On decay and blow-up of solutions for a system of nonlinear wave equations

Wu, Shun-Tang

doi:10.1016/j.jmaa.2012.04.054

Cited by 23 publications

(36 citation statements)

References 19 publications

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“…Recently, Wu [38] considered the initial boundary value problem for system (1)- (5) (A1) The nonlinear source terms 1 ( , V), 2 ( , V) satisfy…”

Section: Historical Researchmentioning

confidence: 99%

Global Well-Posedness for a Class of Kirchhoff-Type Wave System

Jiang

2017

Advances in Mathematical Physics

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In recent years, the small initial boundary value problem of the Kirchhoff-type wave system attracts many scholars' attention. However, the big initial boundary value problem is also a topic of theoretical significance. In this paper, we devote oneself to the well-posedness of the Kirchhoff-type wave system under the big initial boundary conditions. Combining the potential well method with an improved convex method, we establish a criterion for the well-posedness of the system with nonlinear source and dissipative and viscoelastic terms. Based on the criteria, the energy of the system is divided into different levels. For the subcritical case, we prove that there exist the global solutions when the initial value belongs to the stable set, while the finite time blow-up occurs when the initial value belongs to the unstable set. For the supercritical case, we show that the corresponding solution blows up in a finite time if the initial value satisfies some given conditions.

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“…Recently, Wu [38] considered the initial boundary value problem for system (1)- (5) (A1) The nonlinear source terms 1 ( , V), 2 ( , V) satisfy…”

Section: Historical Researchmentioning

confidence: 99%

Global Well-Posedness for a Class of Kirchhoff-Type Wave System

Jiang

2017

Advances in Mathematical Physics

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“…When ρ = 0 and there are no dispersion terms, in the absence of strong damping ( γ i = 0), the system has been investigated by several authors and results concerning existence, decay, and blow-up were obtained [1,2,15,22,25,26,28,29,32,36]. In the case of g i = 0 , Agre and Rammaha [1] proved several results on local and global existence of weak solutions with the nonlinear functions f 1 (u, v) and…”

Section: Introductionmentioning

confidence: 99%

“…Recently, based on the potential well method and a lemma by Nakao [28], Said-Houari [29] established global existence, and polynomial and exponential decay rates for the energy of the same problem with different nonlinear source terms. In [36], Wu studied the following nonlinear wave equations:…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, they established the blow-up properties of the local solution if r > p and the initial energy is less than the potential well depth. In [17], inspiring the work [36], they also considered the following Petrovsky equations:…”

Section: Introductionmentioning

confidence: 99%

“…Improving on the method of [36], they showed the blow-up of solutions and the lifespan estimates when p = 1, q = 1. In the case 1 < p, q < 3, they also obtained a blow-up result when the initial energy is negative or nonnegative at less than the mountain pass level value.…”

Section: Introductionmentioning

confidence: 99%

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Global existence, uniform decay, and exponential growth of solutions for a system of viscoelastic Petrovsky equations

Tahamtani

2014

Turk J Math

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In this paper, we study the initial-boundary value problem for a system of nonlinear viscoelastic Petrovsky equations. Introducing suitable perturbed energy functionals and using the potential well method we prove uniform decay of solution energy under some restrictions on the initial data and the relaxation functions. Moreover, we establish a growth result for certain solutions with positive initial energy.

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Uniform decay and blow‐up of solutions for coupled nonlinear Klein–Gordon equations with nonlinear damping terms

Pışkın

2013

Math Methods in App Sciences

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On decay and blow-up of solutions for a system of nonlinear wave equations

Cited by 23 publications

References 19 publications

Global Well-Posedness for a Class of Kirchhoff-Type Wave System

Global Well-Posedness for a Class of Kirchhoff-Type Wave System

Global existence, uniform decay, and exponential growth of solutions for a system of viscoelastic Petrovsky equations

Uniform decay and blow‐up of solutions for coupled nonlinear Klein–Gordon equations with nonlinear damping terms

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