Potential theory and optimal convergence rates in fast nonlinear diffusion

Angenent, 1988, Large time asymptotics for the porous medium equation, vol. 12, 21 Aronson, 1979, Regularité des solutions de l'équation des milieux poreux dans RN, C. R. Acad. Sci. Paris Ser. A–B, 288, A103 Aronson, 1987, Eventual C∞-regularity and concavity of flows in one-dimensional porous media, Arch. Rational Mech. Anal., 99, 329, 10.1007/BF00282050 Bakry, 1985, Diffusions hypercontractives, vol. 1123, 177 Barenblatt, 1952, On some unsteady motions of a liquid or gas in a porous medium, Akad. Nauk SSSR Prikl. Mat. Mekh., 16, 67 J.A. Carrillo, M. DiFrancesco, G. Toscani, Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering, Proc. Amer. Math. Soc., submitted for publication Carrillo, 2001, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133, 1, 10.1007/s006050170032 Carrillo, 2002, Poincaré inequalities for linearizations of very fast diffusion equations, Nonlinearity, 15, 565, 10.1088/0951-7715/15/3/303 J.A. Carrillo, R.J. McCann, C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Rational Mech. Anal., submitted for publication Carrillo, 2000, Asymptotic L1-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49, 113, 10.1512/iumj.2000.49.1756 Carrillo, 2003, Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations, 28, 1023, 10.1081/PDE-120021185 Chasseigne, 2002, Theory of extended solutions for fast diffusion equations in optimal classes of data. Radiation from singularities, Arch. Rational Mech. Anal., 164, 133, 10.1007/s00205-002-0210-0 Chayes, 1993, On singular diffusion equations with applications to self-organized criticality, Comm. Pure Appl. Math., 46, 1363, 10.1002/cpa.3160461004 Dahlberg, 1984, Nonnegative solutions of the porous medium equation, Comm. Partial Differential Equations, 9, 409, 10.1080/03605308408820336 Dahlberg, 1986, Nonnegative solutions of generalized porous medium equations, Rev. Mat. Iberoamericana, 2, 267, 10.4171/RMI/34 Dahlberg, 1988, Nonnegative solutions of fast diffusion equations, Rev. Mat. Iberoamericana, 4, 11, 10.4171/RMI/61 Dahlberg, 1988, Nonnegative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders, J. Amer. Math. Soc., 1, 401, 10.1090/S0894-0347-1988-0928264-9 Dahlberg, 1993, Weak solutions of the porous medium equation in a cylinder, Trans. Amer. Math. Soc., 336, 701, 10.1090/S0002-9947-1993-1085940-6 Dahlberg, 1993, Weak solutions of the porous medium equation, Trans. Amer. Math. Soc., 336, 711, 10.1090/S0002-9947-1993-1085939-X Daskalopoulos, 1997, The Cauchy problem for variable coefficient porous medium equations, Potential Anal., 7, 485, 10.1023/A:1017967218024 Daskalopoulos, 1999, On the Cauchy problem for ut=Δlogu in higher dimensions, Math. Ann., 131, 189, 10.1007/s002080050257 Daskalopoulos, 2002, Nonradial solvability structure of super-diffusive nonlinear parabolic equations, Trans. Amer. Math. Soc., 354, 1583, 10.1090/S0002-9947-01-02888-4 Daskalopoulos, 2001, All time C∞-regularity of interface in degenerated diffusion: a geometrical approach, Duke Math. J., 108, 295, 10.1215/S0012-7094-01-10824-7 Denzler, 2003, Phase transitions and symmetry breaking in singular diffusion, Proc. Natl. Acad. Sci. USA, 100, 6922, 10.1073/pnas.1231896100 Denzler, 2005, Fast diffusion to self-similarity: complete spectrum, long time asymptotics, and numerology, Arch. Rational Mech. Anal., 175, 301, 10.1007/s00205-004-0336-3 Dolbeault, 2002, Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl., 81, 847, 10.1016/S0021-7824(02)01266-7 Dolbeault, 2005, L1 and L∞ intermediate asymptotics for scalar conservation laws, Asymptot. Anal., 41, 189 Esteban, 1988, A nonlinear heat equation with singular diffusivity, Comm. Partial Differential Equations, 13, 985, 10.1080/03605308808820566 Friedman, 1964 Friedman, 1980, The asymptotic behaviour of a gas in an n-dimensional porous medium, Trans. Amer. Math. Soc., 262, 551 Galaktionov, 1997, Asymptotic behavior near finite time extinction for the fast diffusion equation, Arch. Rational Mech. Anal., 139, 83, 10.1007/s002050050048 Galaktionov, 2000, Asymptotics of the fast-diffusion equation with critical exponent, SIAM J. Math. Anal., 31, 1157, 10.1137/S0036141097328452 Herrero, 1985, The Cauchy problem for ut=Δum when 0<m<1, Trans. Amer. Math. Soc., 291, 145 Kamin (Kamenomostskaya), 1973, The asymptotic behavior of the solution of the filtration equation, Israel J. Math., 14, 76, 10.1007/BF02761536 Kamin, 1988, Fundamental solutions and asymptotic behavior for the p-Laplacian equation, Rev. Mat. Iberoamericana, 4, 339, 10.4171/RMI/77 Kamin, 1991, Asymptotic behavior of the solutions of the porous medium equation with changing signs, SIAM J. Math. Anal., 22, 34, 10.1137/0522003 Kim, 2003, Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to N-waves, J. Differential Equations, 192, 202, 10.1016/S0022-0396(03)00058-5 Kim, 2005, Sharp decay rates for the fastest conservative diffusions, C. R. Acad. Sci. Paris, 341, 157, 10.1016/j.crma.2005.06.025 Kim, 2002, On the rate of convergence and asymptotic profile of solutions to the viscous Burgers equation, Indiana Univ. Math. J., 51, 727, 10.1512/iumj.2002.51.2247 King, 1993, Self-similar behaviour for the equation of fast nonlinear diffusion, Phil. Trans. Roy. Soc. London A, 343, 337, 10.1098/rsta.1993.0052 H. Koch, Non-Euclidean singular integrals and the porous medium equation, Habilitation thesis, Universität Heidelberg, 1999 Ladyženskaja, 1967, Linear and Quasilinear Equations of Parabolic Type, vol. 23 Lederman, 2001, On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass, Comm. Partial Differential Equations, 28, 301, 10.1081/PDE-120019384 Lee, 2003, Geometrical properties of solutions of the porous medium equation for large times, Indiana Univ. Math. J., 52, 991, 10.1512/iumj.2003.52.2200 Lieb, 1997, Analysis, vol. 14 R.J. McCann, D. Slepčev, Second-order asymptotics for the fast-diffusion equation, Int. Math. Res. Not., in press Newman, 1984, A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. I, J. Math. Phys., 25, 3120, 10.1063/1.526028 Otto, 2001, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26, 101, 10.1081/PDE-100002243 Otto, 2000, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173, 361, 10.1006/jfan.1999.3557 Pattle, 1959, Diffusion from an instantaneous point source with concentration dependent coefficient, Quart. J. Mech. Appl. Math., 12, 407, 10.1093/qjmam/12.4.407 Pierre, 1982, Uniqueness of the solutions of ut−Δφ(u)=0 with initial datum a measure, Nonlinear Anal., 6, 175, 10.1016/0362-546X(82)90086-4 Pierre, 1987, Nonlinear fast diffusion with measures as data, vol. 149, 179 del Pino, 2001, On the extinction profile for solutions of ut=Δu(N−2)/(N+2), Indiana Univ. Math. J., 50, 612, 10.1512/iumj.2001.50.1876 Ralston, 1984, A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II, J. Math. Phys., 25, 3124, 10.1063/1.526029 Vázquez, 1982, Symétrisation pour ut=Δϕ(u) et applications, C. R. Acad. Sci. Paris Sér. I Math., 295, 71 Vázquez, 1983, Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium, Trans. Amer. Math. Soc., 277, 507, 10.2307/1999221 Vázquez, 1992, An introduction to the mathematical theory of the Porous Medium Equation, vol. 380, 347 Vázquez, 2003, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equations, 3, 67, 10.1007/s000280300004 Vázquez, 2002, Complexity of large time behaviour of evolution equations with bounded data, Chinese Ann. Math. Ser. B, 23, 293, 10.1142/S0252959902000274 Witelski, 1998, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math., 100, 153, 10.1111/1467-9590.00074 Zel'dovich, 1958, The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations, Sov. Phys. Dokl., 3, 44