Abstract
We begin this chapter with some general results on the existence and regularity of solutions to semilinear parabolic PDE, first treating the pure initial-value problem in §1, for PDE of the form
, where u is defined on [0, T) × M, and M has no boundary. Some of the results established in §1 will be useful in the next chapter, on nonlinear, hyperbolic equations. We also give a precursor to results on the global existence of weak solutions, which will be examined further in Chapter 17, in the context of the Navier-Stokes equations for fluids.
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References
D. Aronson, Density-dependent reaction-diffusion systems, pp. 161–176 in Dynamics and Modelling of Reaction Systems (W. Stout, W. Ray, and C. Conley, eds.), Academic Press, New York, 1980.
D. Aronson, Regularity of flows in porus media, a survey, pp. 35–49 in [NPS], Parti.
D. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal. 25(1967), 81–122.
D. Aronson and H. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propagation, pp. 5–49 in LNM #446, Springer-Verlag, New York, 1975.
D. Aronson and H. Weinberger, Multidimensional nonlinear difusion arising in population genetics, Advances in Math. 30(1978), 37–76.
J. T. Beale and C. Greengard, Convergence of Euler-Stokes splitting of the Navier-Stokes equations, Preprint, 1992.
M. Bramson, Convergence of travelling waves for systems of Kolmogorov-like parabolic equations, pp. 179–190 in [NPS], Part I.
H. Brezis and A. Pazy, Semigroups of nonlinear contractions on convex sets, J. Func. Anal. 6(1970), 237–281.
F. Browder, A priori estimates for elliptic and parabolic equations, Proc. Symp. Pure Math. IV(1961), 73–81.
J. Cannon, J. Douglas, and C. D. Hill, A multi-phase Stefan problem and the disappearance of phases, J.Math. Mech. 17(1967), 21–34.
J. Cannon and C. D. Hill, Existence, uniqueness, stability, and monotone dependence in a Stefan problem for the heat equation, J. Math. Mech. 17(1967), 1–20.
G. Carpenter, A geometrical approach to singular perturbation problems, with application to nerve impulse equations, J. Diff. Eq. 23(1977), 335–367.
S. S. Chern (ed.), Seminar on Nonlinear Partial Differential Equations, MSRI Publ. #2, Springer-Verlag, New York, 1984.
A. Chorin, T. Hughes, M. McCracken, and J. Marsden, Product formulas and numerical algorithms, CPAM 31(1978), 206–256.
B. Chow, The Ricci flow on the 2-sphere, J. Diff. Geom. 33(1991), 325–334.
C. Conley, On travelling wave solutions of nonlinear diffusion equations, pp. 498–510 in Lecture Notes in Physics #38, Springer-Verlag, New York, 1975.
E. DiBenejietto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357(1985), 1–22.
S. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50(1985), 1–26.
G. Dong, Nonlinear Partial Differential Equations of Second Order, Transi. Math. Monog., AMS, Providence, R. I., 1991.
J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10(1978), 1–68.
J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer.J. Math. 86(1964), 109–160.
W. Everitt (ed.), Spectral Theory and Differential Equations, LNM #448, Springer-Verlag, New York, 1974.
E. Fabes and D. Stroock, A new proof of Moser’s parabolic Hamack inequality via the old ideas of Nash, Arch. Rat. Mech. Anal. 96(1986), 327–338.
P. Fife, Asymptotic states of equations of reaction and diffusion, Bull. AMS 84(1978), 693–724.
M. Freidlin, Functional Integration and Partial Differential Equations, Princeton Univ. Press, Princeton, N. J., 1985.
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, N.J., 1964.
A. Friedman, Variational Principles and Free Boundary Problems, Wiley, New York, 1982.
E. Giusti (ed.), Harmonic Mappings and Minimal Immersions, LNM #1161, Springer-Verlag, New York, 1984.
P. Grindrod, Patterns and Waves, the Theory and Applications of Reaction-Diffusion Equations, Clarendon Press, Oxford, 1991.
R. Hamilton, Harmonic Maps of Manifolds with Boundary, LNS #471, Springer-Verlag, New York, 1975.
R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17(1982), 255–307.
R. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71(1988).
R. Hardt and M. Wolf (eds.), Nonlinear Partial Differential Equations in Differential Geometry, IAS/Park City Math. Sen, Vol. 2, AMS, Providence, R. I., 1995.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, LNM #840, Springer-Verlag, New York, 1981.
S. Hildebrandt, Harmonic mappings of Riemannian manifolds, pp. 1–117 of [Giu].
S. Hildebrandt, H. Raul, and R. Widman, Dirichlet’s boundary value problem for harmonic mappings of Riemannian manifolds, Math. Zeit. 147(1976), 225–236.
S. Hildebrandt, H. Raul, and R. Widman, An existence theory for harmonic mappings of Riemannian manifolds, Acta Math. 138(1977), 1–16.
L. Hörmander, Non-linear Hyperbolic Differential Equations, Lecture Notes, Lund Univ., 1986–87.
A. Ivanov, Quasilinear degenerate and nonuniformly elliptic and parabolic equations of second order, Proc. Steklov Inst. Math. 160(1984), 1–287.
J. Jost, Lectures on harmonic maps, pp. 118–192 of [Giu].
J. Jost, Nonlinear Methods in Riemannian and Kahlerian Geometry, Birkhäuser, Boston, 1988.
T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, pp. 25–70 of [Ev].
T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, CPAM 41(1988), 891–907.
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalitiesand Their Applications, Academic Press, New York, 1980.
A. Kolmogorov, I. Petrovskii, and N. Piskunov, A study of the equations of diffusion with increase in the quantity of matter, and its applications to a biological problem, Moscow Univ. Bull. Math. 1(1937), 1–26.
S. Krushkov, A priori estimates for weak solutions of elliptic and parabolic differential equations of second order, Dokl. Akad. Nauk. SSSR 150(1963), 748–751. Engl, transi. Soviet Math. 4(1963), 757–761.
N. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order, D.Reidel, Boston, 1987.
N. Krylov and M. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR Izv. 16(1981), 151–164.
K. Kunisch, K. Murphy, and G. Peichl, Estimates on the conductivity in the one-phase Stefan problem I: basic results, Preprint 1991.
O. Ladyzhenskaya, B. Solonnikov, and N. Ural’tseva, Linear and QuasilinearEquations of Parabolic Type, AMS Transi. 23, Providence, 1968.
A. Leung, Systems of Nonlinear Partial Differential Equations, Kluwer, Boston, 1989.
G. Lieberman, The first initial-boundary value problem for quasilinear second order parabolic equations, Anw. Sc. Norm. Sup. Pisa 13(1986), 347–387.
J. Marsden, On product formulas for nonlinear semigroups, J. Func. Anal. 13(1973), 51–72.
H. McKean, Application of Brownian motion to the equation of Kolmogorov- Petrovskii-Piskunov, CPAM 28(1975), 323–331.
A. Meirmanov, The Stefan Problem, W. de Gruyter, New York, 1992.
J. Moser, A new proof of DeGiorgi’s theorem concerning the regularity problem for elliptic differential equations, CPAM 13(1960), 457–468.
J. Moser, A Hamack inequality for parabolic differential equations, CPAM 15(1964), 101–134.
J. Moser, On a pointwise estimate for parabolic differential equations, CPAM 24(1971), 727–740.
J. Moser, A rapidly convergent iteration method and nonlinear partial differential equations, I, Ann. Sc. Norm. Sup. Pisa 20(1966), 265–315.
J. Murray, Mathematical Biology, Springer-Verlag, New York, 1989.
J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80(1958), 931–954.
W.-M. Ni, L. Peletier, and J. Serrin (eds.), Nonlinear Diffusion Equations and Their Equilibrium States, MSRI Publ., Vols. 12–13, Springer-Verlag, New York, 1988.
J. Polking, Boundary value problems for parabolic systems of partial differential equations, Proc. Symp. Pure Math. X(1967), 243–274.
J. Rabinowitz, A graphical approach for finding travelling wave solutions to reaction-diffusion equations, Senior thesis, Math. Dept., University of North Carolina, 1994.
J. Rauch, Global existence for the Fitzhugh-Nagumo Equations, Comm. PDE 1(1976), 609–621.
J. Rauch and J. Smoller, Qualitative theory of the Fitzhugh-Nagumo equations, Advances in Math. 27(1978), 12–44.
F. Rome, Global Solutions of Reaction-Diffusion Equations, LNM #1072, Springer-Verlag, New York, 1984.
L. Rubenstein, The Stefan Problem, Transi. Math. Monogr. #27, AMS, Providence, R. I., 1971.
R. Schoen, Analytic aspects of the harmonic map problem, pp. 321–358 in [Cher].
R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff.Geom. 17(1982), 307–335.
R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom. 18(1983), 253–268.
Y.-T. Siu, Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics, Birkhäuser, Basel, 1987.
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.
M. Struwe, Variational Methods, Springer-Verlag, New York, 1990.
M. Struwe, Geometrie evolution problems, pp. 259–339 in [HW].
M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.
A. Tromba, Teichmuller Theory in Riemannian Geometry, ETH Lectures in Math., Birkhäuser, Basel, 1992.
K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, CPAM 38(1985), 867–882.
W.-P. Wang, Multiple impulse solutions to McKean’s caricature of the nerve equation, CPAM 41(1988), 71–103; 997–1025.
R. Ye, Global existence and convergence of Yamabe flow, J. Diff. Geom. 39(1994), 35–50.
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Taylor, M.E. (1996). Nonlinear Parabolic Equations. In: Partial Differential Equations III. Applied Mathematical Sciences, vol 117. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4190-2_3
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DOI: https://doi.org/10.1007/978-1-4757-4190-2_3
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