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Global existence of solutions to a wave equation with damping and source terms. (English) Zbl 1018.35053
This paper deals with the following mixed problem: \[ \begin{cases} u''-\Delta u+g(u')= |u|^{q-1} u\quad & \text{in }\Omega \times\mathbb{R}_+,\\ u=0 \quad & \text{on } \Gamma\times\mathbb{R}_+,\\ u(x,0)=u_0(x),\;u'(x,0)=u_1(x)\quad & \text{in } \Omega,\end{cases} \tag{1} \] where \(\Omega\subset \mathbb{R}^n\) is a bounded domain with smooth boundary \(\Gamma=\partial \Omega\) and \(g\) is a real-valued function. The author proves global existence of solution for (1) and studies the asymptotic behaviour of solution to (1) when \(g\) does not necessarily have polynomial growth near zero.

MSC:
35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
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