Lifespan estimate of solution to the semilinear wave equation with damping term and mass term

Xiongmei Fan; Sen Ming; Wei Han; Zikun Liang; Xiongmei Fan; Sen Ming; Wei Han; Zikun Liang

doi:10.3934/math.2023910

AIMS Mathematics

2023, Volume 8, Issue 8: 17860-17889. doi: 10.3934/math.2023910

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Lifespan estimate of solution to the semilinear wave equation with damping term and mass term

1.
Data Science And Technology, North University of China, Taiyuan 030051, China
2.
Department of Mathematics, North University of China, Taiyuan 030051, China
3.
Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, China

Received: 11 April 2023 Revised: 09 May 2023 Accepted: 14 May 2023 Published: 24 May 2023
MSC : 35L70, 58J45

This paper is mainly concerned with the initial boundary value problems of semilinear wave equations with damping term and mass term as well as Neumann boundary conditions on exterior domain in three dimensions. Blow-up and upper bound lifespan estimates of solutions to the problem with damping term and mass term are derived by applying test function technique and iterative method, where nonlinear terms are power nonlinearity $ |u|^p $, derivative nonlinearity $ |u_{t}|^p $, combined nonlinearities $ |u_t|^p+|u|^q $, respectively. Moreover, upper bound lifespan estimate of solution to the problem with scale invariant damping term, non-negative mass term and combined nonlinearities $ |u_t|^p+|u|^q $ is obtained. The proofs are based on the test function method and iterative approach. The main new contribution is that upper bound lifespan estimates of solutions are associated with the Strauss exponent and Glassey exponent. In addition, the variation trend of wave is achieved by taking advantage of numerical simulation.
- semilinear wave equations,
- mass term,
- lifespan estimates,
- test function technique,
- iterative method
Citation: Xiongmei Fan, Sen Ming, Wei Han, Zikun Liang. Lifespan estimate of solution to the semilinear wave equation with damping term and mass term[J]. AIMS Mathematics, 2023, 8(8): 17860-17889. doi: 10.3934/math.2023910

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Abstract

This paper is mainly concerned with the initial boundary value problems of semilinear wave equations with damping term and mass term as well as Neumann boundary conditions on exterior domain in three dimensions. Blow-up and upper bound lifespan estimates of solutions to the problem with damping term and mass term are derived by applying test function technique and iterative method, where nonlinear terms are power nonlinearity $ |u|^p $, derivative nonlinearity $ |u_{t}|^p $, combined nonlinearities $ |u_t|^p+|u|^q $, respectively. Moreover, upper bound lifespan estimate of solution to the problem with scale invariant damping term, non-negative mass term and combined nonlinearities $ |u_t|^p+|u|^q $ is obtained. The proofs are based on the test function method and iterative approach. The main new contribution is that upper bound lifespan estimates of solutions are associated with the Strauss exponent and Glassey exponent. In addition, the variation trend of wave is achieved by taking advantage of numerical simulation.

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