This paper is concerned with the study of the nonlinearly damped system of wave
equations with Dirichlét boundary conditions:
$u_{t t}$ $- \Delta u + |u_t|^{m-1}u_t = F_u(u,v) \text{ in }\Omega\times ( 0,\infty )$,
$v_{t t}$$ - \Delta v + |v_t|^{r-1}v_t = F_v(u,v) \text{ in }\Omega\times( 0,\infty )$,
where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n=1,2,3$ with a smooth boundary
$\partial\Omega=\Gamma$ and $F$ is a $C^1$ function given by
$
F(u,v)=\alpha|u+v|^{p+1}+ 2\beta |uv|^{\frac{p+1}{2}}.
$
Under some conditions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain several results on the global existence, uniform decay rates, and blow up of solutions in finite time when the initial energy is nonnegative.
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