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Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy. (English) Zbl 0956.35087
The paper deals with the long time behaviour of solutions to the initial value problem for semilinear wave equations \(u_{tt}+a|u_t|^{m-1}u_t-\triangle u=b|u|^{p-1}u-q(x)^2u\) in \([0,\infty)\times \mathbb{R}^n,\) where \(a,b>0\), \(p>m\geq 1.\) Under some conditions on \(p,m,n,q(x)\) the authors prove, that for every positive initial energy there exist initial data such that the solutions of the above problem blow up in a finite time.

MSC:
35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
35Q72 Other PDE from mechanics (MSC2000)
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