BLOW-UP OF SOLUTIONS TO THE SEMILINEAR WAVE EQUATIONS WITH FRICTIONAL AND VISCOELASTIC DAMPING TERMS

Xiongmei Fan; Sen Ming; Wei Han; Yeqin Su

doi:10.11948/20210427

2023 Volume 13 Issue 3

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Xiongmei Fan, Sen Ming, Wei Han, Yeqin Su. BLOW-UP OF SOLUTIONS TO THE SEMILINEAR WAVE EQUATIONS WITH FRICTIONAL AND VISCOELASTIC DAMPING TERMS[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1190-1208. doi: 10.11948/20210427

Citation:

Xiongmei Fan, Sen Ming, Wei Han, Yeqin Su. BLOW-UP OF SOLUTIONS TO THE SEMILINEAR WAVE EQUATIONS WITH FRICTIONAL AND VISCOELASTIC DAMPING TERMS[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1190-1208. doi: 10.11948/20210427

BLOW-UP OF SOLUTIONS TO THE SEMILINEAR WAVE EQUATIONS WITH FRICTIONAL AND VISCOELASTIC DAMPING TERMS

1.
Data Science And Technology, North University of China, College Road, Taiyuan, 030051, China
2.
Department of Mathematics, North University of China, College Road, Taiyuan, 030051, China
3.
Department of Securities and Futures, Southwestern University of Finance and Economics, Liutai Road, Chengdu, 611130, China

Author Bio: Email: xiongmeifan1997@163.com (X. Fan); Email: hanwei@nuc.edu.cn (W. Han); Email: suyeqin2008@163.com (Y. Su)
Corresponding author: Email: senming1987@163.com (S. Ming)
Fund Project: The authors were supported by Natural Science Foundation of Shanxi Province of China (201901D211276), the Fundamental Research Program of Shanxi Province (20210302123021, 20210302123045, 20210302123182), the Program for the Innovative Talents of Higher Education Institutions of Shanxi Province, the Innovative Research Team of North University of China (TD201901), National Natural Science Foundation of China (11601446)

Abstract

Abstract

Our starting point in this paper is to investigate the weakly coupled system of semilinear wave equations with two types of damping terms and combined nonlinearities $|v_{t}|^{p_{1}}+|v|^{q_{1}}$, $|u_{t}|^{p_{2}}+|u|^{q_{2}}$ on exterior domain in $n\, (n\geq1)$ dimensions. Local existence and uniqueness of mild solutions to the problem are established. Moreover, non-existence of global solutions to the problem with power nonlinearities $|v|^{p}, \, |u|^{q}$ in the case of coupled system and the problem with power nonlinearity $|u|^{p}$ in the case of single equation are verified. The proofs are based on the test function technique. It is worth noticing that the frictional damping $u_{t}$ is more dominant than the viscoelastic damping $\Delta u_{t}$ when the time trend to infinity. To the best of our knowledge, the blow-up results in Theorems 1.2-1.3 are new.
- Semilinear wave equations /
- local existence and uniqueness /
- exterior domain /
- test function technique /
- blow-up
MSC: 35L70, 58J45

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