COMPLEXITY OF ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE POROUS MEDIUM EQUATION WITH ABSORPTION

doi:10.1016/S0252-9602(10)60179-8

Abstract

Abstract:

In this paper we analyze the large time behavior of nonnegative solutions of the Cauchy problem of the porous medium equation with
absorption u_t-Δu^m+γu^p=0, where γ≥0, m>1and p>m+2/N. We will show that if γ=0 and 0<μ<2N/N(m-1)+2, or γ>0 and 1/p-1<μ<2N/N(m-1)+2, then for any nonnegative function φ in a nonnegative countable subset F of the Schwartz space S(R^N), there exists an initial-value u₀∈C(R^N) with lim_x→∞u₀(x)=0 such that φ is an ω-limit point of the rescaled solutions t ^u/²u(t ^β, t), where β=2-u(m-1)/4.

Key words: complexity, asymptotic behavior, porous medium equation

CLC Number:

35B40

YIN Jing-Xue, WANG Liang-Wei, HUANG Rui. COMPLEXITY OF ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE POROUS MEDIUM EQUATION WITH ABSORPTION[J].Acta mathematica scientia,Series B, 2010, 30(6): 1865-1880.

Trendmd

References

[1] Bènilan P, Crandall M G, Pierre M. Solutions of the porous medium in R^N under optimal conditions on the initial-values. Indiana Univ Math J, 1984, 33: 51--87

[2] Bertsch M, Kersner R, Peletier L A. Positivity versus localization in degenerate diffusion equations. Nonlinear Anal TMA, 1985, 9: 987--1008

[3] Carrillo J A, Toscani G. Asymptotic L¹-decay of solutions of the porous medium equation to self-similarity. Indiana Univ Math J, 2000, 49: 113--141

[4] Carrillo J A, V\'azquez J L. Asymptotic complexity in filtration equations. J Evolution Equations, 2007, 7: 471--495

[5] Cazenave T, Dickstein F, Weissler F B. Universal solutions of the heat equation on R^N. Discrete Contin Dyn Sys, 2003, 9: 1105--1132

[6] Cazenave T, Dickstein F, Weissler F B. Nonparabolic asymptotic limits of solutions of the heat equation on RN. J Dyn Differ Equations, 2007, 19: 789--818

[7] Cazenave T, Dickstein F, Weissler F B. Chaotic behavior of solutions of the Navier-Stokes system in R^N. Adv Differ Equations, 2005, 10: 361--398

[8] Cazenave T, Dickstein F, Weissler F B. A solution of the constant coefficient heat equation on R with exceptional asymptotic behavior: an explicit constuction. J Math Pures Appl, 2006, 85(1): 119--150

[9] Cazenave T, Dickstein F, Weissler F B. Multiscale asymptotic behavior of a solution of the heat equation in R^N//Nonlinear Differential Equations: A Tribute to D.~G.~de Figueiredo, Progress in Nonlinear Differential Equations and their Applications, Vol 66. Basel: Birkh\"{a}user Verlag, 2005: 185--194

[10] Cazenave T, Dickstein F, Weissler F B. Universal solutions of a nonlinear heat equation on R^N. Ann Scuola Norm Sup Pisa Cl Sci, 2003, f 5: 77--117

[11] DiBenedetto E. Continuity of weak solutions to a general porous media equation. Indiana Univ Math J, 1983, 32: 83--118

[12]  DiBenedetto E. Degenerate parabolic equations. New York: Springer-Verlag, 1993

[13] Herraiz L. Asymptotic behaviour of solutions of some semilinear parabolic problems. Ann Inst H Poincar\'e Anal Non Linéaire, 1999, 16: 49--105

[14] Kamenomostskaya (Kamin) S. The asymptotic behavior of the solution of the filtration equation. Israel J Math, 1973, 14: 76--87

[15] Kamin S, V\'azquez J L. Fundamental solutions and asymptotic behaviour for the p-Laplacian equation. Rev Mat Iberoamericana, 1988, 4: 339--354

[16] Kamin S, Peletier L A. Large time behaviour of solutions of the porous medium equation with absorption. Israel J Math, 1986, 55: 129--146

[17] Lee Ki, Petrosyan A, Vazquez J L. Large-time geometric properties of solutions of the evolution p-Laplacian equation. J Differ Equs, 2006, 229: 389--411

[18] V\'azquez J L. Asymptotic behavior for the porous medium equation in the whole space. J Evolution Equations, 2003, 3: 67--118

[19] V\ázquez J L. Smoothing and Decay Estimates for Nonlinear Parabolic Equations, Equations of Porous Medium Type. Oxford: Oxford University Press, 2006

[20] Vázquez J L, Zuazua E. Complexity of large time behaviour of evolution equations with bounded data. Chin Ann Math Ser B, 2002, 23:  293--310

[21] Vázquez J L. The Porous Medium Equation, Mathematical Theory, Oxford Mathematical Monographs. Oxford, New York: The Clarendon
Press/Oxford University Press, 2007

[22] Wu Z Q, Yin J X, Li H L, Zhao J N. Nonlinear Diffusion Equations. Singapore: World Scientific, 2001

[23] Yin J X, Wang L W, Huang R. Complexity of asymptotic behavior of the porous medium equation in R^N. Preprint, 2009

[24] Zhao J N, Yuan H J. Lipschitz continuity of solutions and interfaces of the evolution p-Laplacian equation. Northeast Math J, 1992, 8(1):   21--37