Decay of solutions of a nonlinear hyperbolic system in noncylindrical domain

Rabello, Tania Nunes

doi:10.1155/s0161171294000815

Cited by 6 publications

(3 citation statements)

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“…Notice that the proof of Lemma 3.1, developed here, it is different from the one given by Nakao and Narazaki [16] and Rabelo [17]. Note also that is crucial in proofs above the fact that Q is increasing.…”

Section: Let Us Denote By O An Open Bounded and Nonempty Subset Ofmentioning

confidence: 78%

Remarks on equations of Benjamin–Bona–Mahony type

Límaco

Clark

Medeiros

2007

Journal of Mathematical Analysis and Applications

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We investigate an initial-boundary value problem for equations of Benjamin-Bona-Mahony (BBM) type in two different physical situations. In the first, the mixed problem is considered on a cylinder domain Q of R n × R t . In the second one, the mixed problem is studied inside of an increasing noncylindrical domain Q of R n × R t . In both situations we show the existence of a unique nonlocal solution. In cylindrical case it is proved the existence of weak and strong solutions, regularity of strong solutions, and in noncylindrical case weak solutions. One of the goals of this paper is to show that the noncylindrical problem is well-posed by using the penalty method idealized by Lions [J.L. Lions, Une remarque sur les problèmes d'évolution non linéaires dans des domaines non cylindriques, Rev. Roumaine Math. Pures Appl. 9 (1964) 11-18].

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Section: Let Us Denote By O An Open Bounded and Nonempty Subset Ofmentioning

confidence: 78%

Remarks on equations of Benjamin–Bona–Mahony type

Límaco

Clark

Medeiros

2007

Journal of Mathematical Analysis and Applications

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“…Later, this technique is also used to study the decay of solutions and it is sufficient to make an estimate for approximate solutions independent of parameters (see e.g. [13,14]). The advantage of this method is that there are not many requirements on geometry of domains and the fundamental logarithmic Sobolev inequality developed in fixed area can be applied directly.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Logarithmic wave equations in non-cylindrical domains

Liu

2021

Preprint

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This paper is devoted to studying a type of logarithmic wave equation in non-cylindrical domains. Firstly, by the penalty method, we prove the existence of weak solutions to such kind of equations. Secondly, different from the dissipative wave equation, the energy defined in this problem is not always positive. Thus, some suitable initial data are selected to let the energy be positive. Finally, by a difference inequality, we derive a exponential decay estimate for the positive energy.

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“…The penalty method was also employed by Cooper-Bardos [2] for the same operator u → u − ∆u + |u| ρ u for noncylindrical domains Q, but which are "time like" instead of increasing as in Lions [8]. In Medeiros [11,12], it is considered the operator u → u − ∆u + F(u) by penalty method in increasing domains, see also Strauss [6], Cooper-Medeiros [3], Inoue [7], Nakao-Narazaki [14], and Rabello [5]. We employ certain techniques for cylindrical domains as in Hosoya and Yamada [4].…”

Section: ∇U(x T)mentioning

confidence: 99%