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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/130/0033

    • Keywords

       

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

    • Abstract

       

      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.

    • Author Affiliations

       

      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

    • Dates

       
      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

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