Existence and nonexistence of a global solution for coupled nonlinear wave equations with damping and source

Wu, Jianhua; Li, Shengjia; Chai, Shugen

doi:10.1016/j.na.2010.01.028

Cited by 11 publications

(11 citation statements)

References 9 publications

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“…Furthermore, they proved that the weak solutions blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms. Moreover, many studies of the global well-posedness for wave systems with dissipative terms have been researched in [24][25][26][27][28].…”

Section: Historical Researchmentioning

confidence: 99%

“…By reviewing above known results and also [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38], we will face the fact that the following unsolved problems arise naturally. Firstly, from [38] we know the global existence for the definitely positive energy, but we know less for the initial energy which may be negative.…”

Section: Unsolved Problemsmentioning

confidence: 99%

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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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Section: Historical Researchmentioning

confidence: 99%

Section: Unsolved Problemsmentioning

confidence: 99%

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

Jiang

2017

Advances in Mathematical Physics

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“…In [1], Wu et al considered the global existence and the blow up of the solution of the problem (1.2). Later, Fei and Hongjun [2] improved the blow up result in [1].…”

Section: Introductionmentioning

confidence: 99%

“…In [1], Wu et al considered the global existence and the blow up of the solution of the problem (1.2). Later, Fei and Hongjun [2] improved the blow up result in [1]. Finally, in [3], Pişkin and Polat studied the existence, the decay and the blow up of the solutions for the problem (1.2).…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds for the Blow up Time to a Coupled Nonlinear Hyperbolic Type Equations

Pışkın

Dinç

Tunc

2020

Communications in Advanced Mathematical Sciences

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“…When k = l = θ = ϱ = 0, equations () reduces to the following form,

({, \begin{matrix} u_{tt} + | u_{t} |^{p - 1} u_{t} = div (ρ (| \nabla u |^{2}) \nabla u) + f_{1} (u, v), \\ v_{tt} + | v_{t} |^{q - 1} v_{t} = div (ρ (| \nabla v |^{2}) \nabla v) + f_{2} (u, v) . \end{matrix})

Wu et al . obtained the global existence and blow‐up of the solution of problem (1.6) under some suitable conditions. Also, Liang and Gao considered problem () and improved the blow‐up result obtained in to a large class of initial data with positive initial energy.…”

Section: Introductionmentioning

confidence: 99%

On the decay and blow‐up of solution for coupled nonlinear wave equations with nonlinear damping and source terms

Hao

Wang

2016

Math Methods in App Sciences

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In this work, we consider a nonlinear coupled wave equations with initial‐boundary value conditions and nonlinear damping and source terms. Under suitable assumptions on the damping terms and source terms and initial data in the stable set, we obtain that the decay estimates of the energy function is exponential or polynomial by using Nakao's method. By using the energy method, we obtain the blow‐up result of solution with some positive or nonpositive initial energy. Copyright © 2016 John Wiley & Sons, Ltd.

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Existence and nonexistence of a global solution for coupled nonlinear wave equations with damping and source

Cited by 11 publications

References 9 publications

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

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

Lower Bounds for the Blow up Time to a Coupled Nonlinear Hyperbolic Type Equations

On the decay and blow‐up of solution for coupled nonlinear wave equations with nonlinear damping and source terms

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