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Global existence and decay of energy to systems of wave equations with damping and supercritical sources. (English) Zbl 1275.35142
The paper deals with the initial boundary value problem to the system \[ \begin{cases} u_{tt}-\Delta u+g_1(u_t)=f_1(u,v),\\ v_{tt}-\Delta v+g_2(v_t)=f_2(u,v) \end{cases} \] in \(\Omega \times (0,\infty )\) with boundary conditions \(\partial _{\nu }u+u+g(u_t)=h(u),\) \(v=0\) on \(\partial \Omega \times (0,\infty ).\) Under some restrictions on the parameters of the problem a global existence of a unique weak solution and exponential uniform decay rate of of its energy are proved. Moreover, a blow-up result for weak solutions with nonnegative initial energy is established. The authors use the results from their papers [“Systems of nonlinear wave equations with damping and supercritical boundary and interior sources”. Trans. Am. Math. Soc., (in press)] and [Appl. Anal. 92, No. 6, 1101–1115 (2013; Zbl 1278.35151)].

MSC:
35L53 Initial-boundary value problems for second-order hyperbolic systems
35B40 Asymptotic behavior of solutions to PDEs
35B44 Blow-up in context of PDEs
35L71 Second-order semilinear hyperbolic equations
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