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

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 rst order system, we de ne a class of approximate solutions that employ tipical 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 ∆x = 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 rst order system towards a stationary solution, as t → +∞, as well as uniform error estimates for the approximate solutions.