Global existence of solutions for a system of nonlinear damped wave equations

Abstract

We consider the Cauchy problem of the semilinear damped wave system: \begin{equation} \notag \begin{cases} \partial_{t}^2 u_{j} - \Delta u_{j} + \partial_{t} u_{j} = F_{j}(u), & t > 0, \quad x\in \mathbb R^{n},\\ u_{j}(0,x)=a_{j}(x),\quad \partial_{t} u_{j}(0,x) = b_{j}(x), & x\in \mathbb R^{n}, \end{cases} \end{equation} where $m \ge 2$ and $j = 1$, $\cdots$, $m$. We show the existence of a global-in-time solution for a small initial data under a sharp condition on the nonlinear exponents, which is a natural extension of the results for the single nonlinear damped wave equations ([28], [30]). The proof is based on $L^{p}$-$L^{q}$ type estimates of the fundamental solutions of the linear damped wave equations ([9]) and systematic choice of the function scale to adjust the nonlinear growth order.