The authors investigate the initial value problem to the coupled equations \[ \begin{cases} u_{tt}+u_t+(-1)^s\Delta ^su=\lambda _1| u| ^{p-1}| v| ^{q+1}u,\\ v_{tt}+v_t+(-1)^s\Delta ^sv=\lambda _2| u| ^{p+1}| v| ^{q-1}v \end{cases} \] for \(t>0\) and \(x\in \mathbb{R}^n\), where \(\lambda _1,\lambda _2>0\) and \(p,q\geq 0.\) Under the assumption \(p+q>2ms/n\), they prove that there exists a unique weak solution of the problem for sufficiently small initial data and its asymptotic decay for \(t\rightarrow \infty .\) Moreover, if \(0<p+q<2ms/n\) there exist such initial data of the arbitrary small norms for which no global weak solution exists.