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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 $$\align u_{tt}-\Delta u+|u_t|^{m-1} u_t&= F_u(u,v)\quad\text{in }\Omega\times (0,\infty),\\ v_{tt}-\Delta v+|v_t|^{r-1} v_t&= F_v(u,v) \quad\text{in }\Omega\times (0,\infty), \endalign$$ where $\Omega$ is a bounded domain in $\Bbb R^n$, $n=1,2,3$ 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.

35B40Asymptotic behavior of solutions of PDE
74H35Singularities, blowup, stress concentrations (dynamical problems in solid mechanics)
35L71Semilinear second-order hyperbolic equations
35B44Blow-up (PDE)
35L53Second-order hyperbolic systems, initial-boundary value problems
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