THEORETICAL AND NUMERICAL STUDY OF THE BLOW UP IN A NONLINEAR VISCOELASTIC PROBLEM WITH VARIABLE-EXPONENT AND ARBITRARY POSITIVE ENERGY

doi:10.1007/s10473-022-0107-y

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (1): 141-154.doi: 10.1007/s10473-022-0107-y

THEORETICAL AND NUMERICAL STUDY OF THE BLOW UP IN A NONLINEAR VISCOELASTIC PROBLEM WITH VARIABLE-EXPONENT AND ARBITRARY POSITIVE ENERGY

Ala A. TALAHMEH¹, Salim A. MESSAOUDI², Mohamed ALAHYANE³

1. Department of Mathematics, Birzeit University, West Bank, Birzeit, Palestine;
2. Department of Mathematics, University of Sharjah, P.O. Box 27272, Sharjah, UAE;
3. Department of Mathematics, RISE, University of Sharjah, P.O. Box 27272, Sharjah, UAE

收稿日期:2020-08-06 修回日期:2021-01-04 出版日期:2022-02-25 发布日期:2022-02-24
通讯作者: Salim A. MESSAOUDI,E-mail:smessaoudi@sharjah.ac.ae E-mail:smessaoudi@sharjah.ac.ae
作者简介:Ala A. TALAHMEH,E-mail:atalahmeh@birzeit.edu;Mohamed ALAHYANE,E-mail:malahyane@sharjah.ac.ae

THEORETICAL AND NUMERICAL STUDY OF THE BLOW UP IN A NONLINEAR VISCOELASTIC PROBLEM WITH VARIABLE-EXPONENT AND ARBITRARY POSITIVE ENERGY

Ala A. TALAHMEH¹, Salim A. MESSAOUDI², Mohamed ALAHYANE³

1. Department of Mathematics, Birzeit University, West Bank, Birzeit, Palestine;
2. Department of Mathematics, University of Sharjah, P.O. Box 27272, Sharjah, UAE;
3. Department of Mathematics, RISE, University of Sharjah, P.O. Box 27272, Sharjah, UAE

Received:2020-08-06 Revised:2021-01-04 Online:2022-02-25 Published:2022-02-24
Contact: Salim A. MESSAOUDI,E-mail:smessaoudi@sharjah.ac.ae E-mail:smessaoudi@sharjah.ac.ae

摘要/Abstract

摘要： In this paper, we consider the following nonlinear viscoelastic wave equation with variable exponents:\[u_{tt}-\Delta u+\int_{0}^{t}g(t-\tau)\Delta u(x,\tau){\rm d}\tau +\mu u_t=|u|^{p(x)-2}u,\] where $\mu$ is a nonnegative constant and the exponent of nonlinearity $p(\cdot)$ 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.

关键词: nonlinear damping, blow up, finite time, variable nonlinearity, arbitrary positive energy

Abstract: In this paper, we consider the following nonlinear viscoelastic wave equation with variable exponents:\[u_{tt}-\Delta u+\int_{0}^{t}g(t-\tau)\Delta u(x,\tau){\rm d}\tau +\mu u_t=|u|^{p(x)-2}u,\] where $\mu$ is a nonnegative constant and the exponent of nonlinearity $p(\cdot)$ 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.

Key words: nonlinear damping, blow up, finite time, variable nonlinearity, arbitrary positive energy

中图分类号:

35A01

Ala A. TALAHMEH, Salim A. MESSAOUDI, Mohamed ALAHYANE. THEORETICAL AND NUMERICAL STUDY OF THE BLOW UP IN A NONLINEAR VISCOELASTIC PROBLEM WITH VARIABLE-EXPONENT AND ARBITRARY POSITIVE ENERGY[J]. 数学物理学报(英文版), 2022, 42(1): 141-154.

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THEORETICAL AND NUMERICAL STUDY OF THE BLOW UP IN A NONLINEAR VISCOELASTIC PROBLEM WITH VARIABLE-EXPONENT AND ARBITRARY POSITIVE ENERGY

THEORETICAL AND NUMERICAL STUDY OF THE BLOW UP IN A NONLINEAR VISCOELASTIC PROBLEM WITH VARIABLE-EXPONENT AND ARBITRARY POSITIVE ENERGY

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