Decay of approximate solutions for the damped semilinear wave equation on a bounded 1d domain

Abstract

In this paper we study the long time behavior for a semilinear wave equation with space-dependent and nonlinear damping term.After rewriting the equation as a first order system, we define a class of approximate solutions employing typical tools of hyperbolic systems of conservation laws, such as the Riemann problem. By recasting the problem as a discrete-time nonhomogeneous system, which is related to a probabilistic interpretation of the solution, we provide a strategy to study its long-time behavior uniformly with respect to the mesh size parameter Δx=1/N→0. The proof makes use of the Birkhoff decomposition of doubly stochastic matrices and of accurate estimates on the iteration system as N→∞.Under appropriate assumptions on the nonlinearity, we prove the exponential convergence in L∞ of the solution to the first order system towards a stationary solution, as t→+∞, as well as uniform error estimates for the approximate solutions.