Decay in W^{1,\infty} for the 1D semilinear damped wave equation on a bounded domain

In this Ph.D. thesis, we study a semilinear wave equation with nonlinear and time-dependent 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. We prove that the initial-boundary value problem is well-posed for initial data in $L^\infty$ space. 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 $\DX=1/N\to 0$. 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.