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

TL;DR

This paper investigates the long-term decay of approximate solutions to a damped semilinear wave equation on a bounded 1D domain, using hyperbolic conservation law tools and probabilistic interpretations to establish exponential convergence to stationary solutions.

Contribution

It introduces a novel approach combining hyperbolic system techniques and probabilistic methods to analyze the decay of approximate solutions for the damped semilinear wave equation.

Findings

Exponential convergence of solutions to stationary states.

Uniform error estimates for approximate solutions.

Effective long-time behavior analysis under nonlinear damping.

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 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 $\Delta x=1/N\to 0$. 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.