Rate of decay to equilibrium in some semilinear parabolic equations

Alain Haraux (), Mohamed Ali Jendoubi and Otared Kavian
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Alain Haraux: Université P. & M. Curie, Laboratoire. J. L. Lions
Mohamed Ali Jendoubi: Université de Versailles, Laboratoire de Mathématiques Appliquées UMR 7641
Otared Kavian: Université de Versailles, Laboratoire de Mathématiques Appliquées UMR 7641

A chapter in Nonlinear Evolution Equations and Related Topics, 2003, pp 463-484 from Springer

Abstract: Abstract In this paper we prove, under various conditions, the so-called Lojasiewicz inequality $$ \left\| {E'(u + \varphi )} \right\| \geqslant \gamma {\left| {E(u + \varphi ) - E(\varphi )} \right|^{{1 - \theta }}} $$ , where θ ∈ (0, 1/2], and γ > 0, while ‖u‖ is sufficiently small and ϕ is a critical point of the energy functional E supposed to be only C2 instead of analytic in the classical settings. Here E can be for instance the energy associated to the semilinear heat equation u t = Δu-f(x,u) on a bounded domain Ω ⊂ ℝ N . As a corollary of this inequality we give the rate of convergence of the solution u(t) to an equilibrium, and we exhibit examples showing that the given rate of convergence (which depends on the exponent θ and on the critical point ϕ through the nature of the kernel of the linear operator E″(ϕ)) is optimal.

Keywords: Unique Continuation; Caratheodory Function; Unique Continuation Principle; Lojasiewicz Inequality; Semi Linear Parabolic Equation (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-7924-8_25

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DOI: 10.1007/978-3-0348-7924-8_25

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