Blow-up of solutions for coupled wave equations with damping terms and derivative nonlinearities

Sen Ming; Xiaodong Wang; Xiongmei Fan; Xiao Wu; Sen Ming; Xiaodong Wang; Xiongmei Fan; Xiao Wu

doi:10.3934/math.20241307

AIMS Mathematics

2024, Volume 9, Issue 10: 26854-26876. doi: 10.3934/math.20241307

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Research article

Blow-up of solutions for coupled wave equations with damping terms and derivative nonlinearities

1.
Department of Mathematics, North University of China, Taiyuan 030051, China
2.
Data Science and Technology, North University of China, Taiyuan 030051, China

Received: 06 August 2024 Revised: 30 August 2024 Accepted: 03 September 2024 Published: 14 September 2024
MSC : 35L15, 35L70

This work was concerned with the weakly coupled system of semi-linear wave equations with time dependent speeds of propagation, damping terms, and derivative nonlinear terms in generalized Einstein-de Sitter space-time on $ \mathbb{R}^n $. Under certain assumptions about the indexes $ k_1, \, k_2 $, coefficients $ \mu_1, \, \mu_2 $, and nonlinearity exponents $ p, \, q $, applying the iteration technique, finite time blow-up of local solutions to the small initial value problem of the coupled system was investigated. Blow-up region and upper bound lifespan estimate of solutions to the problem were established. Compared with blow-up results in the previous literature, the new ingredient relied on that the blow-up region of solutions obtained in this work varies due to the influence of coefficients $ k_1, \, k_2 $.
- coupled wave equations,
- damping terms,
- iteration method,
- blow-up,
- lifespan estimate
Citation: Sen Ming, Xiaodong Wang, Xiongmei Fan, Xiao Wu. Blow-up of solutions for coupled wave equations with damping terms and derivative nonlinearities[J]. AIMS Mathematics, 2024, 9(10): 26854-26876. doi: 10.3934/math.20241307

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Abstract

This work was concerned with the weakly coupled system of semi-linear wave equations with time dependent speeds of propagation, damping terms, and derivative nonlinear terms in generalized Einstein-de Sitter space-time on $ \mathbb{R}^n $. Under certain assumptions about the indexes $ k_1, \, k_2 $, coefficients $ \mu_1, \, \mu_2 $, and nonlinearity exponents $ p, \, q $, applying the iteration technique, finite time blow-up of local solutions to the small initial value problem of the coupled system was investigated. Blow-up region and upper bound lifespan estimate of solutions to the problem were established. Compared with blow-up results in the previous literature, the new ingredient relied on that the blow-up region of solutions obtained in this work varies due to the influence of coefficients $ k_1, \, k_2 $.

References

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