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On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. (English) Zbl 1183.35033
Summary: This paper is concerned with the study of the nonlinearly damped system of wave equations with Dirichlet boundary conditions
\[ \begin{aligned} u_{tt}-\Delta u+|u_t|^{m-1} u_t&= F_u(u,v)\quad\text{in }\Omega\times (0,\infty),\\ v_{tt}-\Delta v+|v_t|^{r-1} v_t&= F_v(u,v) \quad\text{in }\Omega\times (0,\infty), \end{aligned} \]
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 of finite time when the initial energy is nonnegative.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
74H35 Singularities, blow-up, stress concentrations for dynamical problems in solid mechanics
35L71 Second-order semilinear hyperbolic equations
35B44 Blow-up in context of PDEs
35L53 Initial-boundary value problems for second-order hyperbolic systems
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