Asymptotics for some nonlinear damped wave equation: finite time convergence versus exponential decay results

Abstract

Given a bounded open set Omega subset of R-n and a continuous convex function Phi: L-2(Omega) -> R, let us consider the following damped wave equation u(tt) - Delta u + partial derivative Phi(u(t)) 0, (t, x) is an element of (0, +infinity) x Omega, (S) under Dirichlet boundary conditions. The notation partial derivative Phi refers to the subdifferential of Phi in the sense of convex analysis. The nonlinear term partial derivative Phi allows to modelize a large variety of friction problems. Among them, the case Phi = vertical bar.vertical bar L-1 corresponds to a Coulomb friction, equal to the opposite of the velocity sign. After we have proved the existence and uniqueness of a solution to (S), our main purpose is to study the asymptotic properties of the dynamical system (S). In two significant situations, we bring to light an interesting phenomenon of dichotomy: either the solution converges in a finite time or the speed of convergence is exponential as t -> +infinity. We also give conditions which ensure the finite time stabilization of (S) toward some stationary solution.