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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 \(1<m<p\).

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