# The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

• Juan Luis Vázquez
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2186)

## Abstract

We describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the last two centuries. It was followed by the theory of parabolic equations of different types. In a parallel development, the theory of stochastic partial differential equations gives a foundation to the probabilistic study of diffusion.

Nonlinear diffusion equations have played an important role not only in theory but also in physics and engineering, and we focus on a relevant aspect, the existence and propagation of free boundaries. Due to our research, we use the porous medium and fast diffusion equations as case examples.

A large part of the paper is devoted to diffusion driven by fractional Laplacian operators and other nonlocal integro-differential operators representing nonlocal, long-range diffusion effects. Three main models are examined (one linear, two nonlinear), and we report on recent progress in which the author is involved.

## Notes

### Acknowledgements

This work was partially supported by Spanish Project MTM2014-52240-P. The text is based on series of lectures given at the CIME Summer School held in Cetraro, Italy, in July 2016. The author is grateful to the CIME foundation for the excellent organization. The author is also very grateful to his collaborators mentioned in the text for an effort of many years. Special thanks are due to F. del Teso, N. Simonov and D. Stan for a careful reading and comments on the text.

## References

1. 1.
N. Alibaud, S. Cifani, E. Jakobsen, Continuous dependence estimates for nonlinear fractional convection-diffusion equations. SIAM J. Math. Anal. 44, 603–632 (2012)
2. 2.
M. Allen, L. Caffarelli, A. Vasseur, Porous medium flow with both a fractional potential pressure and fractional time derivative. Chin. Ann. Math. Ser. B 38(1), 45–82 (2017)
3. 3.
A. Alphonse, C.M. Elliott, Well-posedness of a fractional porous medium equation on an evolving surface. Nonlinear Anal. 137, 3–42 (2016)
4. 4.
L. Ambrosio, S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity. Commun. Pure Appl. Math. 61(11), 1495–1539 (2008)
5. 5.
L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. (Birkhäuser, Basel, 2008)
6. 6.
L. Ambrosio, E. Mainini, S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices. Ann. IHP, Analyse Non linéaire 28(2), 217–246 (2011)Google Scholar
7. 7.
F. Andreu, J.M. Mazón, J. Rossi, J. Toledo, Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, vol. 165 (American Mathematical Society, Providence, RI, 2010)Google Scholar
8. 8.
F. Andreu-Vaillo, V. Caselles, J. Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Progress in Mathematics, vol. 223 (Birkhäuser Verlag, Basel, 2004)Google Scholar
9. 9.
S.B. Angenent, D.G. Aronson, The focusing problem for the radially symmetric porous medium equation. Commun. Partial Differ. Equ. 20, 1217–1240 (1995)
10. 10.
D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 116 (Cambridge University Press, Cambridge, 2009)Google Scholar
11. 11.
A. Arnold, P. Markowich, G. Toscani, A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Commun. Partial Differ. Equ. 26(1–2), 43–100 (2001)
12. 12.
D.G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems (Montecatini Terme, 1985). Lecture Notes in Mathematics, vol. 1224 (Springer, Berlin, 1986), pp. 1–46Google Scholar
13. 13.
D.G. Aronson, P. Bénilan, Régularité des solutions de l’équation des milieux poreux dans R n. C. R. Acad. Sci. Paris Ser. A-B 288, 103–105 (1979)
14. 14.
D.G. Aronson, J.A. Graveleau, Self-similar solution to the focusing problem for the porous medium equation. Eur. J. Appl. Math. 4(1), 65–81 (1993)
15. 15.
D.G. Aronson, J. Serrin, Local behavior of solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 25, 81–122 (1967)
16. 16.
D.G. Aronson, J.L. Vázquez, Anomalous exponents in nonlinear diffusion. J. Nonlinear Sci. 5(1), 29–56 (1995)
17. 17.
D.G. Aronson, H.F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, vol. 446 (Springer, Berlin, 1975), pp. 5–49Google Scholar
18. 18.
D.G. Aronson, H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30(1), 33–76 (1978)
19. 19.
D.G. Aronson, L.A. Caffarelli, J.L. Vazquez, Interfaces with a corner point in one-dimensional porous medium flow. Commun. Pure Appl. Math. 38(4), 375–404 (1985)
20. 20.
D.G. Aronson, O. Gil, J.L. Vázquez, Limit behaviour of focusing solutions to nonlinear diffusions. Commun. Partial Differ. Equ. 23(1–2), 307–332 (1998)
21. 21.
I. Athanasopoulos, L.A. Caffarelli, Continuity of the temperature in boundary heat control problem. Adv. Math. 224(1), 293–315 (2010)
22. 22.
A. Audrito, J.L. Vázquez, The Fisher-KPP problem with doubly nonlinear diffusion. arxiv:1601.05718v2 [math.AP]Google Scholar
23. 23.
G.I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge University Press, Cambridge, 1996). Updated version of Similarity, Self-Similarity, and Intermediate Asymptotics (Consultants Bureau, New York, 1979)Google Scholar
24. 24.
B. Barrios, I. Peral, F. Soria, E. Valdinoci, A Widder’s type theorem for the heat equation with nonlocal diffusion. Arch. Ration. Mech. Anal. 213(2), 629–650 (2014)
25. 25.
R.F. Bass, Diffusions and Elliptic Operators. Probability and Its Applications (Springer, New York, 1998)Google Scholar
26. 26.
P. Bénilan, H. Brezis, M.G. Crandall, A semilinear equation in L 1(R N). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2, 523–555 (1975)Google Scholar
27. 27.
J.G. Berryman, C.J. Holland, Nonlinear diffusion problem arising in plasma physics. Phys. Rev. Lett. 40, 1720–1722 (1978)
28. 28.
J. Bertoin, Lévy Processes. Cambridge Tracts in Mathematics, vol. 121 (Cambridge University Press, Cambridge, 1996)Google Scholar
29. 29.
A. Bertozzi, T. Laurent, F. Léger, Aggregation via Newtonian potential and aggregation patches. M3AS 22(suppl. 1), 1140005, 39 pp. (2012)Google Scholar
30. 30.
P. Biler, G. Wu, Two-dimensional chemotaxis models with fractional diffusion. Math. Methods Appl. Sci. 32(1), 112–126 (2009)
31. 31.
P. Biler, G. Karch, R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions. Commun. Math. Phys. 294(1), 145–168 (2010)
32. 32.
P. Biler, C. Imbert, G. Karch, Barenblatt profiles for a nonlocal porous medium equation. C.R. Math. 349(11), 641–645 (2011)Google Scholar
33. 33.
P. Biler, C. Imbert, G. Karch, Nonlocal porous medium equation: Barenblatt profiles and other weak solutions. Arch. Ration. Mech. Anal. 215(2), 497–529 (2015)
34. 34.
C. Bjorland, L. Caffarelli, A. Figalli, Non-local gradient dependent operators. Adv. Math. 230(4–6), 1859–1894 (2012)
35. 35.
A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, J.L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates. Arch. Rat. Mech. Anal. 191, 347–385 (2009)
36. 36.
R.M. Blumenthal, R.K. Getoor, Some theorems on stable processes. Trans. Am. Math. Soc. 95(2), 263–273 (1960)
37. 37.
K. Bogdan, K. Burdzy, Z.-Q. Chen, Censored stable processes. Probab. Theory Relat. Fields 127, 89–152 (2003)
38. 38.
M. Bonforte, A. Figalli, Total variation flow and sign fast diffusion in one dimension. J. Differ. Equ. 252(8), 4455–4480 (2012)
39. 39.
M. Bonforte, G. Grillo, Asymptotics of the porous media equation via Sobolev inequalities. J. Funct. Anal. 225(1), 33–62 (2005)
40. 40.
M. Bonforte, J.L. Vázquez, Global positivity estimates and Harnack inequalities for the fast diffusion equation. J. Funct. Anal. 240(2), 399–428 (2006)
41. 41.
M. Bonforte, J.L. Vázquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations. Adv. Math. 223(2), 529–578 (2010)
42. 42.
M. Bonforte, J.L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations. Adv. Math. 250, 242–284 (2014). arXiv:1210.2594Google Scholar
43. 43.
M. Bonforte, J.L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains. Arch. Ration. Mech. Anal. 218(1), 317–362 (2015)
44. 44.
M. Bonforte, J.L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains. Part I. Existence, uniqueness and upper bounds. Nonlinear Anal. 131, 363–398 (2016)Google Scholar
45. 45.
M. Bonforte, J. Dolbeault, G. Grillo, J.L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities. Proc. Natl. Acad. Sci. USA 107(38), 16459–16464 (2010)
46. 46.
M. Bonforte, G. Grillo, J.L. Vázquez, Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold. Arch. Ration. Mech. Anal. 196(2), 631–680 (2010)
47. 47.
M. Bonforte, Y. Sire, J.L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete Contin. Dyn. Syst. A 35(12), 5725–5767 (2015)
48. 48.
M. Bonforte, A. Segatti, J.L. Vázquez, Non-existence and instantaneous extinction of solutions for singular nonlinear fractional diffusion equations. Calc. Var. Partial Differ. Equ. 55(3), 23 pp. (2016) Art. 68Google Scholar
49. 49.
M. Bonforte, A. Figalli, X. Ros-Otón, Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains. Commun. Pure Appl. Math. 70(8), 1472–1508 (2017)
50. 50.
M. Bonforte, A. Figalli, J.L. Vázquez, Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains. arXiv:1610.09881Google Scholar
51. 51.
M. Bonforte, Y. Sire, J.L. Vázquez, Optimal existence and uniqueness theory for the fractional heat equation. Nonlinear Anal. 153, 142–168 (2017)
52. 52.
N. Bournaveas, V. Calvez, The one-dimensional Keller-Segel model with fractional diffusion of cells. Nonlinearity 23(4), 923–935 (2010)
53. 53.
J. Boussinesq, Recherches théoriques sur l’écoulement des nappes d’eau infiltrées dans le sol et sur le débit des sources. Comp. Rend. Acad. Sci. J. Math. Pure. Appl. 10, 5–78 (1903/1904)
54. 54.
C. Brändle, A. de Pablo, Nonlocal heat equations: decay estimates and Nash inequalities. arXiv:1312.4661Google Scholar
55. 55.
C. Brändle, J.L. Vázquez, Viscosity solutions for quasilinear degenerate parabolic equations of porous medium type. Indiana Univ. Math. J. 54(3), 817–860 (2005)
56. 56.
C. Brändle, E. Colorado, A. de Pablo, U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 143(1), 39–71 (2013)
57. 57.
R. Brown, A brief account of microscopical observations …. Philos. Mag. 4, 161–173 (1828)Google Scholar
58. 58.
C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol. 20 (Springer; Unione Matematica Italiana, Bologna, 2016)Google Scholar
59. 59.
X. Cabré, L. Caffarelli, Fully Nonlinear Elliptic Equations (American Mathematical Society, Providence, RI, 1995)
60. 60.
X. Cabré, J.M. Roquejoffre, Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire. C. R. Math. Acad. Sci. Paris 347(23–24), 1361–1366 (2009)
61. 61.
X. Cabré, J.M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations. Commun. Math. Phys. 320(3), 679–722 (2013)
62. 62.
X. Cabré, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224(5), 2052–2093 (2010)
63. 63.
L.A. Caffarelli, The obstacle problem, Lezioni Fermiane. [Fermi Lectures] Acc. Nazionale dei Lincei; Scuola Normale Superiore, Pisa (1998)Google Scholar
64. 64.
L.A. Caffarelli, Non-local diffusions, drifts and games, in Nonlinear Partial Differential Equations. Abel Symposium, vol. 7 (Springer, Heidelberg, 2012), pp. 37–52Google Scholar
65. 65.
L.A. Caffarelli, A. Friedman, Continuity of the density of a gas flow in a porous medium. Trans. Am. Math. Soc. 252, 99–113 (1979)
66. 66.
L.A. Caffarelli, S. Salsa, A Geometric Approach to Free Boundary Problems (American Mathematical Society, Providence, RI, 2005)
67. 67.
L.A. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)
68. 68.
L.A. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. 171, 1903–1930 (2010)
69. 69.
L. Caffarelli, J.L. Vázquez, Viscosity solutions for the porous medium equation, in Differential Equations: La Pietra 1996 (Florence). Proceedings of Symposia in Pure Mathematics, vol. 65 (American Mathematical Society, Providence, 1999), p. 1326Google Scholar
70. 70.
L.A. Caffarelli, J.L. Vázquez, Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537–565 (2011)
71. 71.
L.A. Caffarelli, J.L. Vázquez, Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Contin. Dyn. Syst. A 29(4), 1393–1404 (2011)
72. 72.
L.A. Caffarelli, N.I. Wolanski, C 1,α regularity of the free boundary for the N-dimensional porous media equation. Commun. Pure Appl. Math. 43, 885–902 (1990)
73. 73.
L.A. Caffarelli, J.L. Vázquez, N.I. Wolanski, Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. J. 36, 373–401 (1987)
74. 74.
L. Caffarelli, C.H. Chan, A. Vasseur, Regularity theory for parabolic nonlinear integral operators. J. Am. Math. Soc. 24(3), 849–869 (2011)
75. 75.
L.A. Caffarelli, F. Soria, J.L. Vázquez, Regularity of solutions of the fractional porous medium flow. J. Eur. Math. Soc. 15(5), 1701–1746 (2013). arXiv 1201.6048v1 (2012)Google Scholar
76. 76.
V. Calvez, J. A. Carrillo, F. Hoffmann, The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime, Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions (Springer, Berlin, 2017)Google Scholar
77. 77.
M. Caputo, Linear model of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13(5), 529–539 (1967)
78. 78.
J.A. Carrillo, G. Toscani, Asymptotic L 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49, 113–141 (2000)
79. 79.
J.A. Carrillo, A. Jüngel, P.A. Markowich, G. Toscani, A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133(1), 1–82 (2001)
80. 80.
J.A. Carrillo, Y. Huang, M.C. Santos, J.L. Vázquez, Exponential convergence towards stationary states for the 1D porous medium equation with fractional pressure J. Differ. Equ. 258(3), 736–763 (2015)
81. 81.
A. Castro, D. Córdoba, Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math. 219 (6), 1916–1936 (2008)
82. 82.
A. Castro, D. Córdoba, F. Gancedo, R. Orive, Incompressible flow in porous media with fractional diffusion. Nonlinearity 22 (8), 1791–1815 (2009)
83. 83.
A. Chang, M.D.M. González, Fractional Laplacian in conformal geometry. Adv. Math. 226(2), 1410–1432 (2011)
84. 84.
H. Chang-Lara, G. Dávila, Regularity for solutions of nonlocal parabolic equations II. J. Differ. Equ. 256(1), 130–156 (2014)
85. 85.
E. Chasseigne, J.L. Vázquez, Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities. Arch. Ration. Mech. Anal. 164(2), 133–187 (2002)
86. 86.
Z.Q. Chen, R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal. 226 (1), 90–113 (2005)
87. 87.
Z.Q. Chen, P. Kim, R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian. J. Eur. Math. Soc. 12(5), 1307–1329 (2010)
88. 88.
Z.Q. Chen, P. Kim, R. Song, Two-sided heat kernel estimates for censored stable-like processes. Probab. Theory Relat. Fields 146(3–4), 361–399 (2010)
89. 89.
B. Chow, P. Lu, L. Ni, Hamilton’s Ricci Flow. Graduate Studies in Mathematics, vol. 77 (American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006)Google Scholar
90. 90.
S. Cifani, E.R. Jakobsen, Entropy solution theory for fractional degenerate convection-diffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(3), 413–441 (2011)
91. 91.
R. Cont, P. Tankov, Financial Modelling with Jump Processes (Chapman & Hall/CRC, Boca Raton, 2004)Google Scholar
92. 92.
M. Cozzi, A. Figalli, Regularity theory for local and nonlocal minimal surfaces: an overview, in Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions (Springer, Berlin, 2017)Google Scholar
93. 93.
M.G. Crandall, T.M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces. Am. J. Math. 93, 265–298 (1971)
94. 94.
M.G. Crandall, L.C. Evans, P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 282(2), 487–502 (1984)
95. 95.
J. Crank, The Mathematics of Diffusion, 2nd edn. (Clarendon Press, Oxford, 1975)
96. 96.
J. Crank, Free and Moving Boundary Problems (The Clarendon Press, Oxford University Press, New York, 1987)
97. 97.
N. Cusimano, F. Del Teso, L. Gerardo-Giorda, G. Pagnini, Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions. Preprint (2017)Google Scholar
98. 98.
H. Darcy, Les Fontaines Publiques de la ville de Dijon (V. Dalmont, Paris, 1856), pp. 305–401Google Scholar
99. 99.
P. Daskalopoulos, C. Kenig, Degenerate Diffusions. Initial Value Problems and Local Regularity Theory. EMS Tracts in Mathematics, vol. 1 (European Mathematical Society (EMS), Zürich, 2007)Google Scholar
100. 100.
P. Daskalopoulos, Y. Sire, J.L. Vázquez, Weak and smooth solutions for a fractional Yamabe flow: the case of general compact and locally conformally flat manifolds. Communications in Partial Differential Equations (to appear)Google Scholar
101. 101.
E.B. Davies, Heat kernel bounds for second order elliptic operators on Riemannian manifolds. Am. J. Math. 109(3), 545–569 (1987)
102. 102.
E.B. Davies, Heat Kernels and Spectral Theory Cambridge Tracts in Mathematics, vol. 92 (Cambridge University Press, Cambridge, 1990)Google Scholar
103. 103.
E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. (Italian) Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3, 25–43 (1957)Google Scholar
104. 104.
A. de Pablo, A. Sánchez, Travelling wave behaviour for a porous-Fisher equation. Eur. J. Appl. Math. 9(3), 285–304 (1998)
105. 105.
A. de Pablo, J.L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation. J. Differ. Equ. 93(1), 19–61 (1991)
106. 106.
A. De Pablo, F. Quirós, A. Rodríguez, J.L. Vázquez, A fractional porous medium equation. Adv. Math. 226(2), 1378–1409 (2011)
107. 107.
A. De Pablo, F. Quirós, A. Rodríguez, J.L. Vázquez, A general fractional porous medium equation. Commun. Pure Appl. Math. 65(9), 1242–1284 (2012)
108. 108.
A. de Pablo, F. Quirós, A. Rodríguez, J.L. Vázquez, Classical solutions for a logarithmic fractional diffusion equation. J. Math. Pures Appl. (9) 101(6), 901–924 (2014)Google Scholar
109. 109.
A. De Pablo, F. Quirós, A. Rodríguez, Nonlocal filtration equations with rough kernels. Nonlinear Anal. TMA 137, 402–425 (2016)
110. 110.
M. del Pino, Bubbling blow-up in critical parabolic problems, in Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions (Springer, Berlin, 2017)Google Scholar
111. 111.
F. del Teso, Finite difference method for a fractional porous medium equation. Calcolo 51(4), 615–638 (2014)
112. 112.
F. del Teso, J. Endal, E.R. Jakobsen, Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type. Adv. Math. 305, 78–143 (2017)
113. 113.
J.I. Díaz, Nonlinear partial differential equations and free boundaries. Vol. I. Elliptic equations. Research Notes in Mathematics, vol. 106 (Pitman Advanced Publishing Program, Boston, MA, 1985)Google Scholar
114. 114.
E. diBenedetto, Degenerate Parabolic Equations. Series Universitext (Springer, New York, 1993)Google Scholar
115. 115.
E. DiBenedetto, U. Gianazza, V. Vespri, Harnack’s Inequality for Degenerate and Singular Parabolic Equations. Springer Monographs in Mathematics (Springer, New York, 2012)Google Scholar
116. 116.
E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Preprint (2011)Google Scholar
117. 117.
S. Dipierro, E. Valdinoci, A simple mathematical model inspired by the Purkinje cells: from delayed travelling waves to fractional diffusion. arXiv:1702.05553Google Scholar
118. 118.
C.M. Elliott, V. Janovský, An error estimate for a finite-element approximation of an elliptic variational inequality formulation of a Hele-Shaw moving-boundary problem. IMA J. Numer. Anal. 3(1), 1–9 (1983)
119. 119.
C. Escudero, The fractional Keller-Segel model. Nonlinearity 19 (12), 2909–2918 (2006)
120. 120.
L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, RI, 1998)Google Scholar
121. 121.
L.C. Evans, The 1-Laplacian, the -Laplacian and differential games. Perspectives in nonlinear partial differential equations. Contemporary Mathematics, vol. 446, (American Mathematical Society, Providence, RI, 2007), pp. 245–254Google Scholar
122. 122.
L.C. Evans, An introduction to Stochastic Differential Equations (American Mathematical Society, Providence, RI, 2013)
123. 123.
E.B. Fabes, C.E. Kenig, R.P. Serapioni, The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7(1), 77–116 (1982)
124. 124.
P. Fabrie, Solutions fortes et comportement asymptotique pour un modèle de convection naturelle en milieu poreux (French) Acta Appl. Math. 7, 49–77 (1986)
125. 125.
M. Felsinger, M. Kassmann, Local regularity for parabolic nonlocal operators. Commun. Partial Differ. Equ. 38(9), 1539–1573 (2013)
126. 126.
A. Fick, Ueber diffusion (in German) [On Diffusion]. Ann. Phys. 94, 59–86 (1855)
127. 127.
R.A. Fisher, The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937)
128. 128.
J. Fourier, Théorie analytique de la Chaleur; reprint of the 1822 original: Éditions Jacques Gabay, Paris, 1988. English version: The Analytical Theory of Heat (Dover, New York, 1955)Google Scholar
129. 129.
A. Friedman, Partial Differential Equations of Parabolic Type (Prentice-Hall, Englewood Cliffs, NJ, 1964)
130. 130.
A. Friedman, Stochastic Differential Equations and Applications, vols. 1–2 (Academic, New York, 1976)
131. 131.
A. Friedman, Variational Principles and Free Boundaries (Wiley, New York, 1982)
132. 132.
A. Friedman, S. Kamin, The asymptotic behavior of gas in an N-dimensional porous medium. Trans. Am. Math. Soc. 262, 551–563 (1980)
133. 133.
H. Fujita, On the blowing up of solutions of the Cauchy problem for $$u_{t} = \Delta u + u^{1+\alpha }$$. J. Fac. Sci. Tokyo Sect. IA Math. 13, 109–124 (1966)
134. 134.
V.A. Galaktionov, J.L. Vázquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions. Commun. Pure Appl. Math. 50(1), 1–67 (1997)
135. 135.
V.A. Galaktionov, J.L. Vázquez, The problem of blow-up in nonlinear parabolic equations. Current developments in partial differential equations (Temuco, 1999). Discrete Contin. Dyn. Syst. 8(2), 399–433 (2002)Google Scholar
136. 136.
G. Giacomin, J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys. 87 (1–2), 37–61 (1997)
137. 137.
G. Giacomin, J.L. Lebowitz, R. Marra, Macroscopic evolution of particle systems with short and long-range interactions. Nonlinearity 13(6), 2143–2162 (2000)
138. 138.
I.I. Gihman, A.V. Skorohod, The Theory of Stochastic Processes. III. Grundlehren der Mathematischen Wissenschaften, vol. 232 (Springer, Berlin, 1979) [Russian Edition 1975]Google Scholar
139. 139.
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 1988)
140. 140.
G. Gilboa, S. Osher, Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)
141. 141.
M.D.M. González, Recent progress on the fractional Laplacian in conformal geometry. arxiv:1609.08988v1Google Scholar
142. 142.
M.D.M. González, J. Qing, Fractional conformal Laplacians and fractional Yamabe problems. Anal. Partial Differ. Equ. 6(7), 1535–1576 (2013)
143. 143.
A.A. Grigor’yan, On the fundamental solution of the heat equation on an arbitrary Riemannian manifold. (Russian) Mat. Zametki 41 (5), 687–692, 765 (1987). English translation: Math. Notes 41(5–6), 386–389 (1987)Google Scholar
144. 144.
A.A. Grigor’yan, Heat kernels on weighted manifolds and applications. Contemp. Math. 398, 93–191 (2006)
145. 145.
G. Grillo, M. Muratori, Radial fast diffusion on the hyperbolic space. Proc. Lond. Math. Soc. (3) 109(2), 283–317 (2014)Google Scholar
146. 146.
G. Grillo, M. Muratori, M.M. Porzio, Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincaré inequalities. Discrete Contin. Dyn. Syst. 33(8), 3599–3640 (2013)
147. 147.
G. Grillo, M. Muratori, F. Punzo, Fractional porous media equations: existence and uniqueness of weak solutions with measure data. Calc. Var. Partial Differ. Equ. 54(3), 3303–3335 (2015)
148. 148.
G. Grillo, M. Muratori, J.L. Vázquez, The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour. arXiv:1604.06126 [math.AP]Google Scholar
149. 149.
Q.Y. Guan, Z.M. Ma, Reflected symmetric α-stable processes and regional fractional Laplacian. Probab. Theory Relat. Fields 134, 649–694 (2006)
150. 150.
M.E. Gurtin, R.C. MacCamy, On the diffusion of biological populations. Math. Biosci. 33(1–2), 35–49 (1977)
151. 151.
R.S. Hamilton, The Ricci flow on surfaces. Contemp. Math. 71, 237–262 (1988)
152. 152.
A.K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation. Philos. Mag. 26, 65–72 (1972)Google Scholar
153. 153.
H.S. Hele-Shaw, The flow of water. Nature 58, 34–36 (1898)
154. 154.
M.A. Herrero, J.J.L. Velázquez, A blow-up mechanism for a chemotaxis model. Ann. Sc. Norm. Super. Pisa. Cl. Sci. IV 24(4), 633–683 (1997)
155. 155.
S.D. Howison, Complex variable methods in Hele-Shaw moving boundary problems. Eur. J. Appl. Math. 3, 209–224 (1992)
156. 156.
Y.H. Huang, Explicit Barenblatt profiles for fractional porous medium equations. Bull. Lond. Math. Soc. 4(46), 857–869 (2014)
157. 157.
R. Hynd, E. Lindgren, Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution. Anal. Partial Differ. Equ. 9(6), 1447–1482 (2016)
158. 158.
R.G. Iagar, A. Sánchez, J.L. Vázquez, Radial equivalence for the two basic nonlinear degenerate diffusion equations. J. Math. Pures Appl. (9) 89(1), 1–24 (2008)Google Scholar
159. 159.
M. Jara, Hydrodynamic limit of particle systems with long jumps. http://arxiv.org/abs/0805.1326v2
160. 160.
M.D. Jara, T. Komorowski, S. Olla, Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab. 19(6), 2270–2300 (2009)
161. 161.
M. Jara, C. Landim, S. Sethuraman, Nonequilibrium fluctuations for a tagged particle in mean-zero one-dimensional zero-range processes. Probab. Theory Relat. Fields 145, 565–590 (2009)
162. 162.
T.L. Jin, J.G. Xiong, A fractional Yamabe ow and some applications. J. Reine Angew. Math. 696, 187–223 (2014)
163. 163.
A. Jüngel, Cross diffusions, Chap. 4 in Entropy Methods for Diffusive Partial Differential Equations. Springer Briefs in Mathematics (Springer, Cham, 2016)Google Scholar
164. 164.
S. Kamenomostskaya (Kamin), On the Stefan problem. Mat. Sb. 53, 489–514 (1961)Google Scholar
165. 165.
S. Kamin, P. Rosenau, Propagation of thermal waves in an inhomogeneous medium. Commun. Pure Appl. Math. 34, 831–852 (1981)
166. 166.
S. Kamin, J.L. Vázquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation. Rev. Mat. Iberoam. 4(2), 339–354 (1988)
167. 167.
S. Kamin, G. Reyes, J.L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density. Discrete Contin. Dyn. Syst. 26, 521–549 (2010)
168. 168.
S. Kaplan, On the growth of solutions of quasi-linear parabolic equations. Commun. Pure Appl. Math. 16, 305–330 (1963)
169. 169.
G. Karch, Nonlinear evolution equations with anomalous diffusion, in Qualitative Properties of Solutions to Partial Differential Equations. Jindrich Nečas Center for Mathematical Modelling Lecture Notes, vol. 5 (Matfyzpress, Prague, 2009), pp. 25–68Google Scholar
170. 170.
M. Kassmann, A priori estimates for integro-differential operators with measurable kernels. Calc. Var. 34, 1–21 (2009)
171. 171.
C. Kienzler, Flat fronts and stability for the porous medium equation. Dissertation, 2013. See also arxiv.org 1403.5811 (2014)Google Scholar
172. 172.
C. Kienzler, H. Koch, J.L. Vázquez, Flatness implies smoothness for solutions of the porous medium equation. arXiv:1609.09048.v1Google Scholar
173. 173.
S. Kim, K.-A. Lee, Hölder estimates for singular non-local parabolic equations. J. Funct. Anal. 261(12), 3482–3518 (2011)
174. 174.
D. Kinderlehrer, G. Stampacchia, An introduction to Variational Inequalities and Their Applications. Pure and Applied Mathematics, vol. 88 (Academic, New York, 1980)Google Scholar
175. 175.
J.R. King, Extremely high concentration dopant diffusion in silicon. IMA J. Appl. Math. 40(3), 163–181 (1988)
176. 176.
J.R. King, Self-similar behaviour for the equation of fast nonlinear diffusion. Philos. Trans. R. Soc. Lond. A 343, 337–375 (1993)
177. 177.
J. King, P. McCabe, On the Fisher-KPP equation with fast nonlinear diffusion. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459(2038), 2529–2546 (2003)
178. 178.
J. King, A.A. Lacey, J.L. Vázquez, Persistence of corners in free boundaries in Hele-Shaw flow. Complex analysis and free boundary problems (St. Petersburg, 1994). European J. Appl. Math. 6(5), 455–490 (1995)Google Scholar
179. 179.
A. Kiselev, F. Nazarov, A. Volberg. Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Inv. Math. 167, 445–453 (2007)
180. 180.
A. Kiselev, F. Nazarov, R. Shterenberg, Blow up and regularity for fractal Burgers equation. Dyn. Partial Differ. Equ. 5, 211–240 (2008)
181. 181.
H. Koch, Non-Euclidean singular integrals and the porous medium equation. University of Heidelberg, Habilitation Thesis (1999). http://www.iwr.uniheidelberg.de/groups/amj/koch.html Google Scholar
182. 182.
A.N. Kolmogorov, I.G. Petrovskii, N.S. Piskunov, Etude de l’équation de diffusion avec accroissement de la quantité de matière, et son application à un problème biologique. Bjul. Moskowskogo Gos. Univ. 17, 1–26 (1937)Google Scholar
183. 183.
S.N. Kruzhkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) (Russian) 81(123), 228–255 (1970)Google Scholar
184. 184.
O.A. Ladyzhenskaya, N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations (Academic, New York, 1968) [Translated from the Russian]
185. 185.
O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23 (American Mathematical Society, Providence, RI, 1968)Google Scholar
186. 186.
G. Lamé, B.P. Clapeyron, Mémoire sur la solidification par refroidissement d’un globe liquid. Ann. Chimie Phys. 47, 250–256 (1831)Google Scholar
187. 187.
N.S. Landkof, Foundations of Modern Potential Theory. Die Grundlehren der mathematischen Wissenschaften, Band 180 (Springer, New York, 1972) [Translated from the Russian by A.P. Doohovskoy]Google Scholar
188. 188.
L.S. Leibenzon, The Motion of a Gas in a Porous Medium, Complete Works, vol. 2 (Acad. Sciences URSS, Moscow, 1953) (Russian). First published in Neftanoe i slantsevoe khozyastvo, 10, 1929, and Neftanoe khozyastvo, 8–9, 1930 (Russian)Google Scholar
189. 189.
H.A. Levine, The role of critical exponents in blowup theorems. SIAM Rev. 32(2), 262–288 (1990)
190. 190.
D. Li, J.L. Rodrigo, X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem. Rev. Mat. Iberoam. 26(1), 295–332 (2010)
191. 191.
G.M. Lieberman, Second Order Parabolic Differential Equations (World Scientific, River Edge, NJ, 1996)
192. 192.
F.H. Lin, P. Zhang, On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete Contin. Dyn. Syst. 6, 121–142 (2000)
193. 193.
J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications (French). vol. 1. Travaux et Recherches Mathématiques, vol. 17 (Dunod, Paris 1968); vol. 2. Travaux et Recherches Mathématiques, vol. 18 (Dunod, Paris 1968); vol. 3. Travaux et Recherches Mathématiques, vol. 20 (Dunod, Paris, 1970)Google Scholar
194. 194.
S. Lisini, E. Mainini, A. Segatti, A gradient flow approach to the porous medium equation with fractional pressure. arXiv:1606.06787Google Scholar
195. 195.
P. Lu, L. Ni, J.L. Vázquez, C. Villani, Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds. J. Math. Pures Appl. (9) 91(1), 1–19 (2009)Google Scholar
196. 196.
A. Majda, E. Tabak, A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow, nonlinear phenomena in ocean dynamics (Los Alamos, NM, 1995). Phys. D. 98(2–4), 515–522 (1996)
197. 197.
J.M. Mazón, J.D. Rossi, J. Toledo, Fractional p-Laplacian evolution equations. J. Math. Pures Appl. (9) 105(6), 810–844 (2016)Google Scholar
198. 198.
A.M. Meirmanov, The Stefan Problem. de Gruyter Expositions in Mathematics, vol. 3 (Walter de Gruyter & Co., Berlin, 1992) [translated from the Russian]Google Scholar
199. 199.
A. Mellet, S. Mischler, C. Mouhot, Fractional diffusion limit for collisional kinetic equations. Arch. Ration. Mech. Anal. 199, 493–525 (2011)
200. 200.
R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)
201. 201.
J.W. Morgan, G. Tian, Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs (American Mathematical Society, Providence, RI, 2007)Google Scholar
202. 202.
J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Commun. Pure Appl. Math. 13, 457–468 (1960)
203. 203.
J. Moser, A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17, 101–134 (1964)
204. 204.
R. Musina, A.I. Nazarov, On fractional Laplacians. Commun. Partial Differ. Equ. 39(9), 1780–1790 (2014)
205. 205.
M. Muskat, The Flow of Homogeneous Fluids Through Porous Media (McGraw-Hill, New York, 1937)
206. 206.
J. Nash, Parabolic equations. Proc. Natl. Acad. Sci. USA 43, 754–758 (1957)
207. 207.
J. Nash, Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
208. 208.
R.H. Nochetto, E. Otarola, A.J. Salgado, A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15(3), 733–791 (2015)
209. 209.
R.H. Nochetto, E. Otarola, A.J. Salgado, A PDE approach to space-time fractional parabolic problems. SIAM J. Numer. Anal. 54(2), 848–873 (2016)
210. 210.
K. Nyström, O. Sande, Extension properties and boundary estimates for a fractional heat operator. Nonlinear Anal. 140, 29–37 (2016)
211. 211.
H. Okuda, J.M. Dawson, Theory and numerical simulation on plasma diffusion across a magnetic field. Phys. Fluids 16, 408–426 (1973)
212. 212.
O.A. Oleinik, A.S. Kalashnikov, Y.-I. Chzou, The Cauchy problem and boundary problems for equations of the type of unsteady filtration. Izv. Akad. Nauk SSR Ser. Math. 22, 667–704 (1958) [in Russian]
213. 213.
C. Pozrikidis, The Fractional Laplacian (Chapman and Hall/CRC, Boca Raton, 2016)
214. 214.
D. Puhst, On the evolutionary fractional p-Laplacian. Appl. Math. Res. Express 2015(2), 253–273 (2015)
215. 215.
G. Reyes, J.L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Commun. Pure Appl. Anal. 8, 493–508 (2009)
216. 216.
S. Richardson, Some Hele Shaw flows with time-dependent free boundaries. J. Fluid Mech. 102, 263–278 (1981)
217. 217.
A. Rodríguez, J.L. Vázquez, Obstructions to existence in fast-diffusion equations J. Differ. Equ. 184(2), 348–385 (2002)
218. 218.
X. Ros-Oton, J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101, 275–302 (2014)
219. 219.
L.I. Rubinstein, The Stefan Problem. Translations of Mathematical Monographs, vol. 27 (American Mathematical Society, Providence, RI, 1971)Google Scholar
220. 220.
P.G. Saffman, G.I. Taylor. The penetration of fluid into a porous medium Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A 245, 312–329 (1958)
221. 221.
S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, 3rd edn. Unitext, vol. 99 (Springer, Berlin, 2016)Google Scholar
222. 222.
R.W. Schwab, L. Silvestre, Regularity for parabolic integro-differential equations with very irregular kernels. Anal. Partial Differ. Equ. 9(3), 727–772 (2016)
223. 223.
C. Seis, Long-time asymptotics for the porous medium equation: The spectrum of the linearized operator. J. Differ. Equ. 256(3), 1191–1223 (2014)
224. 224.
S. Serfaty, J.L. Vazquez, A mean field equation as limit of nonlinear diffusion with fractional Laplacian operators. Calc. Var. Partial Differ. Equ. 49(3–4), 1091–1120 (2014)
225. 225.
J. Serrin, Local behavior of solutions of quasi-linear equations. Acta Math. 111, 247–302 (1964)
226. 226.
R. Servadei, E. Valdinoci, On the spectrum of two different fractional operators. Proc. R. Soc. Edinb. A 144(4), 831–855 (2014)
227. 227.
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator. Ph. D. thesis, University of Texas at Austin (2005)Google Scholar
228. 228.
L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace Indiana Univ. Math. J. 55(3), 1155–1174 (2006)
229. 229.
Y. Sire, J. L.Vázquez, B. Volzone, Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application. Chin. Ann. Math. Ser. B 38(2), 661–686 (2017)
230. 230.
J.A. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, New York, 1982)
231. 231.
H.M. Soner, Stochastic representations for nonlinear parabolic PDEs, in Handbook of Differential Equations: Evolutionary Equations. Handbook of Differential Equations, vol. III (Elsevier/North-Holland, Amsterdam, 2007), pp. 477–526Google Scholar
232. 232.
D. Stan, J.L. Vázquez, Asymptotic behaviour of the doubly nonlinear diffusion equation $$u_{t} = \Delta _{p}(u^{m})$$ on bounded domains. Nonlinear Anal. 77, 1–32 (2013)
233. 233.
D. Stan, J.L. Vázquez, The Fisher-KPP equation with nonlinear fractional diffusion. SIAM J. Math. Anal. 46(5), 3241–3276 (2014)
234. 234.
D. Stan, F. del Teso, J.L. Vázquez, Finite and infinite speed of propagation for porous medium equations with fractional pressure. Comptes Rendus Mathématique (Comptes Rendus Acad. Sci. Paris) 352(2), 123–128 (2014). arXiv:1311.7007Google Scholar
235. 235.
D. Stan, F. del Teso, J.L. Vázquez, Transformations of Self-Similar Solutions for porous medium equations of fractional type. Nonlinear Anal. 119, 62–73 (2015)
236. 236.
D. Stan, F. del Teso, J.L. Vázquez, Finite and infinite speed of propagation for porous medium equations with fractional pressure. J. Differ. Equ. 260(2), 1154–1199 (2016)
237. 237.
D. Stan, F. del Teso, J.L. Vázquez, Existence of weak solutions for porous medium equations with nonlocal pressure. arXiv:1609.05139Google Scholar
238. 238.
J. Stefan, Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Ann. Phys. Chemie 42, 269–286 (1891)
239. 239.
E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30 (Princeton University Press, Princeton, NJ, 1970)Google Scholar
240. 240.
P.R. Stinga, J.L. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Comm. Partial Differ. Equ. 35, 2092–2122 (2010)
241. 241.
P.R. Stinga, J.L. Torrea, Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation. arXiv:1511.01945Google Scholar
242. 242.
A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. B 237(641), 37–72 (1952)
243. 243.
E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. 49, 33–44 (2009)
244. 244.
S.R.S. Varadhan, Lectures on Diffusion Problems and Partial Differential Equations. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 64 (Tata Institute of Fundamental Research, Bombay, 1980)Google Scholar
245. 245.
N.T. Varopoulos, Random walks and Brownian motion on manifolds. Symposia Mathematica, vol. XXIX (Cortona, 1984), 97–109, Sympos. Math., vol. XXIX (Academic, New York, 1987)Google Scholar
246. 246.
J.L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type. J. Math. Pures Appl. (9) 71(6), 503–526 (1992)Google Scholar
247. 247.
J.L. Vázquez, Asymptotic behaviour for the Porous Medium Equation posed in the whole space. J. Evol. Equ. 3, 67–118 (2003)
248. 248.
J.L. Vázquez, Asymptotic behaviour for the PME in a bounded domain. The Dirichlet problem. Monatshefte für Math. 142(1–2), 81–111 (2004)
249. 249.
J.L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type. Oxford Lecture Series in Mathematics and Its Applications, vol. 33 (Oxford University Press, Oxford, 2006)Google Scholar
250. 250.
J.L. Vázquez, Perspectives in Nonlinear Diffusion: Between Analysis, Physics and Geometry. Proceedings of International Congress of Mathematicians. vol. I (European Mathematical Society, Zürich, 2007), pp. 609–634Google Scholar
251. 251.
J.L. Vázquez, The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs (The Clarendon Press/Oxford University Press, Oxford, 2007)Google Scholar
252. 252.
J.L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in Nonlinear Partial Differential Equations: the Abel Symposium 2010, ed. by H. Holden, K.H. Karlsen (Springer, Berlin, 2012), pp. 271–298
253. 253.
J.L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Nonlinear Elliptic and Parabolic Differential Equations. Discrete Contin. Dyn. Syst. S 7(4), 857–885 (2014)Google Scholar
254. 254.
J.L. Vázquez, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type. J. Eur. Math. Soc. 16(4), 769–803 (2014)
255. 255.
J.L. Vázquez, The mesa problem for the fractional porous medium equation. Interfaces Free Bound. 17(2), 261–286 (2015)
256. 256.
J.L. Vázquez, Fundamental solution and long time behaviour of the Porous Medium Equation in hyperbolic space. J. Math. Pures Appl. (9) 104(3), 454–484 (2015)Google Scholar
257. 257.
J.L. Vázquez, The Dirichlet Problem for the fractional p-Laplacian evolution equation. J. Differ. Equ. 260(7), 6038–6056 (2016)
258. 258.
J.L. Vázquez, Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D. J. Evol. Equ. 16, 723–758 (2016)
259. 259.
J.L. Vázquez, B. Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type. J. Math. Pures Appl. (9) 101(5), 553–582 (2014)Google Scholar
260. 260.
J.L. Vázquez, B. Volzone, Optimal estimates for fractional fast diffusion equations. J. Math. Pures Appl. (9) 103(2), 535–556 (2015)Google Scholar
261. 261.
J.L. Vázquez, A. de Pablo, F. Quirós, A. Rodríguez, Classical solutions and higher regularity for nonlinear fractional diffusion equations. J. Eur. Math. Soc. 19(7), 1949–1975 (2017)
262. 262.
L. Vlahos, H. Isliker, Y. Kominis, K. Hizonidis, Normal and anomalous diffusion: a tutorial, in Order and Chaos, ed. by T. Bountis, vol. 10 (Patras University Press, Patras, 2008)Google Scholar
263. 263.
E. Weinan, Dynamics of vortex-liquids in Ginzburg-Landau theories with applications to superconductivity. Phys. Rev. B 50(3), 1126–1135 (1994)Google Scholar
264. 264.
D.V. Widder, The Heat Equation (Academic, New York, 1975)
265. 265.
Wikipedia, article Diffusion, February (2017)Google Scholar
266. 266.
P. Wilmott, S. Howison, J. Dewynne, The Mathematics of Financial Derivatives. A Student Introduction (Cambridge University Press, Cambridge, 1995)Google Scholar
267. 267.
W.A. Woyczyński, Lévy processes in the physical sciences, in Lévy Processes – Theory and Applications, ed. by T. Mikosch, O. Barndorff-Nielsen, S. Resnick (Birkhäuser, Boston, 2001), pp. 241–266
268. 268.
S.T. Yau, On the heat kernel of a complete Riemannian manifold. J. Math. Pures Appl. (9) 57(2), 191–201 (1978)Google Scholar
269. 269.
Ya.B. Zel’dovich, Yu.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena II (Academic, New York, 1966)Google Scholar
270. 270.
X.H. Zhou, W.L. Xiao, J.C. Chen, Fractional porous medium and mean field equations in Besov spaces. Electron. J. Differ. Equ. 2014(199), 1–14 (2014)