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 first order system, we define 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 $Δx=1/N\to 0$. The proof makes use of the Birkhoff decomposition of doubly stochastic matrices and of accurate estimates on the iteration system as $N\to\infty$. Under appropriate assumptions on the nonlinearity, we prove the exponential convergence in $L^\infty$ of the solution to the first order system towards a stationary solution, as $t\to+\infty$, as well as uniform error estimates for the approximate solutions.