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Higher-order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach. (English) Zbl 1315.35004
Mem. Am. Math. Soc. 1101, v, 81 p. (2015).
Summary: This paper quantifies the speed of convergence and higher-order asymptotics of fast diffusion dynamics on \(\mathbb R^n\) to the Barenblatt (self similar) solution. Degeneracies in the parabolicity of this equation are cured by re-expressing the dynamics on a manifold with a cylindrical end, called the cigar. The nonlinear evolution becomes differentiable in Hölder spaces on the cigar. The linearization of the dynamics is given by the Laplace-Beltrami operator plus a transport term (which can be suppressed by introducing appropriate weights into the function space norm), plus a finite-depth potential well with a universal profile. In the limiting case of the (linear) heat equation, the depth diverges, the number of eigenstates increases without bound, and the continuous spectrum recedes to infinity. We provide a detailed study of the linear and nonlinear problems in Hölder spaces on the cigar, including a sharp boundedness estimate for the semigroup, and use this as a tool to obtain sharp convergence results toward the Barenblatt solution, and higher order asymptotics. In finer convergence results (after modding out symmetries of the problem), a subtle interplay between convergence rates and tail behavior is revealed. The difficulties involved in choosing the right functional spaces in which to carry out the analysis can be interpreted as genuine features of the equation rather than mere annoying technicalities.

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
76S05 Flows in porous media; filtration; seepage
35C06 Self-similar solutions to PDEs
58J35 Heat and other parabolic equation methods for PDEs on manifolds
Full Text: DOI arXiv
[1] Sigurd Angenent, Large time asymptotics for the porous media equation, Nonlinear diffusion equations and their equilibrium states, I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 12, Springer, New York, 1988, pp. 21-34.
[2] Sigurd Angenent, Local existence and regularity for a class of degenerate parabolic equations, Math. Ann. 280 (1988), no. 3, 465-482. · Zbl 0619.35114
[3] S. B. Angenent and D. G. Aronson, Optimal asymptotics for solutions to the initial value problem for the porous medium equation, Nonlinear problems in applied mathematics, SIAM, Philadelphia, PA, 1996, pp. 10-19. · Zbl 0886.35112
[4] G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 67-78 (Russian). · Zbl 0049.41902
[5] Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). · Zbl 0223.53034
[6] Adrien Blanchet, Matteo Bonforte, Jean Dolbeault, Gabriele Grillo, and Juan-Luis Vázquez, Hardy-Poincaré inequalities and applications to nonlinear diffusions, C. R. Math. Acad. Sci. Paris 344 (2007), no. 7, 431-436 (English, with English and French summaries). · Zbl 1190.35119
[7] Adrien Blanchet, Matteo Bonforte, Jean Dolbeault, Gabriele Grillo, and Juan Luis Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal. 191 (2009), no. 2, 347-385. · Zbl 1178.35214
[8] M. Bonforte, J. Dolbeault, G. Grillo, and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proc. Natl. Acad. Sci. USA 107 (2010), no. 38, 16459-16464. · Zbl 1256.35026
[9] Matteo Bonforte, Gabriele Grillo, and Juan Luis Vázquez, Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold, Arch. Ration. Mech. Anal. 196 (2010), no. 2, 631-680. · Zbl 1209.35069
[10] J. A. Carrillo, M. Di Francesco, and G. Toscani, Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering, Proc. Amer. Math. Soc. 135 (2007), no. 2, 353-363. · Zbl 1125.35053
[11] J. A. Carrillo, C. Lederman, P. A. Markowich, and G. Toscani, Poincaré inequalities for linearizations of very fast diffusion equations, Nonlinearity 15 (2002), no. 3, 565-580. · Zbl 1011.35025
[12] J. A. Carrillo and G. Toscani, Asymptotic \(L^1\)-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J. 49 (2000), no. 1, 113-142. · Zbl 0963.35098
[13] José A. Carrillo and Juan L. Vázquez, Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations 28 (2003), no. 5-6, 1023-1056. · Zbl 1036.35100
[14] Bennett Chow and Dan Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004. · Zbl 1086.53085
[15] Björn E. J. Dahlberg and Carlos E. Kenig, Nonnegative solutions to fast diffusions, Rev. Mat. Iberoamericana 4 (1988), no. 1, 11-29. · Zbl 0709.35054
[16] Panagiota Daskalopoulos and Natasa Sesum. Eternal solutions to the Ricci flow on \mathbb {R}^2. Int. Math. Res. Not., pages Art. ID 83610, 20, 2006. · Zbl 1204.53051
[17] Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. · Zbl 0559.47040
[18] Manuel Del Pino and Jean Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9) 81 (2002), no. 9, 847-875 (English, with English and French summaries). · Zbl 1112.35310
[19] Jochen Denzler and Robert J. McCann, Phase transitions and symmetry breaking in singular diffusion, Proc. Natl. Acad. Sci. USA 100 (2003), no. 12, 6922-6925 (electronic). · Zbl 1076.35055
[20] Jochen Denzler and Robert J. McCann, Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology, Arch. Ration. Mech. Anal. 175 (2005), no. 3, 301-342. · Zbl 1083.35074
[21] Jochen Denzler and Robert J. McCann, Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, & nonradial focusing geometries, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 5, 865-888 (English, with English and French summaries). · Zbl 1146.76053
[22] Jean Dolbeault and Giuseppe Toscani, Fast diffusion equations: matching large time asymptotics by relative entropy methods, Kinet. Relat. Models 4 (2011), no. 3, 701-716. · Zbl 1252.35065
[23] Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. · Zbl 0952.47036
[24] Mikhail V. Fedoryuk, Asymptotic analysis, Springer-Verlag, Berlin, 1993. Linear ordinary differential equations; Translated from the Russian by Andrew Rodick. · Zbl 0782.34001
[25] Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. · Zbl 0092.31002
[26] Avner Friedman and Shoshana Kamin, The asymptotic behavior of gas in an \(n\)-dimensional porous medium, Trans. Amer. Math. Soc. 262 (1980), no. 2, 551-563. · Zbl 0447.76076
[27] Th. Gallay, A center-stable manifold theorem for differential equations in Banach spaces, Comm. Math. Phys. 152 (1993), no. 2, 249-268. · Zbl 0776.34051
[28] Thierry Gallay and C. Eugene Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on \(\mathbf R^2\), Arch. Ration. Mech. Anal. 163 (2002), no. 3, 209-258. · Zbl 1042.37058
[29] Miguel A. Herrero and Michel Pierre, The Cauchy problem for \(u_t=\Delta u^m\) when \(0<m<1\), Trans. Amer. Math. Soc. 291 (1985), no. 1, 145-158. · Zbl 0583.35052
[30] Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. · Zbl 0342.47009
[31] Yong Jung Kim and Robert J. McCann, Potential theory and optimal convergence rates in fast nonlinear diffusion, J. Math. Pures Appl. (9) 86 (2006), no. 1, 42-67 (English, with English and French summaries). · Zbl 1112.35025
[32] Herbert Koch. Non-Euclidean Singular Integrals and the Porous Medium Equation. Habilitation Thesis, Unversität Heidelberg, Germany, 1999.
[33] N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996. · Zbl 0865.35001
[34] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural\(^{\prime}\)ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian).
[35] George L. Lamb Jr., Elements of soliton theory, John Wiley & Sons, Inc., New York, 1980. Pure and Applied Mathematics; A Wiley-Interscience Publication.
[36] Claudia Lederman and Peter A. Markowich, On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass, Comm. Partial Differential Equations 28 (2003), no. 1-2, 301-332. · Zbl 1024.35040
[37] Robert J. McCann and Dejan Slepčev, Second-order asymptotics for the fast-diffusion equation, Int. Math. Res. Not. , posted on (2006), Art. ID 24947, 22. · Zbl 1130.35080
[38] Felix Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001), no. 1-2, 101-174. · Zbl 0984.35089
[39] R. E. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407-409. · Zbl 0119.30505
[40] Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. · Zbl 0549.35002
[41] Thomas Runst and Winfried Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. · Zbl 0873.35001
[42] S. S. Titov and V. A. Ustinov, Investigation of polynomial solutions of the two-dimensional Leĭbenzon filtration equation with an integral exponent of the adiabatic curve, Approximate methods for solving boundary value problems of continuum mechanics (Russian), Akad. Nauk SSSR, Ural. Nauchn. Tsentr, Sverdlovsk, 1985, pp. 64-70, 91 (Russian).
[43] Juan Luis Vázquez, Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium, Trans. Amer. Math. Soc. 277 (1983), no. 2, 507-527. · Zbl 0528.76096
[44] Juan Luis Vázquez, Asymptotic beahviour for the porous medium equation posed in the whole space, J. Evol. Equ. 3 (2003), no. 1, 67-118. Dedicated to Philippe Bénilan. · Zbl 1036.35108
[45] Juan Luis Vázquez, The porous medium equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. Mathematical theory. · Zbl 1107.35003
[46] C. Eugene Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rational Mech. Anal. 138 (1997), no. 3, 279-306. · Zbl 0882.35061
[47] Thomas P. Witelski and Andrew J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math. 100 (1998), no. 2, 153-193. · Zbl 1001.35056
[48] Ya.B. Zel’dovich and G.I. Barenblatt. The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations. Sov. Phys. Doklady, 3:44-47, 1958.
[49] Ya.B. Zel’dovich and A.S. Kompaneets. Theory of heat transfer with temperature dependent thermal conductivity. In Collection in Honour of the 70th Birthday of Academician A.F. Ioffe, pages 61-71. Izdvo. Akad. Nauk. SSSR, Moscow, 1950.
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