Thresholds for low regularity solutions to wave equations with structural damping

Fukushima, Tomonori; Ikehata, Ryo; Michihisa, Hironori

doi:10.1016/j.jmaa.2020.124669

Cited by 14 publications

(9 citation statements)

References 22 publications

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“…Proof. One can find the proof of the estimates ( 19) and (20) in [10,7] except for the case (20) with pn, j, αq P t1u ˆt0u ˆp0, 1 2 s. On the other hand, applying the same arguments as in [10,7] we can easily obtain the following estimates for n " 1:…”

Section: 2mentioning

confidence: 86%

“…In this section, at first, we are going to prove the decay estimates for the fundamental solutions to (2). Some of them are already derived in the previous papers [10,7]. However, for the convenience of the readers, we rephrase them to our notation.…”

Section: The Treatment Of the Linear Equationmentioning

confidence: 99%

“…Moreover, they also proved that the low frequency part is dominant and the solution is approximated by the diffusion wave as t Ñ 8. Quite recently, Fukushima-Ikehata-Michihisa [7] have studied the higher order asymptotic expansion of solutions to (2), which corresponds to the additional regularity assumptions on the initial data.…”

Section: Introductionmentioning

confidence: 99%

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Global existence results for semi-linear structurally damped wave equations with nonlinear convection

Dao¹,

Takeda²

2021

Preprint

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In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term νp´∆q 2 ut, where ν ą 0 is a constant. As being mentioned in [8,10], the linear principal part brings both the diffusion phenomenon and the regularity loss of solutions. This implies that, for the nonlinear problems, the choice of solution spaces plays an important role to obtain the global solutions with the sharp decay properties in time. Our main purpose of this paper is to prove the global (in time) existence of solutions for the small data and their decay properties for the supercritical nonlinearities.

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Section: 2mentioning

confidence: 86%

Section: The Treatment Of the Linear Equationmentioning

confidence: 99%

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Global existence results for semi-linear structurally damped wave equations with nonlinear convection

Dao¹,

Takeda²

2021

Preprint

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“…The profiles include an oscillation property coming from the condition θ > 1 2 . In this connection, even if θ is large enough, say θ > 1, the equation (1.7) does not have any regularity-loss structures, while in the case when θ > 1 the equation (1.5) with σ = 1 has such a regularity-loss structure as was pointed out in [26,20]. A research in [26] has been motivated by an interesting paper by [22].…”

Section: Introductionmentioning

confidence: 85%

Double diffusion structure of logarithmically damped wave equations with a small parameter

Piske¹,

Charão²,

Ikehata³

2021

Preprint

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We consider a wave equation with a nonlocal logarithmic damping depending on a small parameter 0 < θ < 1 2 . This research is a counter part of that was initiated by Charão-D'Abbicco-Ikehata considered in [5] for the large parameter case θ > 1 2 . We study the Cauchy problem for this model in R n to the case θ ∈ (0, 1 2 ), and we obtain an asymptotic profile and optimal estimates in time of solutions as t → ∞ in L 2 -sense. An important discovery in this research is that in the case when n = 1, we can present a threshold θ * = 1 4 of the parameter θ ∈ (0, 1 2 ) such that the solution of the Cauchy problem decays with some optimal rate for θ ∈ (0, θ * ) as t → ∞, while the L 2 -norm of the corresponding solution never decays for θ ∈ [θ * , 12 ), and in particular, in the case θ ∈ [θ * , 1 2 ) it shows an infinite time L 2 -blow up of the corresponding solutions. The former (i.e., θ ∈ (0, θ * ) case) indicates an usual diffusion phenomenon, while the latter (i.e., θ ∈ [θ * , 1 2 ) case) implies, so to speak, a singular diffusion phenomenon. Such a singular diffusion in the one dimensional case is a quite novel phenomenon discovered through our new model produced by logarithmic damping with a small parameter θ. It might be already prepared in the usual structural damping case such as (−∆) θ ut with θ ∈ (0, 1/2), however unfortunately nobody has ever just pointed out even in the structural damping case.

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“…In [24] and [20], the so-called regularity-loss structure of the solution in the high frequency zone can be studied to (1.6) with σ = 1 and θ > 1, and these researches are strongly inspired from the abstract theory due to [21].…”

Section: Introductionmentioning

confidence: 99%

A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

Charão¹,

Piske²,

Ikehata³

2021

Preprint

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We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in R n , and study the asymptotic profile and optimal decay rates of solutions as t → ∞ in L 2 -sense. The operator L considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charão-Ikehata [5]. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.

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Thresholds for low regularity solutions to wave equations with structural damping

Cited by 14 publications

References 22 publications

Global existence results for semi-linear structurally damped wave equations with nonlinear convection

Global existence results for semi-linear structurally damped wave equations with nonlinear convection

Double diffusion structure of logarithmically damped wave equations with a small parameter

A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

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