Global Solutions for a System of Klein-Gordon Equations with Memory

Mognon, Angela; Andrade, Doherty

doi:10.5269/bspm.v21i1-2.7512

Cited by 10 publications

(3 citation statements)

References 8 publications

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“…Further generalizations are also given in [27,28] by using Galerkin method, where the authors consider nonlinearities of the form |v| ρ+2 |u| ρ u, |u| ρ+2 |v| ρ v and some hypotheses about the coefficient ρ, which are related to the dimension d of the space. Related to global existence and uniqueness of solutions we also have the article [1], where the authors prove the existence of solutions to a Klein-Gordon system with memory and nonlinearities similar to those considered in [27,28]. Another generalization for this system can be founded in [12], where the authors consider a k × k system os Klein-Gordon equations with acoustic boundary conditions.…”

Section: Remarkmentioning

confidence: 99%

Uniform stabilization of the Klein-Gordon system

Cavalcanti¹,

Delatorre²,

Soares³

et al. 2020

Communications on Pure &Amp; Applied Analysis

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We consider the Klein-Gordon system posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gérard [5]. Although the present problem has some similarity to the reference [6] it is important to mention that due to the Kelvin-Voigt dissipation character associated with the nonlinearity of the problem the approach used is completely new, which is the main purpose of this paper.

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Section: Remarkmentioning

confidence: 99%

Uniform stabilization of the Klein-Gordon system

Cavalcanti¹,

Delatorre²,

Soares³

et al. 2020

Communications on Pure &Amp; Applied Analysis

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show abstract

“…The unknowns u and v represent the displacements of waves. This system can be considered as a generalization of the well-known Klein-Gordon system that appears in the quantum field theory [3,29],…”

mentioning

confidence: 99%

“…al [5] considered the following problem 5) and established the existence of the solution and proved an explicit and general decay rate results, using the multiplier method and some properties of the convex functions. For viscoelastic systems, Andrade and Mognon [3] treated the following problem…”

mentioning

confidence: 99%