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Rate of decay to equilibrium in some semilinear parabolic equations. (English) Zbl 1036.35035
The classical assertion due to S. Lojasiewicz [I.H.E.S. notes 1965] claims, that for any analytic function $$F: \mathbb R^n \to \mathbb R$$ with $$F(0) = 0$$, $$F'(0) = 0$$ there exist constants $$\gamma > 0, \; \theta \in (0, 1{/}2]$$ such that $\| F'(x) \| \geq \gamma | F(x)| ^{1- \theta} \tag{1}$ in some neighbourhood of the origin. In the paper under review an analogon of (1) is proved for the energy functional $$E$$ connected with an operator $$A$$, assuming only that $$E$$ is of class $$C^2$$. Here $$A$$ is a self-adjoint operator acting on $$L^2(\Omega)$$ ($$\Omega$$ a domain in $$\mathbb R^N$$), having a compact resolvent. This result is a tool to estimate in the sequel the rate of convergence to an equilibrium for the global solution $$u(t)$$ of the equation $u_t + A u + f(\cdot, u) = 0$ when $$t \to \infty$$. In particular $$A = - \Delta$$ or $$A = -u_{xx}$$ in the one-dimensional case.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K90 Abstract parabolic equations 35B45 A priori estimates in context of PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
##### Keywords:
Lojasiewicz inequality
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