On Global Solutions and Energy Decay for the Wave Equations of Kirchhoff Type with Nonlinear Damping Terms

Matsuyama, Tokio; Ikehata, Ryo

doi:10.1006/jmaa.1996.0464

Cited by 102 publications

(45 citation statements)

References 14 publications

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“…In other words, C(Ω, q + 1) = sup{ u q+1 ∇u 2 |, u ∈ H 1 0 (Ω), u = 0} is positive and finite. Lemma 2.2 (Gagliardo-Nirenberg [4]). Let 1 r < q +∞ and p q.…”

Section: Preliminariesmentioning

confidence: 99%

On solutions of quasilinear wave equations with nonlinear damping terms

Park

Bae

2000

Czechoslovak Mathematical Journal

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“…In other words, C(Ω, q + 1) = sup{ u q+1 ∇u 2 |, u ∈ H 1 0 (Ω), u = 0} is positive and finite. Lemma 2.2 (Gagliardo-Nirenberg [4]). Let 1 r < q +∞ and p q.…”

Section: Preliminariesmentioning

confidence: 99%

On solutions of quasilinear wave equations with nonlinear damping terms

Park

Bae

2000

Czechoslovak Mathematical Journal

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“…For instance, we can see [18,19,21,22,23,30,31,34,35]. It is interesting to observe that problems with condition M (s) = 1 and a feedback occurs on the boundary were studied by many authors (see [3,4,5,6,7,12,13,14,20]).…”

Section: Introductionmentioning

confidence: 99%

On Solvability of the Dissipative Kirchhoff Equation With Nonlinear Boundary Damping

Zhang¹,

Huang²

2014

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, we prove the global existence and uniqueness of the dissipative Kirchhoff equationwith nonlinear boundary damping by Galerkin approximation benefited from the ideas of Zhang et al. [33]. Furthermore,we overcome some difficulties due to the presence of nonlinear terms M ( ∇u 2 ) and g(ut) by introducing a new variables and we can transform the boundary value problem into an equivalent one with zero initial data by argument of compacity and monotonicity.

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“…Kirchhoff [14] was the first one to study the oscillations of stretched strings and plates. The question of existence and nonexistence of solutions have been discussed by many authors (see [15,16,[18][19][20][21][22]). The model in hand, with Balakrishnan-Taylor damping (σ > 0) and h = 0, was initially proposed by Balakrishnan and Taylor in 1989 [2] and Bass and Zes [4].…”

Section: Introductionmentioning

confidence: 99%

Exponential stability and blow up for a problem with Balakrishnan–Taylor damping

Tatar

Zaraï

2011

Demonstratio Mathematica

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Abstract. This work is devoted to the study of a nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping. We show that the weak dissipation produced by the memory term is strong enough to stabilize solutions exponentially. Also, we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a stronger damping. IntroductionThe objective of this work is to study the following initial-boundary value problemwhere Ω is a bounded domain in R n with smooth boundary Γ. Here h represents the kernel of the memory term. All the parameters ξ 0 , ξ 1 , and σ are assumed to be positive constants. When ξ 1 = σ = h = 0, the equation (1) reduces to a nonlinear wave equation which has been extensively studied and several results concerning existence and nonexistence have been established [3,[9][10][11]13]. When ξ 0 ,ξ 1 = 0, σ = h = 0, the equation in (1) reduces to the well-known Kirchhoff equation which has been introduced in [14] in order 2000 Mathematics Subject Classification: 35L20, 35B40, 45K05.

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On Global Solutions and Energy Decay for the Wave Equations of Kirchhoff Type with Nonlinear Damping Terms

Cited by 102 publications

References 14 publications

On solutions of quasilinear wave equations with nonlinear damping terms

On solutions of quasilinear wave equations with nonlinear damping terms

On Solvability of the Dissipative Kirchhoff Equation With Nonlinear Boundary Damping

Exponential stability and blow up for a problem with Balakrishnan–Taylor damping

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