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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
##### Keywords:
non-polynomial growth of the nonlinearity