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Nonexistence of global solutions to wave Equations with structural damping and nonlinear memory
  • Mokhtar Kirane,
  • Abderrazak NABTi,
  • Mohamed Jleli
Mokhtar Kirane
University de La Rochelle
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Abderrazak NABTi
Universite de Tebessa
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Mohamed Jleli
kING SAUD UNIVERSITY Riyadh, Saudi Arabia
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Peer review status:UNDER REVIEW

25 Jun 2020Submitted to Mathematical Methods in the Applied Sciences
04 Jul 2020Assigned to Editor
04 Jul 2020Submission Checks Completed
07 Jul 2020Reviewer(s) Assigned

Abstract

For the following wave equations with structural damping and nonlinear memory source terms \[ u_{tt}+(-\Delta)^{\frac{\alpha}{2}}u +(-\Delta)^{\frac{\beta}{2}}u_t =\int_{0}^{t}(t-s)^{\gamma-1} \vert u (s)\vert^{p}\,\text{d}s, \] and \[ u_{tt}+(-\Delta)^{\frac{\alpha}{2}}u +(-\Delta)^{\frac{\beta}{2}}u_t = \int_{0}^{t}(t-s)^{\gamma-1} \vert u_s (s)\vert^{p}\,\text{d}s, \] posed in $(x,t) \in \mathbb{R}^N \times [0,\infty) $, where $u=u(x,t)$ is real-value unknown function, $p>1$, $\alpha,\beta\in (0, 2]$, $\gamma\in (0,1)$, we prove the nonexistence of global solutions. Moreover, we give an upper bound estimate of the life span of solutions.