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Self-Similar Solutions of Nonlinear PDE

DOI: 10.4064/bc74-0-8

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Large time behaviour of solutions to nonhomogeneous diffusion equations

Journal article published in 2006 by Jean Dolbeault ORCID, Grzegorz Karch
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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Abstract

This note is devoted to the study of the long time behaviour of the solutions to the heat and the porous medium equations in the presence of an external source term, using entropy methods and self-similar variables. Intermediate asymptotics and convergence results are shown using interpolation inequalities, Gagliardo-Nirenberg-Sobolev inequalities and Csiszar-Kullback type estimates. 1. Introduction. In this note, we study the large time behavior in L1(RN) of solutions to the Cauchy problem for the porous media equation (m > 1) and the heat equation (m = 1) in the presence of an external source term: vt = �vm + G(x,t) x ∈ RN, t > 0 , (1) v(x,0) = v0(x) . (2)