2015
DOI: 10.1016/j.jmaa.2015.06.051
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On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions

Abstract: Please cite this article in press as: M.R. Ebert et al., On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions, AbstractWe consider the energy estimates for the wave equation with time dependent propagation speed. It is known that the asymptotic behavior of the energy is determined by the interactions of the properties of the propagation speed: smoothness, oscillation and the difference from an auxiliary function. The main purpose of the article is… Show more

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Cited by 13 publications
(4 citation statements)
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“…for suitable real constants µ 1 , µ 2 , H, and α ∈ (0, 1). Many papers have been devoted to equations of this type (see for example the classical references [4,7] or the more recent ones [1,2,3,5,6,8,9,10,12,13,15,16]). Let us briefly discuss the previous results that are more relevant to our presentation.…”
Section: Introductionmentioning
confidence: 99%
“…for suitable real constants µ 1 , µ 2 , H, and α ∈ (0, 1). Many papers have been devoted to equations of this type (see for example the classical references [4,7] or the more recent ones [1,2,3,5,6,8,9,10,12,13,15,16]). Let us briefly discuss the previous results that are more relevant to our presentation.…”
Section: Introductionmentioning
confidence: 99%
“…If a(t) does not satisfy (1.12), then GEC does not hold in general. However, if the initial data is a function of the Gevrey class of suitable order and (1.9) holds, then there exists a positive constant C(u 0 , u 1 ), which depends on not only the initial energy E(0) but also a norm of the initial data in the Gevrey class, such that the total energy E(t) is bounded as follows (see [4]):…”
Section: Resultsmentioning
confidence: 99%
“…The conditions (M1)-(M3) seem to be artificial, but they can be actually natural from the viewpoint of previous studies. (M1) and (M2) correspond to stabilization property and C 2 -property with very fast oscillation, respectively, which were introduced in [3,7] for the energy estimate of the wave equation with time dependent propagation speed. (2.6) and (2.8) are corresponding to generalized zero mean condition, which was introduced in [8].…”
Section: Main Theoremmentioning
confidence: 99%