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

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}-Δu+g\astΔu+u_t=|\partial_t^{\ell}v|^p,\\ &v_{tt}-Δv+g\astΔv+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.

We think this topic is not good enough. The first author will develop another new topic by himself only, which is related to this topic. In order to guarantee the unique of the paper, we want to remove it