The diffusion phenomenon for dissipative wave equations in metric measure spaces

Taylor, Montgomery; Todorova, Grozdena

doi:10.1016/j.jde.2020.07.018

Cited by 2 publications

(6 citation statements)

References 30 publications

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“…Remark 2.6. Assumption 2.5 is actually needed only to prove the critical Sobolev inequality (41) in Lemma A.3 below. To simplify the description, we always impose this assumption in the semilinear problem (see also Remark 2.11 below).…”

Section: 2mentioning

confidence: 99%

“…The first purpose of this paper is to prove the Matsumura type estimates for solutions to the linear problem (5). The Matsumura type estimates have already beed studied by [33,41] in the abstract setting. However, our approach is different from these and is based on a direct argument analogous to the Fourier analysis; thereby, it is applicable to several evolution equations (see Remark 2.3 below).…”

mentioning

confidence: 99%

“…and Chill and Haraux [6] improved the convergence rates in the asymptotic behavior in [18]. The Matsumura type estimates for (5) have been studied under some assumptions on X, µ, and A (see Radu, Todorova and Yordanov [33], and Taylor and Todorova [41]). The study of asymptotic behavior for (5) has been further developed (see [28,34,39]).…”

mentioning

confidence: 99%

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Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces

Ikeda,

Taniguchi,

Wakasugi

2024

EECT

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The purpose of this paper is to prove the small data global existence of solutions to the semilinear damped wave equation ∂ 2 t u + Au + ∂tu = |u| p−1 u on a measure space X with a self-adjoint operator A on L 2 (X). Under a certain decay estimate on the corresponding heat semigroup, we establish the linear estimates which generalize the so-called Matsumura estimates. Our approach is based on a direct spectral analysis analogous to the Fourier analysis. The self-adjoint operators treated in this paper include some important examples such as the Laplace operators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary open set, the Robin Laplacian on an exterior domain, the Schrödinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian.

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Section: 2mentioning

confidence: 99%

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confidence: 99%

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Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces

Ikeda,

Taniguchi,

Wakasugi

2024

EECT

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“…and Chill and Haraux [6] improved the convergence rates in the asymptotic behavior in [16]. The Matsumura type estimates for (1.4) have been studied under some assumptions on X, µ, and A (see Radu, Todorova and Yordanov [30], and Taylor and Todorova [37]). The study of asymptotic behavior for (1.4) has been further developed (see [26,31,35]).…”

Section: Introductionmentioning

confidence: 99%

“…The first purpose of this paper is to prove the Matsumura type estimates for solutions to the linear problem (1.4). The Matsumura type estimates have already beed studied by [30,37] in the abstract setting. However, our approach is different from these and is based on a direct argument analogous to the Fourier analysis; thereby, it is applicable to several evolution equations (see Remark 2.3 below).…”

Section: Introductionmentioning

confidence: 99%

Global existence and asymptotic behavior for nonlinear damped wave equations on measure spaces

Ikeda¹,

Taniguchi²,

Wakasugi³

2021

Preprint

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The purpose of this paper is to prove the small data global existence of solutions to the nonlinear damped wave equationu on a measure space X with a self-adjoint operator A on L 2 (X). Under a certain decay estimate on the corresponding heat semigroup, we establish the linear estimates which generalize the so-called Matsumura estimates. Our approach is based on a direct spectral analysis analogous to the Fourier analysis. The self-adjoint operators treated in this paper include some important examples such as the Laplace operators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary open set, the Robin Laplacian on an exterior domain, the Schrödinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian.

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The diffusion phenomenon for dissipative wave equations in metric measure spaces

Cited by 2 publications

References 30 publications

Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces

Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces

Global existence and asymptotic behavior for nonlinear damped wave equations on measure spaces

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