Rate of decay to equilibrium in some semilinear parabolic equations

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.