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Theoretical and numerical study of the blow up in a nonlinear viscoelastic problem with variable-exponent and arbitrary positive energy

Abstract

In this paper, we consider the following nonlinear viscoelastic wave equation with variable exponents:

$$u_{tt}-\Delta u+\int_{0}^{t} g(t-\tau)\Delta u(x,\tau)\rm{d}\tau+\mu u_{t}=\vert u\vert^{p(x)-2}u,$$

where μ is a nonnegative constant and the exponent of nonlinearity p(·) and g are given functions. Under arbitrary positive initial energy and specific conditions on the relaxation function g, we prove a finite-time blow-up result. We also give some numerical applications to illustrate our theoretical results.

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Acknowledgements

The authors thank Birzeit University and Sharjah University for their support. The second and the third authors are sponsored by MASEP Research Group in the Research Institute of Sciences and Engineering at University of Sharjah. Grant No. 2002144089, 2019–2020.

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Correspondence to Salim A. Messaoudi.

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Talahmeh, A.A., Messaoudi, S.A. & Alahyane, M. Theoretical and numerical study of the blow up in a nonlinear viscoelastic problem with variable-exponent and arbitrary positive energy. Acta Math Sci 42, 141–154 (2022). https://doi.org/10.1007/s10473-022-0107-y

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  • DOI: https://doi.org/10.1007/s10473-022-0107-y

Key words

  • nonlinear damping
  • blow up
  • finite time
  • variable nonlinearity
  • arbitrary positive energy

2010 MR Subject Classification

  • 35A01
  • 35B44
  • 65M60