Blow up property for viscoelastic evolution equations on manifolds with conical degeneration

Mohsen Alimohammady, Morteza Koozehgar Kalleji

Published 2016-02-05Version 1

This paper is concerned with the study of the nonlinear viscoelastic evolution equation with strong damping and source terms, described by \[u_{tt} - \Delta_{\mathbb{B}}u + \int_{0}^{t}g(t-\tau)\Delta_{\mathbb{B}}u(\tau)d\tau + f(x)u_{t}|u_{t}|^{m-2} = h(x)|u|^{p-2}u , \hspace{1 cm} x\in int\mathbb{B}, t > 0,\] where $\mathbb{B}$ is a stretched manifold. First, we prove the solutions of problem {1.1} in cone Sobolev space $\mathcal{H}^{1,\frac{n}{2}}_{2,0}(\mathbb{B}),$ admit a blow up in finite time for $p > m$ and positive initial energy. Then, we construct a lower bound for obtained blow up time under appropriate assumptions on data.