2018/07/05 by Debora Amadori, Amadori, Debora, Fatima Aqel +3
Earth and Planetary Sciences · Economics, Econometrics and Finance · Engineering · #35B40 #35L20 #35L50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Meteorological Phenomena and Simulations #Stability and Controllability of Differential Equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1807.01968
openalex publication_date 2018/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the long time behavior for a semilinear wave equation\nwith space-dependent and nonlinear damping term. After rewriting the equation\nas a first order system, we define a class of approximate solutions that employ\ntipical tools of hyperbolic systems of conservation laws, such as the Riemann\nproblem. By recasting the problem as a discrete-time nonhomogeneous system,\nwhich is related to a probabilistic interpretation of the solution, we provide\na strategy to study its long-time behavior uniformly with respect to the mesh\nsize parameter \Δ x=1/N\→ 0. The proof makes use of the Birkhoff\ndecomposition of doubly stochastic matrices and of accurate estimates on the\niteration system as N\→\∞.\n Under appropriate assumptions on the nonlinearity, we prove the exponential\nconvergence in L^\∞ of the solution to the first order system towards a\nstationary solution, as t\→+\∞, as well as uniform error estimates for\nthe approximate solutions.\n