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

Abstract

In this paper we study the long time behavior for a semilinear wave equation with spacedependent 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 discretetime 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 first order system towards a stationary solution, as t → +∞, as well as uniform error estimates for the approximate solutions.

Stability analysis of a 1D wave equation with a nonmonotone distributed damping

“…Moreover, an optimal decay rate is obtained for this equation. Recently, in Amadori et al (2019), a similar result have been obtained, using techniques coming from conservation laws theory. Note that both of these results hold true only for monotone nonlinear damping, which is not the case of our paper.…”

L-asymptotic stability analysis of a 1D wave equation with a nonlinear damping

“…In [4], the authors provide an analysis of the asymptotic behavior of trajectories of (1) assuming that the damping function a is bounded away from zero and the nonlinearity σ belongs to a positive sector, i.e.,…”

“…The techniques used in this paper are based on the theory of scalar conservation laws, which makes indeed the analysis in those spaces really natural. In constrast with these papers, our results apply for more general nonlinearities, with, in some cases, less regular initial conditions, and we even treat the case where the damping action is localized, i.e., it acts on a subdomain of [0, 1], which does not hold true for [12], neither for [4]. Note that, assuming that the initial conditions are in H ∞ , we provide some exponential stability results with decay rate estimate in the case where σ is not monotone, which is a non trivial task in general.…”

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

“…The damped wave equation and its time-asymptotic stability properties have been studied in several papers, see for instance [14] and references therein, in terms of the decay of energy (L 2 norm of the derivatives of u). The L p framework, with p ∈ [2, ∞] was considered in [9,1,6].…”

“…-second, we address the time-asymptotic stability of the solution ρ = 0 = J; by following the approach introduced in [1], we obtain a result on the exponential decay of the L ∞ -norm of the solution to (1.1), under the assumption that the damping term is linear and time-independent; see Theorem 1.2. In this specific context, this result extends the main result obtained in [1], where BV (Bounded Variation) initial data were assumed; since the constant values in the time-asymptotic estimate were depending on the total variation of the solution, a density argument was not sufficient to extend the result to the class of L ∞ initial data.…”