Global existence and uniform decay of solutions for a coupled system of nonlinear viscoelastic wave equations with not necessarily differentiable relaxation functions
The authors deal with the initial-boundary value problem for the coupled system of nonlinear viscoelastic wave equations NEWLINE\[NEWLINE\begin{aligned} &|u_t|^\rho u_{tt}-\Delta u-\Delta u_{tt}+\int_0^t g(t-\tau)\Delta u(x,\tau)d\tau -\Delta u_t=f(u,v),\;(x,t)\in \Omega\times (0,\infty),\\ &|v_t|^\rho v_{tt}-\Delta v-\Delta v_{tt}+\int_0^t g(t-\tau)\Delta v(x,\tau)d\tau -\Delta v_t=k(u,v),\;(x,t)\in \Omega\times (0,\infty),\\ &u(x,t)=v(x,t)=0,\;(x,t)\in \partial \Omega\times (0,\infty),\\ &u(x,0)=u_0(x),\;u_t(x,0)=u_1(x),\;x\in \Omega,\end{aligned} NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in \(\mathbb{R}^n,\;n\geq 1;\) with a smooth boundary,\ \(\rho\) is a real number such that \(0<\rho<2/(n-2)\) if \(n\geq 3\) or \(\rho > 0\) if \(n=1,2.\) The purpose of the paper is to establish the global existence and exponential decay of solutions under weaker conditions on the relaxation functions that are not necessarily differentiable.
scientific article; zbMATH DE number 6047242
scientific article