J.evol.equ. 3 (2003) 463 – 484 1424–3199/03/030465 – 22 DOI 10.1007/s00028-003-1112-8 © Birkh¨ auser Verlag, Basel, 2003 Rate of decay to equilibrium in some semilinear parabolic equations Alain Haraux, Mohamed Ali Jendoubi and Otared Kavian Dedicated to Philippe B´ enilan Abstract. In this paper we prove, under various conditions, the so-called Lojasiewicz inequality ‖E ′ (u + ϕ)‖≥ γ |E(u + ϕ) − E(ϕ)| 1−θ , 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 C 2 , instead of analytic in the classical settings. Here E can be for instance the energy associated to the semilinear heat equation u t = u − f (x, u) on a bounded domain  ⊂ 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. 1. Introduction and main results Let  be a bounded, connected open subset of R N with a Lipschitz continuous boundary and let us consider the semilinear parabolic equation u t − u + f (u) = 0 in R + × , u = 0 on R + × ∂ (1.1) where f : R −→ R is a locally Lipschitz continuous function. According to the well known La Salle’s invariance principle, a solution u of (1.1) which is uniformly bounded on R + , approaches the set of stationary solutions of (1.1) as t →∞, see for instance C. M. Dafermos [9], A. Haraux [12]. Convergence to an equilibrium has been established in many cases, for instance T. I. Zelenyak [26], H. Matano [22], L. Simon [25], P. L. Lions [19], J. Hale & G. Raugel [11], A. Haraux & P. Pol´ aˇ cik [16], P. Brunovsk´ y & P. Pol´ aˇ cik [5], but remains an open question in general (convergence may fail if the nonlinearity f depends on x , as shown in P. Pol´ aˇ cik & K. Rybakowski [23], see also P. Pol´ aˇ cik & F. Simondon [24]). One of the means used in the proof of such convergence results to equilibrium is an inequality due to S. Lojasiewicz [20] asserting that for any analytic function F : R n −→ R with F(0) = 0 and F ′ (0) = 0, there exist constants γ > 0 and θ ∈ (0, 1/2] and a neighbourhood ω of the origin such that: for any x ∈ ω one has ‖F ′ (x)‖≥ γ |F(x)| 1−θ .