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Decay of approximate solutions for the damped semilinear wave equation on a bounded 1d domain. (English. French summary) Zbl 1437.35062

Summary: 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 \(\Delta x=1/N\rightarrow 0\). The proof makes use of the Birkhoff decomposition of doubly stochastic matrices and of accurate estimates on the iteration system as \(N\rightarrow\infty\).
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 \(t\rightarrow+\infty\), as well as uniform error estimates for the approximate solutions.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L50 Initial-boundary value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
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