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Existence of a solution of the wave equation with nonlinear damping and source terms. (English) Zbl 0803.35092
The authors study the problem of nonlinear wave equation $$u_{tt} - \Delta u + au_ t | u_ t |^{m-1} = bu | u |^{p-1}$$ in $$\Omega \subseteq\mathbb{R}^ n$$ with Dirichlet boundary conditions on $$\partial \Omega$$, where $$p>1$$ for $$n \leq 2$$, $$1 \leq p \leq {n \over n - 2}$$ for $$n \geq 3$$ and $$a,b>0$$. They prove a global existence of the solution to the above problem for large initial data and $$1,p \leq m$$ and blowing-up of $$L_ \infty$$-norm of solution for suitably large initial data and $$1<m<p$$.

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations
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