Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction

Said‐Houari, Belkacem; Nascimento, Flávio A. Falcão

doi:10.3934/cpaa.2013.12.375

Cited by 9 publications

(3 citation statements)

References 22 publications

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“…a large number of papers on this subject are available in the literature, where various decay estimates for the solutions of (4) were obtained; see in this regard [6,8,9,22,27,28,29,30] and the references cited therein. For the particular case of the wave equation with (internal or boundary) finite memory; see [2,1,7,43] and [44]- [52]. See also [21] for the wave equation with complementary finite and infinite memories, and [20] for the Timoshenko systems with complementary finite memory and nonlinear frictional damping.…”

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confidence: 99%

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Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay

Guesmia

Tatar

2015

Communications on Pure &Amp; Applied Analysis

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International audienceIn this paper, we consider a class of second order abstract linear hyperbolic equations with infinite memory and distributed time delay. Under appropriate assumptions on the infinite memory and distributed time delay convolution kernels, we prove well-posedness and stability of the system. Our estimation shows that the dissipation resulting from the infinite memory alone guarantees the asymptotic stability of the system in spite of the presence of distributed time delay. The decay rate of solutions is found explicitly in terms of the growth at infinity of the infinite memory and the distributed time delay convolution kernels. An application of our approach to the discrete time delay case is also given

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confidence: 99%

“…then, for any U 0 ∈ H , there exist positive constants δ 1 and δ 2 (depending on U 0 H , a, b, d, g 0 , g(0), ξ, µ and δ 0 ) such that the weak solution of ( 12) associated with (134)-( 135) satisfies (41) if lim t→+∞ ξ(t) > 0, and it satisfies (43) if lim t→+∞ ξ(t) = 0.…”

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confidence: 99%

Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay

Guesmia

Tatar

2015

Communications on Pure &Amp; Applied Analysis

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“…when the operator

u_{t} - Δ u

in is replaced by the wave operator

u_{t t} - Δ u

. Some related problems concerning wave equations with nonlinear damping and source terms have been considered in . In particular, Cavalcanti et al.…”

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confidence: 99%

Global existence, asymptotic stability and blow‐up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

Zhang

Liu

2019

Mathematische Nachrichten

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In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms.We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively. K E Y W O R D S blow-up, decay, generalized Boussinesq equation, global existence, nonlinear boundary condition M S C ( 2 0 1 0 ) 35K05, 35K61, 35K70

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General decay rates for the wave equation with mixed-type damping mechanisms on unbounded domain with finite measure

Silva

Nascimento

Rodrigues

2015

Z. Angew. Math. Phys.

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Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction

Cited by 9 publications

References 22 publications

Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay

Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay

Global existence, asymptotic stability and blow‐up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

General decay rates for the wave equation with mixed-type damping mechanisms on unbounded domain with finite measure

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