An application of $L^m-L^r$ estimates to weakly coupled systems of semilinear viscoelastic wave equations

TL;DR

This paper develops $L^m-L^r$ estimates for linear viscoelastic wave equations and uses them to establish global existence results for weakly coupled nonlinear systems with initial data in various $L^r$ spaces.

Contribution

It introduces new $L^m-L^r$ estimates for linear viscoelastic wave equations and applies these to prove global solutions for coupled nonlinear systems with diverse initial data regularities.

Findings

Established $L^m-L^r$ estimates for linear viscoelastic wave equations.

Proved global existence of solutions for coupled systems with initial data in different $L^r$ spaces.

Extended analysis to systems with different nonlinear power sources.

Abstract

We consider weakly coupled systems of semilinear viscoelastic wave equations with different power source nonlinearities in $\mathbb{R}^n$, $n\geq1$ as follows: \begin{equation*} \left\{\begin{aligned} &u_{tt}-\Delta u+g\ast\Delta u+u_t=|\partial_t^{\ell}v|^p,\\ &v_{tt}-\Delta v+g\ast\Delta v+v_t=|\partial_t^{\ell}u|^q,\\ \end{aligned}\right. \end{equation*} with $\ell=0,1$ and $p,q>1$. After presenting $L^m-L^r$ estimates with $1\leq m\leq r\leq \infty$ of solutions to the corresponding linearized problem with vanishing right-hand side, we prove the existence of global in time solutions to the weakly coupled systems, where the initial data are supposed to belong to different $L^r$ spaces with different additional $L^m$ regularities.