Rate of decay to equilibrium in some semilinear parabolic equations. (Q1412091)

The classical assertion due to \textit{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.