Global existence of solutions for a system of nonlinear damped wave equations. (Q636912)

The initial value problem to the semilinear damped wave system \(\partial ^2_{t}u_{j}- \Delta u_{j}+\partial _{t}u_{j}=F_{j}(u)\), \(j=1,2,\dots ,m\), \(t>0\), \(x\in \mathbb {R}^{n}\), where \(| F_{j}(u)| \leq C\prod _{k=1}^{m}| u_{k}| ^{p_{jk}}\) is dealt with. The authors generalize the method of \textit{H. Takeda} [J. Math. Anal. Appl. 360, No. 2, 631--650 (2009; Zbl 1183.35192)], who considered \(F_{j}(u)=| u_{j-1}| ^{p_{j}}\), to construct a global solution of the problem. The proof is based on \(L^{p}-L^{q}\)- type estimates of the fundamental solution to the linear damped equation and systematic choice of the function scale to adjust the nonlinear growth order.