Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms
Viscoelastic damping phenomena are ubiquitous in diverse kinds of wave motions of nonlinear media. This arouses extensive interest in studying the existence, the finite time blow-up phenomenon and various large time behaviors of solutions to viscoelastic wave equations. In this paper, we are concern...
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MDPI AG
2023-08-01
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| author | Chengqiang Wang Can Wang Xiangqing Zhao Zhiwei Lv |
| author_facet | Chengqiang Wang Can Wang Xiangqing Zhao Zhiwei Lv |
| author_sort | Chengqiang Wang |
| collection | DOAJ |
| description | Viscoelastic damping phenomena are ubiquitous in diverse kinds of wave motions of nonlinear media. This arouses extensive interest in studying the existence, the finite time blow-up phenomenon and various large time behaviors of solutions to viscoelastic wave equations. In this paper, we are concerned with a class of variable coefficient coupled quasi-linear wave equations damped by viscoelasticity with a long-term memory fading at very general rates and possibly damped by friction but provoked by nonlinear interactions. We prove a local existence result for solutions to our concerned coupled model equations by applying the celebrated Faedo-Galerkin scheme. Based on the newly obtained local existence result, we prove that solutions would exist globally in time whenever their initial data satisfy certain conditions. In the end, we provide a criterion to guarantee that some of the global-in-time-existing solutions achieve energy decay at general rates uniquely determined by the fading rates of the memory. Compared with the existing results in the literature, our concerned model coupled wave equations are more general, and therefore our theoretical results have wider applicability. Modified energy functionals (can also be viewed as certain Lyapunov functionals) play key roles in proving our claimed general energy decay result in this paper. |
| first_indexed | 2024-03-11T00:07:37Z |
| format | Article |
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| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T00:07:37Z |
| publishDate | 2023-08-01 |
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| spelling | doaj.art-90daebc104994eb9b820ec2839a4d35c2023-11-19T00:15:09ZengMDPI AGAxioms2075-16802023-08-0112878010.3390/axioms12080780Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source TermsChengqiang Wang0Can Wang1Xiangqing Zhao2Zhiwei Lv3School of Mathematics, Chengdu Normal University, Chengdu 611130, ChinaSchool of Mathematics, Chengdu Normal University, Chengdu 611130, ChinaSchool of Mathematics, Suqian University, Suqian 223800, ChinaSchool of Mathematics, Suqian University, Suqian 223800, ChinaViscoelastic damping phenomena are ubiquitous in diverse kinds of wave motions of nonlinear media. This arouses extensive interest in studying the existence, the finite time blow-up phenomenon and various large time behaviors of solutions to viscoelastic wave equations. In this paper, we are concerned with a class of variable coefficient coupled quasi-linear wave equations damped by viscoelasticity with a long-term memory fading at very general rates and possibly damped by friction but provoked by nonlinear interactions. We prove a local existence result for solutions to our concerned coupled model equations by applying the celebrated Faedo-Galerkin scheme. Based on the newly obtained local existence result, we prove that solutions would exist globally in time whenever their initial data satisfy certain conditions. In the end, we provide a criterion to guarantee that some of the global-in-time-existing solutions achieve energy decay at general rates uniquely determined by the fading rates of the memory. Compared with the existing results in the literature, our concerned model coupled wave equations are more general, and therefore our theoretical results have wider applicability. Modified energy functionals (can also be viewed as certain Lyapunov functionals) play key roles in proving our claimed general energy decay result in this paper.https://www.mdpi.com/2075-1680/12/8/780existence resultsgeneral energy decayquasi-linear wave equationsvariable coefficient wave equationsviscoelastic damping |
| spellingShingle | Chengqiang Wang Can Wang Xiangqing Zhao Zhiwei Lv Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms Axioms existence results general energy decay quasi-linear wave equations variable coefficient wave equations viscoelastic damping |
| title | Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms |
| title_full | Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms |
| title_fullStr | Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms |
| title_full_unstemmed | Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms |
| title_short | Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms |
| title_sort | existence and general energy decay of solutions to a coupled system of quasi linear viscoelastic variable coefficient wave equations with nonlinear source terms |
| topic | existence results general energy decay quasi-linear wave equations variable coefficient wave equations viscoelastic damping |
| url | https://www.mdpi.com/2075-1680/12/8/780 |
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