P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem, Math. Ann. 320 (2001), 299–305.MATHMathSciNetCrossRefGoogle Scholar
  437. [439]
    P. Quittner, A priori estimates of global solutions and multiple equilibria of a parabolic problem involving measure, Electron. J. Differential Equations 2001, ti 29 (2001), 1–17.Google Scholar
  438. [440]
    P. Quittner, Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math. 29 (2003), 757–799.MATHMathSciNetGoogle Scholar
  439. [441]
    P. Quittner, Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, NoDEA Nonlinear Differential Equations Appl. 11 (2004), 237–258.MATHMathSciNetGoogle Scholar
  440. [442]
    P. Quittner, Complete and energy blow-up in superlinear parabolic problems, Recent Advances in Elliptic and Parabolic Problems (Chiun-Chuan Chen, Michel Chipot and Chang-Shou Lin, eds.), World Scientific Publ., Hackensack, NJ, 2005, pp. 217–229.CrossRefGoogle Scholar
  441. [443]
    P. Quittner, The decay of global solutions of a semilinear parabolic equation, Discrete Contin. Dyn. Syst. (to appear).Google Scholar
  442. [444]
    P. Quittner, Qualitative theory of semilinear parabolic equations and systems, Lectures on evolutionary partial differential equations, Lecture notes of the Jindrich Necas Center for Mathematical Modeling, vol. 2, Matfyzpress, Praha, 2007 (to appear).Google Scholar
  443. [445]
    P. Quittner and A. Rodriguez-Bernal, Complete and energy blow-up in parabolic problems with nonlinear boundary conditions, Nonlinear Anal. 62 (2005), 863–875.MATHMathSciNetCrossRefGoogle Scholar
  444. [446]
    P. Quittner and Ph. Souplet, Admissible L p norms for local existence and for continuation in semilinear parabolic systems are not the same, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 1435–1456.MATHMathSciNetCrossRefGoogle Scholar
  445. [447]
    P. Quittner and Ph. Souplet, Global existence from single-component L p estimates in a semilinear reaction-diffusion system, Proc. Amer. Math. Soc. 130 (2002), 2719–2724.Google Scholar
  446. [448]
    P. Quittner and Ph. Souplet, A priori estimates of global solutions of superlinear parabolic problems without variational structure, Discrete Contin. Dyn. Syst. 9 (2003), 1277–1292.MATHMathSciNetGoogle Scholar
  447. [449]
    P. Quittner and Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Rational Mech. Anal. 174 (2004), 49–81.Google Scholar
  448. [450]
    P. Quittner, Ph. Souplet and M. Winkler, Initial blow-up rates and universal bounds for nonlinear heat equations, J. Differential Equations 196 (2004), 316–339.MATHMathSciNetCrossRefGoogle Scholar
  449. [451]
    P.H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161–202.MATHMathSciNetCrossRefGoogle Scholar
  450. [452]
    P.H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc. 272 (1982), 753–770.MATHMathSciNetCrossRefGoogle Scholar
  451. [453]
    M. Ramos, S. Terracini and C. Troestler, Superlinear indefinite elliptic problems and Pohozaev type identities, J. Funct. Anal. 159 (1998), 596–628.MATHMathSciNetCrossRefGoogle Scholar
  452. [454]
    W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations 161 (2000), 219–243.MATHMathSciNetCrossRefGoogle Scholar
  453. [455]
    F. Rellich, Darstellung der Eigenwerte von Δu + λu = 0 durch ein Randintegral, Math. Z. 46 (1940), 635–636.MATHMathSciNetCrossRefGoogle Scholar
  454. [456]
    O. Rey, A multiplicity result for a variational problem with lack of compactness, Nonlinear Anal. 13 (1989), 1241–1249.MATHMathSciNetCrossRefGoogle Scholar
  455. [457]
    F. Ribaud, Analyse de Littlewood Paley pour la résolution d’équations paraboliques semi-linéaires, Doctoral Thesis, University of Paris XI, 1996.Google Scholar
  456. [458]
    J. Rossi and Ph. Souplet, Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system, Differential Integral Equations 18 (2005), 405–418.MathSciNetMATHGoogle Scholar
  457. [459]
    F. Rothe, Uniform bounds from bounded L p-functionals in reaction-diffusion equations, J. Differential Equations 45 (1982), 207–233.MATHMathSciNetCrossRefGoogle Scholar
  458. [460]
    F. Rothe, Global solutions of reaction-diffusion systems, Lecture Notes in Mathematics 1072, Springer-Verlag, Berlin-Heidelberg-New York, 1984.Google Scholar
  459. [461]
    P. Rouchon, Blow-up of solutions of nonlinear heat equations in unbounded domains for slowly decaying initial data, Z. Angew. Math. Phys. 52 (2001), 1017–1032.MATHMathSciNetCrossRefGoogle Scholar
  460. [462]
    P. Rouchon, Boundedness of global solutions for nonlinear diffusion equations with localized source, Differential Integral Equations 16 (2003), 1083–1092.MATHMathSciNetGoogle Scholar
  461. [463]
    P. Rouchon, Universal bounds for global solutions of a diffusion equation with a nonlocal reaction term, J. Differential Equations 193 (2003), 75–94.MATHMathSciNetCrossRefGoogle Scholar
  462. [464]
    P. Rouchon, Universal bounds for global solutions of a diffusion equation with a mixed local-nonlocal reaction term, Acta Math. Univ. Comenian. (N.S.) 75 (2006), 63–74.MATHMathSciNetGoogle Scholar
  463. [465]
    J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math. 48 (1992), 249–264.MATHMathSciNetCrossRefGoogle Scholar
  464. [466]
    A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov, Blow-up in quasilinear parabolic equations, Nauka, Moscow, 1987, English translation: Walter de Gruyter, Berlin, 1995.Google Scholar
  465. [467]
    D.H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Math. 30 (1968), 148–172.MATHMathSciNetGoogle Scholar
  466. [468]
    D. Schmitt, Existence globale ou explosion pour les systèmes de réactiondiffusion avec contrôle de masse, Doctoral Thesis, University of Nancy 1, 1995.Google Scholar
  467. [469]
    J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, Contributions to nonlinear functional analysis, Academic Press, New York, 1971, pp. 565–601.Google Scholar
  468. [470]
    J. Serrin and H. Zou, Existence and non-existence for ground states of quasilinear elliptic equations, Arch. Rational Mech. Anal. 121 (1992), 101–130.MATHMathSciNetCrossRefGoogle Scholar
  469. [471]
    J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations 9 (1996), 635–653.MATHMathSciNetGoogle Scholar
  470. [472]
    J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena 46 (1998, (suppl.), 369–380.MATHMathSciNetGoogle Scholar
  471. [473]
    J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002), 79–142.MATHMathSciNetCrossRefGoogle Scholar
  472. [474]
    L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. Math. 118 (1983), 525–571.CrossRefGoogle Scholar
  473. [475]
    S. Snoussi, S. Tayachi and F.B. Weissler, Asymptotically self-similar global solutions of a general semilinear heat equation, Math. Ann. 321 (2001), 13–155.MathSciNetCrossRefGoogle Scholar
  474. [476]
    S. Sohr, Beschränkter H -Funktionalkalkül für elliptische Randwertsysteme, Dissertation, Kassel 1999.Google Scholar
  475. [477]
    V. Solonnikov, Green matrices for parabolic boundary value problems, Zap. Naucn. Sem. Leningrad 14 (1969), 132–150.Google Scholar
  476. [478]
    Ph. Souplet, Sur l’asymptotique des solutions globales pour une équation de la chaleur semi-linéaire dans des domaines non bornés, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 877–882.MATHMathSciNetGoogle Scholar
  477. [479]
    Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal. 28 (1998), 1301–1334.MathSciNetCrossRefGoogle Scholar
  478. [480]
    Ph. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations 153 (1999), 374–406.MATHMathSciNetCrossRefGoogle Scholar
  479. [481]
    Ph. Souplet, Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations, Comm. Partial Differential Equations 24 (1999), 951–973.Google Scholar
  480. [482]
    Ph. Souplet, A note on the paper by Qi S. Zhang: “ A priori estimates and the representation formula for all positive solutions to a semilinear parabolic problem ”;, J. Math. Anal. Appl. 243 (2000), 453–457.MATHMathSciNetCrossRefGoogle Scholar
  481. [483]
    Ph. Souplet, Decay of heat semigroups in L and applications to nonlinear parabolic problems in unbounded domains, J. Funct. Anal. 173 (2000), 343–360.MATHMathSciNetCrossRefGoogle Scholar
  482. [484]
    Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electron. J. Differential Equations 2001, ti 20 (2001), 1–19.MathSciNetGoogle Scholar
  483. [485]
    Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic e-quations with general boundary conditions, Differential Integral Equations 15 (2002), 237–256.Google Scholar
  484. [486]
    Ph. Souplet, Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys. 55 (2004), 28–31.MATHMathSciNetGoogle Scholar
  485. [487]
    Ph. Souplet, Uniform blow-up profile and boundary behaviour for a non-local reaction-diffusion equation with critical damping, Math. Methods Appl. Sci. 27 (2004), 1819–1829.MATHMathSciNetCrossRefGoogle Scholar
  486. [488]
    Ph. Souplet, Infinite time blow-up for superlinear parabolic problems with localized reaction, Proc. Amer. Math. Soc. 133 (2005), 431–436.MATHMathSciNetCrossRefGoogle Scholar
  487. [489]
    Ph. Souplet, Optimal regularity conditions for elliptic problems via L δp spaces, Duke Math. J. 127 (2005), 175–192.MATHMathSciNetCrossRefGoogle Scholar
  488. [490]
    Ph. Souplet, The influence of gradient perturbations on blow-up asyrnptotics in semilinear parabolic problems: a survey, Progress in Nonlinear Differential Equations and Their Applications, Vol. 64, Birkhaüser, 2005, pp. 473–496.MathSciNetGoogle Scholar
  489. [491]
    Ph. Souplet, A remark on the large-time behavior of solutions of viscous Hamilton-Jacobi equations, Acta Math. Univ. Comenian. (N.S.) 76 (2007), 11–13.MATHMathSciNetGoogle Scholar
  490. [492]
    Ph. Souplet, A note on diffusion-induced blow-up, J. Dynam. Differential Equations 19 (2007), to appear.Google Scholar
  491. [493]
    Ph. Souplet, Single point blow-up for a semilinear parabolic system, J. Eur. Math. Soc. (to appear).Google Scholar
  492. [494]
    Ph. Souplet and S. Tayachi, Optimal condition for non-simultaneous blow-up in a reaction-diffusion system, J. Math. Soc. Japan 56 (2004), 571–584.MATHMathSciNetCrossRefGoogle Scholar
  493. [495]
    Ph. Souplet, S. Tayachi and F.B. Weissler, Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J. 45 (1996), 655–682.MATHMathSciNetCrossRefGoogle Scholar
  494. [496]
    Ph. Souplet and J.L. Vázquez, Stabilization towards a singular steady state with gradient blow-up for a convection-diffusion problem, Discrete Contin. Dyn. Syst. 14 (2006), 221–234.MATHMathSciNetGoogle Scholar
  495. [497]
    Ph. Souplet and F.B. Weissler, Self-similar subsolutions and blowup for non-linear parabolic equations, J. Math. Anal. Appl. 212 (1997), 60–74.MATHMathSciNetCrossRefGoogle Scholar
  496. [498]
    Ph. Souplet and F.B. Weissler, Poincaré’s inequality and global solutions of a nonlinear parabolic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), 337–373.MathSciNetCrossRefGoogle Scholar
  497. [499]
    Ph. Souplet and F.B. Weissler, Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 213–235.MATHMathSciNetCrossRefGoogle Scholar
  498. [500]
    Ph. Souplet and Q.S. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations, J. Anal. Math. 99 (2006), 355–396.MathSciNetMATHCrossRefGoogle Scholar
  499. [501]
    M.A.S. Souto, A priori estimates and existence of positive solutions of non-linear cooperative elliptic systems, Differential Integral Equations 8 (1995), 1245–1258.MATHMathSciNetGoogle Scholar
  500. [502]
    R. Sperb, Growth estimates in diffusion-reaction problems, Arch. Rational Mech. Anal. 75 (1980), 127–145.MathSciNetGoogle Scholar
  501. [503]
    P.N. Srikanth, Uniqueness of solutions of nonlinear Dirichlet problems, Differential Integral Equations 6 (1993), 663–670.MATHMathSciNetGoogle Scholar
  502. [504]
    B. Straughan, Explosive instabilities in mechanics, Springer, Berlin, 1998.MATHGoogle Scholar
  503. [505]
    M. Struwe, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, Springer, Berlin, 2000.MATHGoogle Scholar
  504. [506]
    M. Struwe, Infinitely many critical points for functionals which are not even and applications to nonlinear boundary value problems, Manuscripta Math. 32 (1980), 335–364.MATHMathSciNetCrossRefGoogle Scholar
  505. [507]
    R. Suzuki, Asymptotic behavior of solutions of quasilinear parabolic equations with slowly decaying initial data, Adv. Math. Sci. Appl. 9 (1999), 291–317.MATHMathSciNetGoogle Scholar
  506. [508]
    G. Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372.MATHMathSciNetGoogle Scholar
  507. [509]
    S. Taliaferro, Local behavior and global existence of positive solutions of au λ ≤-Δu ≤uλ, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 889–901.MATHMathSciNetCrossRefGoogle Scholar
  508. [510]
    J.I. Tello, Stability of steady states of the Cauchy problem for the exponential reaction-diffusion equation, J. Math. Anal. Appl. 324 (2006), 381–396.MATHMathSciNetCrossRefGoogle Scholar
  509. [511]
    E. Terraneo, Non-uniqueness for a critical non-linear heat equation, Comm. Partial Differential Equations 27 (2002), 185–218.MATHMathSciNetCrossRefGoogle Scholar
  510. [512]
    Al. Tersenov and Ar. Tersenov, Global solvability for a class of quasilinear parabolic problems, Indiana Univ. Math. J. 50 (2001), 1899–1913.MATHMathSciNetCrossRefGoogle Scholar
  511. [513]
    H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam-New York-Oxford, 1978.Google Scholar
  512. [514]
    W.C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations 42 (1981), 400–413.MATHMathSciNetCrossRefGoogle Scholar
  513. [515]
    M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japon. 17 (1972), 173–193.MATHMathSciNetGoogle Scholar
  514. [516]
    A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B 237 (1952), 37–72.CrossRefGoogle Scholar
  515. [517]
    R.E.L. Turner, A priori bounds for positive solutions of nonlinear elliptic equations in two variables, Duke Math. J. 41 (1974), 759–774.MATHMathSciNetCrossRefGoogle Scholar
  516. [518]
    D. Tzanetis, Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating, Electron. J. Differential Equations (2002, 11), 1–26.MathSciNetGoogle Scholar
  517. [519]
    J.J.L. Velázquez, Local behaviour near blow-up points for semilinear parabolic equations, J. Differential Equations 106 (1993), 384–415.MATHMathSciNetCrossRefGoogle Scholar
  518. [520]
    J.J.L. Velázquez, Estimates on the (n-1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42 (1993), 445–476.MATHMathSciNetCrossRefGoogle Scholar
  519. [521]
    J.J.L. Velázquez, Blow up for semilinear parabolic equations, Recent advances in partial differential equations (Res. Appl. Math. 30) (M.A. Herrero, E. Zuazua, eds.), Masson, Paris, 1994, pp. 131–145.Google Scholar
  520. [522]
    J.J.L. Velázquez, V.A. Galaktionov and M.A. Herrero, The space structure near a blow-up point for semilinear heat equations: a formal approach, Zh. Vychisl. Mat. i Mat. Fiziki 31 (1991), 399–411.Google Scholar
  521. [523]
    L. Véron, Singularities of solutions of second order quasilinear equations, Pitman Research Notes in Mathematics Series, 353, Longman, Harlow, 1996.Google Scholar
  522. [524]
    L. Wang and Q. Chen, The asymptotic behaviour of blow-up solution of localized nonlinear equation, J. Math. Anal. Appl. 200 (1996), 315–321.MATHMathSciNetCrossRefGoogle Scholar
  523. [525]
    M.-X. Wang and Y. Wang, Properties of positive solutions for non-local reaction-diffusion problems, Math. Methods Appl. Sci. 19 (1996), 1141–1156.MATHMathSciNetCrossRefGoogle Scholar
  524. [526]
    N.A. Watson, Parabolic equations on an infinite strip, Marcel Dekker, Boston, 1989.MATHGoogle Scholar
  525. [527]
    H. Weinberger, An example of blowup produced by equal diffusions, J. Differential Equations 154 (1999), 225–237.MATHMathSciNetCrossRefGoogle Scholar
  526. [528]
    F.B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal. 32 (1979), 277–296.MATHMathSciNetCrossRefGoogle Scholar
  527. [529]
    F.B. Weissler, Local existence and nonexistence for semilinear parabolic equations in L p, Indiana Univ. Math. J. 29 (1980), 79–102.MATHMathSciNetCrossRefGoogle Scholar
  528. [530]
    F.B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), 29–40.MATHMathSciNetGoogle Scholar
  529. [531]
    F.B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations 55 (1984), 204–224.MATHMathSciNetCrossRefGoogle Scholar
  530. [532]
    F.B. Weissler, Asymptotic analysis of an ordinary differential equation and non-uniqueness for a semilinear partial differential equation, Arch. Rational Mech. Anal. 91 (1985), 231–245.MathSciNetCrossRefGoogle Scholar
  531. [533]
    F.B. Weissler, Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations, Arch. Rational Mech. Anal. 91 (1985), 247–266.MathSciNetCrossRefGoogle Scholar
  532. [534]
    F.B. Weissler, An L blow-up estimate for a nonlinear heat equation, Comm. Pure Appl. Math. 38 (1985), 291–295.MATHMathSciNetCrossRefGoogle Scholar
  533. [535]
    F.B. Weissler, L p-energy and blow-up for a semilinear heat equation, In Non-linear functional analysis and its applications, Proc. Symp. Pure Math. 45/2 (1986), 545–551.MathSciNetGoogle Scholar
  534. [536]
    B. Wollenmann, Uniqueness for semilinear parabolic problems, PhD Thesis, Universität Zürich, 2001.Google Scholar
  535. [537]
    R. Xing, A priori estimates for global solutions of semilinear heat equations in ℝ n, Nonlinear Anal. (to appear).Google Scholar
  536. [538]
    E. Yanagida, Uniqueness of rapidly decaying solutions to the Haraux-Weissler equation, J. Differential Equations 127 (1996), 561–570.MATHMathSciNetCrossRefGoogle Scholar
  537. [539]
    X.-F. Yang, Nodal sets and Morse indices of solutions of super-linear elliptic PDEs, J. Funct. Anal. 160 (1998), 223–253.MATHMathSciNetCrossRefGoogle Scholar
  538. [540]
    H. Zaag, A Liouville theorem and blow-up behavior for a vector-valued non-linear heat equation with no gradient structure, Comm. Pure Appl. Math. 54 (2001), 107–133.Google Scholar
  539. [541]
    H. Zaag, On the regularity of the blow-up set for semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 505–542.MATHMathSciNetCrossRefGoogle Scholar
  540. [542]
    H. Zaag, Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation, Duke Math. J. 133 (2006), 499–525.MATHMathSciNetCrossRefGoogle Scholar
  541. [543]
    T.I. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Differential Equations 4 (1968), 17–22.Google Scholar
  542. [544]
    L. Zhang, Uniqueness of positive solutions of Δu + u + u p = 0 in a ball, Comm. Partial Differential Equations 17 (1992), 1141–1164.MATHMathSciNetCrossRefGoogle Scholar
  543. [545]
    Q.S. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians, J. Differential Equations 182 (2002), 416–430.MATHMathSciNetCrossRefGoogle Scholar
  544. [546]
    P. Zhao and C. Zhong, On the infinitely many positive solutions of a super-critical elliptic problem, Nonlinear Anal. 44 (2001), 123–139.MATHMathSciNetCrossRefGoogle Scholar
  545. [547]
    H. Zou, A priori estimates for a semilinear elliptic systems without variational structure and their applications, Math. Ann. 323 (2002), 713–735.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag AG 2007

Personalised recommendations