General decay of solutions of a wave equation with a boundary control of memory type

Messaoudi, Salim A.; Soufyane, Abdelaziz

doi:10.1016/j.nonrwa.2009.10.013

Cited by 44 publications

(36 citation statements)

References 17 publications

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“…In both cases the solution decays with the same rate of the relaxation function. This result was later generalized by Messaoudi and Soufyane [60], where relaxation functions of general decay type were considered. Alabau-Boussouira [1] used some weighted integral inequalities and convexity arguments and proved a semi-explicit formula which leads to decay rates of the energy in terms of the behavior of the nonlinear feedback near the origin, for which the optimal exponential and polynomial decay rate estimates are only special cases.…”

Section: Decay In the Case Of Constant Exponentsmentioning

confidence: 80%

On wave equation: review and recent results

Messaoudi

Talahmeh

2017

Arab. J. Math.

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The aim of this paper is to give an overview of results related to nonlinear wave equations during the last half century. In this regard, we present results concerning existence, decay and blow up for classical nonlinear equations. After that, we discuss briefly some important results of the variable-exponent Lebesgue and Sobolev spaces. Results related to nonexistence and blow up for wave equations with non-standard nonlinearities (nonlinearities involving variable exponents) are given in more detail. Finally, we present some recent decay and blow up results together with their proofs. Mathematics Subject Classification

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Section: Decay In the Case Of Constant Exponentsmentioning

confidence: 80%

On wave equation: review and recent results

Messaoudi

Talahmeh

2017

Arab. J. Math.

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“…Although the subject is important, there are few mathematical results in the presence of the nonlinearity given by h(∇u), see [24][25][26]. In light of this and previous articles [17,22], it is interesting to investigate whether we still have the similar general decay result as in [17] for nondissipative distributed system (1.1) with the memory-type damping acting on a part of the boundary. Hence, the main purpose of this article is to answer the above question for system (1.1)-(1.4).…”

Section: Introductionmentioning

confidence: 82%

“…By adopting and modifying the method proposed by Messaoudi and Soufyane in 2010 [17], we establish a general decay result, from which the usual exponential and polynomial decay rate are only special cases. Further, our result allows certain kernels which are not necessarily of exponential or polynomial decay.…”

Section: Decay Of Solutionsmentioning

confidence: 99%

“…Hence, the main purpose of this article is to answer the above question for system (1.1)-(1.4). Consequently, by following the arguments close to the one in [17] with necessary modification required the nature of our problem, we establish a general decay result which includes the usual exponential and polynomial decay rates. Furthermore, our results allow a larger class of relax functions which are not necessarily of exponential and polynomial decay.…”

Section: Introductionmentioning

confidence: 96%

“…They established general decay rate results, from which the usual exponential and polynomial decay rates are only special cases. Recently, Messaoudi and Soufyane [17] studied the following problem:…”

Section: Introductionmentioning

confidence: 99%

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General decay for a wave equation of Kirchhoff type with a boundary control of memory type

2011

Bound Value Probl

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A nonlinear wave equation of Kirchhoff type with memory condition at the boundary in a bounded domain is considered. We establish a general decay result which includes the usual exponential and polynomial decay rates. Furthermore, our results allow certain relaxation functions which are not necessarily of exponential and polynomial decay. This improves earlier results in the literature. MSC: 35L05; 35L70; 35L75; 74D10.

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Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions

Liu

Chen

2015

Mathematische Nachrichten

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In this paper, we prove the existence and general energy decay rate of global solution to the mixed problem for nondissipative multi-valued hyperbolic differential inclusionswith memory boundary conditions on a portion of the boundary and acoustic boundary conditions on the rest of it. For the existence of solutions, we prove the global existence of weak solution by using Galerkin's method and compactness arguments. For the energy decay rates, we first consider the general nonlinear case of h satisfying a smallness condition, and prove the general energy decay rate by using perturbed modified energy method. Then, we consider the linear case of h: h(∇u) = −∇φ · (a∇u) and prove the general decay estimates of equivalent energy.

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General decay of solutions of a wave equation with a boundary control of memory type

Cited by 44 publications

References 17 publications

On wave equation: review and recent results

On wave equation: review and recent results

General decay for a wave equation of Kirchhoff type with a boundary control of memory type

Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions

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