Blow-up solutions for a nonlinear wave equation with boundary damping and interior source

Feng, Hongyinping; Li, Shengjia; Xia, Zhi

doi:10.1016/j.na.2011.10.027

Cited by 11 publications

(16 citation statements)

References 16 publications

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“…He obtained the local existence, blow up and global existence results of solutions. More results on the initial boundary value problem for the wave equations with nonlinear internal source and boundary velocity feedback, we refer readers to see (Di and Shang [9], Feng and Li [11,12], Liu, Sun and Li [24]) and the papers cited therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source

Shang

2020

era

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In this paper, we consider the initial boundary value problem for the fourth order wave equation with nonlinear boundary velocity feedbacks f 1 (uνt), f 2 (ut) and internal source |u| ρ u. Under some geometrical conditions, the existence and uniform decay rates of the solutions are proved even if the nonlinear boundary velocity feedbacks f 1 (uνt), f 2 (ut) have not polynomial growth near the origin respectively. By the combination of the Galerkin approximation, potential well method and a special basis constructed, we first obtain the global existence and uniqueness of regular solutions and weak solutions. In addition, we also investigate the explicit decay rate estimates of the energy, the ideas of which are based on the construction of a special weight function φ(t) (that depends on the behaviors of the functions f 1 (uνt), f 2 (ut) near the origin), nonlinear integral inequality and the Multiplier method.

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Section: Introductionmentioning

confidence: 99%

Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source

Shang

2020

era

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“…He obtained the local existence, blow up and global existence results of solutions. More results on the initial boundary value problem for the wave equations in the context of nonlinear boundary damping and source terms, we refer readers to see (Aassila et al [1], Cavalcanti et al [6], Feng and Li [9], [10]) and the papers cited therein.…”

Section: Introductionmentioning

confidence: 99%

Global existence and nonexistence of solutions for a viscoelastic wave equation with nonlinear boundary source term

Shang

Peng

2016

Math. Nachr.

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In this paper, we consider the initial boundary value problem for a viscoelastic wave equation with nonlinear boundary source term. First of all, we introduce a family of potential wells and prove the invariance of some sets. Then we establish the existence and nonexistence of global weak solution with small initial energy under suitable assumptions on the relaxation function g(·), nonlinear function f (·), the initial data and the parameters in the equation. Furthermore, we obtain the global existence of weak solution for the problem with critical initial conditions I (u 0 ) ≥ 0 and E(0) = d.

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“…Assumption (1.6) was also skipped in the more recent papers [18,41], which deal with the one-dimensional case n = 1, when β = 0 and α ≡ 1. Blow-up for problem (1.2) is proved there when E(u 0 , u 1 ) < d and either…”

Section: Introductionmentioning

confidence: 99%

“…In this way we extend the blow-up result from [3] to positive initial energy while extending the result from [41] to n 1. Instead of using interpolation theory, we adapt a more elementary estimate, used in [18,41] when n = 1, to the case when n 1.…”

Section: Introductionmentioning

confidence: 99%

Blow-up for the wave equation with nonlinear source and boundary damping terms

Fiscella

Vitillaro

2015

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Abstract. The paper deals with blow-up for the solutions of an evolution problem consisting on a semilinear wave equation posed in a bounded C 1,1 open subset of R n , supplied with a Neumann boundary condition involving a nonlinear dissipation. The typical problem studied iswhereThe initial data are posed in the energy space. The aim of the paper is to improve previous blow-up results concerning the problem.

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Blow-up solutions for a nonlinear wave equation with boundary damping and interior source

Cited by 11 publications

References 16 publications

Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source

Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source

Global existence and nonexistence of solutions for a viscoelastic wave equation with nonlinear boundary source term

Blow-up for the wave equation with nonlinear source and boundary damping terms

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