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Existence and nonexistence of a global solution for coupled nonlinear wave equations with damping and source. (English) Zbl 1190.35145
Summary: We consider the system of nonlinear wave equations $$\left.\aligned u_{tt}+|u_t|^{m-1}u_t&= \text{div}(\rho(|\nabla u|^2)\nabla u)+ f_1(u,v)\\ v_{tt}+|v_t|^{r-1}v_t&= \text{div}(\rho(|\nabla v|^2)\nabla v)+ f_1(u,v) \endaligned\right\} (x,t)\in\Omega\times(0,T),$$ with initial and Dirichlet boundary conditions. Under some suitable assumptions on the functions $f_1$ and $f_2$, the initial data and the parameters in the equations, the theorems of global existence and nonexistence are proved.

35L53Second-order hyperbolic systems, initial-boundary value problems
35L72Quasilinear second-order hyperbolic equations
35B44Blow-up (PDE)
Full Text: DOI
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