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Viscoelastic equation; blow up; cone Sobolev spaces; degenerated differential operator

This paper is concerned with the study of nonlinear viscoelastic evolutionequation with strong damping and source terms, described by $$u_{tt} − \Delta_{\mathbb{B}}u +\int^{t}_{0}\,g(t − \tau)\Delta_{\mathbb{B}}u(\tau)d\tau + f (x)u_{t} |u_{t}|^{m−2}\\ = h(x)|u|^{p−2}u, \,\,\,\,x \in int\,\mathbb{B}, t > 0,$$

where $\mathbb{B}$ is a stretched manifold. First, we prove the solutions of problem (1.1) in the cone Sobolev space $\mathcal{H}^{1,\frac{n}{2}}_{2,0} (\mathbb{B})$, which admit a blow up in finite time for $p$ > $m$ and positive initial energy. Then, we construct a lower bound for obtaining blow up time under appropriate assumptions on data.

MOHSEN ALIMOHAMMADY¹ MORTEZA KOOZEHGAR KALLEJI²

1. Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar 47416-1468, Iran
2. Department of Mathematics, Faculty of Sciences, Arak University, Arak 38156-8-8349, Iran

Published

7 May 2020

Manuscript received

28 January 2019

Manuscript revised

22 June 2019

Accepted

20 November 2019

Early published

Unedited version published online

Final version published online

6 May 2020