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

摘要

Given a bounded open set ΩC Rn and a continuous convex function Φ:L2(Ω)→R, let us consider the following damped wave equation utt- △u+aΦ(ut)∈0, (t,x)∈(0,+∞)×Ω, under Dirichlet boundary conditions. The notation ?Φ refers to the subdifferential of Φ in the sense of convex analysis. The nonlinear term ?Φ allows to modelize a large variety of friction problems. Among them, the case Φ=||L1 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→+∞. We also give conditions which ensure the finite time stabilization of (S) toward some stationary solution.