Convergence to equilibrium for semilinear parabolic problems in ℝn

J. Busca, M. A. Jendoubi, P. Poláčik

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31 Scopus citations

Abstract

We consider the semilinear parabolic equation ut = Δu +f(u) on ℝN, assuming that f is an arbitrary C1 function satisfying f(0) = 0 and f′(0) < 0. We prove that any bounded positive solution that decays to zero at spatial infinity, uniformly with respect to t, converges to a (single) stationary solution as t → ∞. Our proof combines energy and comparison techniques with dynamical system arguments. We first establish an asymptotic symmetrization result: as t → ∞, u(x, t) approaches a set of steady states that are radially symmetric about a common origin in ℝN. To this aim we introduce a new toot that we call first moments of energy. Having established the symmetrization, we apply a general convergence result for gradient-like dynamical systems. This amounts to showing that the dimension of the kernel of the linearized operator around an equilibrium w matches the dimension of a manifold of equilibria passing through w.

Original languageEnglish (US)
Pages (from-to)1793-1814
Number of pages22
JournalCommunications in Partial Differential Equations
Volume27
Issue number9-10
DOIs
StatePublished - 2002

Bibliographical note

Funding Information:
*Supported in part by VEGA Grant 1/7677/20.

Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

Keywords

  • Asymptotic behavior
  • Asymptotic symmetry
  • Cauchy problem
  • Convergence
  • Parabolic equations

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