Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level. (English) Zbl 1447.35077
Summary: In this paper, we study a three-dimensional (3D) viscoelastic wave equation with nonlinear weak damping, supercritical sources and prescribed past history \(u_0(x,t)\), \(t\leq 0\):
\[
u_{tt} -k(0)\Delta u-\int^{+\infty}_0 k' (s)\Delta u (t-s)\mathrm{d}s + |u_t|^{m-1} u_t= |u|^{p-1} u,\quad \text{in }\Omega\times (0,T),
\]
where the relaxation function \(k\) is monotone decreasing with \(k(+\infty)=1\), \(m\geq 1\) and \(1\leq p<6\). When the source is stronger than dissipations, i.e. \(p>\max \{m,\sqrt{k(0)}\}\), we obtain some finite time blow-up results with positive initial energy. In particular, we obtain the existence of certain solutions which blow up in finite time for initial data at arbitrary energy level.
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35L71 | Second-order semilinear hyperbolic equations |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 35R09 | Integro-partial differential equations |
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