Abstract
In this paper, we consider the complexity of large time behavior of solutions to the porous medium equation u t − Δu m = 0 in \({\mathbb{R}^N}\) with m > 1. We first show that for any given \({0<\mu<\frac{2N}{N(m-1)+2}}\) and \({\beta>\frac{2-\mu(m-1)}{4}}\), the ω-limit set of \({t^{\frac{\mu}{2}}u(t^{\beta}\cdot,t)}\) includes all of the nonnegative functions f in the Schwartz space \({\fancyscript{S}(\mathbb{R}^N)}\) with f(0) = 0. Furthermore, we prove that, for a given countable subset E of the interval \({\left(0,\frac{2N}{(N(m-1)+2)(2+\mu(m-1))}\right)}\), there exists an initial value u 0(x) such that for all μ and β satisfying \({0<\mu<\frac{2N}{N(m-1)+2}, \beta></\frac{2N}{N(m-1)+2},>\frac{2-\mu(m-1)}{4}}\) and \({\frac{\mu}{2\beta}\in E}\), the ω-limit set of \({t^{\frac{\mu}{2}}u(t^{\beta}\cdot,t)}\) is equal to \({C_{0}^{+}(\mathbb{R}^N)\equiv\{f\in C_{0}(\mathbb{R}^N);\ f(x)\geq 0,\ f(0)=0\}}\).
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References
Bénilan P., Crandall M.G., Pierre M.: Solutions of the porous medium in RN under optimal conditions on the initial-values. Indiana Univ. Math. J. 33, 51–87 (1984)
Carrillo J.A., Toscani G.: Asymptotic L 1−decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49, 113–141 (2000)
Cazenave T., Dickstein F., Weissler F.B.: Universal solutions of the heat equation on \({\mathbb{R}^N}\). Discrete Contin. Dyn. Sys 9, 1105–1132 (2003)
Cazenave T., Dickstein F., Weissler F.B.: Nonparabolic asymptotic limits of solutions of the heat equation on \({\mathbb{R}^N}\). J. Dyn. Differ. Equations 19, 789–818 (2007)
Cazenave T., Dickstein F., Escobedo M., Weissler F.B.: Self-similar solutions of a nonlinear heat equation. J. Math. Sci. Univ. Tokyo 8, 501–540 (2001)
Di Benedetto E.: Degenerate parabolic equations. Springer-Verlag, New York (1993)
Friedman A., Kamin S.: The asymptotic behavior of gas in an N-dimensional porous medium. Trans. Amer. Math. Soc. 262, 551–563 (1980)
Kamenomostskaya (Kamin) S.: The asymptotic behavior of the solution of the filtration equation. Israel J. Math. 14, 76–87 (1973)
Quirós F., Vázquez J.L.: Asymptotic behaviour of the porous media equation in an exterior domain. Ann. Scuola Norm. Superiore Pisa Cl. Scienze 4(4), 183–227 (1999)
Toscani G.: Entropy dissipation and the rate of convergence to equilibrium for the Fokker-Planck equation. Quart. Appl. Math. LVII, 521–541 (1999)
Vázquez J.L.: Asymptotic behavior for the porous medium equation in the whole space. J. Evolution Equations 3, 67–118 (2003)
Vázquez J. L., Smoothing and decay estimates for nonlinear parabolic equations, Equations of porous medium type, Oxford University Press, (2006).
Vázquez J.L.: The Dirichlet problem for the porous medium equation in bounded domains. Asymptotic behavior, Monatsh Math 142, 81–111 (2004)
Vázquez J.L., Zuazua E.: Complexity of large time behaviour of evolution equations with bounded data. Chin. Ann. Math. Ser. B 23, 293–310 (2002)
Vázquez J.L.: The Porous Medium Equation, Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, Oxford/New York (2007)
Zhuoqun Wu., Jingxue Yin., Huilai Li and Junning Zhao., Nonlinear Diffusion Equations World Scientific, Singapore (2001).
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Yin, J., Wang, L. & Huang, R. Complexity of asymptotic behavior of the porous medium equation in \({\mathbb{R}^N}\) . J. Evol. Equ. 11, 429–455 (2011). https://doi.org/10.1007/s00028-010-0097-4
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DOI: https://doi.org/10.1007/s00028-010-0097-4