Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level

ABSTRACT

In this paper, we study a three-dimensional (3D) viscoelastic wave equation with nonlinear weak damping, supercritical sources and prescribed past history $u_0(x,t)$, $t\le 0$: $$\begin{array}{c}\hfill u_{\mathit{tt}}-k(0)\mathrm{\Delta }u-{\int }_0^{+\infty }{k}^{\prime }(s)\mathrm{\Delta }u(t-s)\mathrm{d}s+|u_t{|}^{m-1}{u}_t={|u|}^{p-1}u,\text{in}\mathrm{\Omega }\times (0,T),\end{array}$$

where the relaxation function k is monotone decreasing with $k(+\infty )=1$, $m\ge 1$ and $1\le p<6$. 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.

KEYWORDS:

• Wave equation
• viscoelasticity
• nonlinear damping
• supercritical source
• blow-up

AMS SUBJECT CLASSIFICATIONS:

• 35L10
• 35L20
• 35L70
• 35B44

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors were supported financially by the National Natural Science Foundation of China [grant number 11371221], [grant number 11571296] and Natural Science Foundation of Shandong Province of China (ZR2017MA036).