Existence and blow up for viscoelastic hyperbolic equations with variable exponents

Ying Chu; Bo Wen; Libo Cheng; Ying Chu; Bo Wen; Libo Cheng

doi:10.3934/cam.2024032

Communications in Analysis and Mechanics

2024, Volume 16, Issue 4: 717-737. doi: 10.3934/cam.2024032

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Research article

Existence and blow up for viscoelastic hyperbolic equations with variable exponents

School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun, 130022, China

Received: 22 November 2023 Revised: 27 February 2024 Accepted: 07 August 2024 Published: 14 October 2024
35L35, 35B40, 35B44

In this article, we consider a nonlinear viscoelastic hyperbolic problem with variable exponents. By using the Faedo$ - $Galerkin method and the contraction mapping principle, we obtain the existence of weak solutions under suitable assumptions on the variable exponents $ m(x) $ and $ p(x) $. Then we prove that a solution blows up in finite time with positive initial energy as well as nonpositive initial energy.
- variable exponents,
- weak solutions,
- viscoelastic,
- existence,
- blow up
Citation: Ying Chu, Bo Wen, Libo Cheng. Existence and blow up for viscoelastic hyperbolic equations with variable exponents[J]. Communications in Analysis and Mechanics, 2024, 16(4): 717-737. doi: 10.3934/cam.2024032

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Abstract

In this article, we consider a nonlinear viscoelastic hyperbolic problem with variable exponents. By using the Faedo$ - $Galerkin method and the contraction mapping principle, we obtain the existence of weak solutions under suitable assumptions on the variable exponents $ m(x) $ and $ p(x) $. Then we prove that a solution blows up in finite time with positive initial energy as well as nonpositive initial energy.

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