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

被引:6
作者
Amadori, Debora [1 ]
Aqel, Fatima Al-Zahra' [1 ]
Dal Santo, Edda [1 ]
机构
[1] Univ Laquila, DISIM Dept, Laquila, Italy
来源
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES | 2019年 / 132卷
关键词
Space-dependent relaxation model; L-infinity error estimate; Damped wave equation; Initial boundary value problem in one dimension;
D O I
10.1016/j.matpur.2019.05.010
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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 -> 0. The proof makes use of the Birkhoff decomposition of doubly stochastic matrices and of accurate estimates on the iteration system as N -> infinity. Under appropriate assumptions on the nonlinearity, we prove the exponential convergence in L-infinity of the solution to the first order system towards a stationary solution, as t -> +infinity, as well as uniform error estimates for the approximate solutions. (C) 2019 Elsevier Masson SAS. All rights reserved.
引用
收藏
页码:166 / 206
页数:41
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