A tracing of the fractional temperature field

Shi, Shaoguang; Xiao, Jie

doi:10.1007/s11425-016-0494-6

Cited by 17 publications

(9 citation statements)

References 26 publications

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“…)) be the parabolic maximal function of a nonnegative Radon measure µ on R n+1 + . We show the embedding for p > q inspired by some ideas from [18], which need first the following L p -boundedness of Mµ.…”

Section: Embeddings Ofmentioning

confidence: 99%

“…For fractional diffusion equations, motivated by Xiao [20], Zhai in [23] explored the embeddings of the homogeneous Sobolev space Ẇβ,p (R n ) into the Lebesgue space L q (R n+1 + , µ). By using the L p −capacities associated with the fractional heat kernel, Chang-Xiao in [8] and Shi-Xiao in [18] established embeddings similar to (1.4). This article will be organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Embeddings of Function Spaces via the Caffarelli-Silvestre Extension, Capacities and Wolff potentials

Li¹,

Shi²,

Hu³

et al. 2020

Preprint

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Let P α f (x, t) be the Caffarelli-Silvestre extension of a smooth function f (x) : R n → R n+1The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure+ , µ). On one hand, these embeddings will be characterized by using a newly introduced L p −capacity associated with the Caffarelli-Silvestre extension. In doing so, the mixed norm estimates of P α f (x, t), the dual form of the L p −capacity, the L p −capacity of general balls, and a capacitary strong type inequality will be established, respectively. On the other hand, when p > q > 1, these embeddings will also be characterized in terms of the Hedberg-Wolff potential of µ. Secondly, we characterize a nonnegative measure µ on R n+1 + such that f (x) → P α f (x, t) induces bounded embedding from the homogeneous Sobolev spaces Ẇβ,p (R n ) to the L q (R n+1 + , µ) in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.

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Section: Embeddings Ofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Embeddings of Function Spaces via the Caffarelli-Silvestre Extension, Capacities and Wolff potentials

Li¹,

Shi²,

Hu³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…To make sense for the integrals, we require , , where and For more background on the fractional Laplacian operator , we refer to [1–4]. We mention that there are also several applications involving the fractional Laplacian in mathematical physics [5–8], finance [9], image processing [10], and so on.…”

Section: Introductionmentioning

confidence: 99%

Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian

Meng

Yang

2018

J Inequal Appl

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In this paper, we consider the following nonlinear Schrödinger system involving the fractional Laplacian operator: where . When Ω is the unit ball or , we prove that the solutions are radially symmetric and decreasing. When Ω is the parabolic domain on , we prove that the solutions are increasing. Furthermore, if Ω is the , then we also derive the nonexistence of positive solutions to the system on the half-space. We assume that the nonlinear terms f, g and the solutions u, v satisfy some amenable conditions in different cases.

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“…In [45], Zhai obtained a Strichartz type estimate for S α F and proved the global existence and uniqueness of regular solutions for the generalized Naiver-Stokes equation. The papers [27,5,38] explored some analytic-geometric properties of the regularity and the capacity associated with ∂ t + (−∆) α .…”

mentioning

confidence: 99%

Fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations

Huang

et al. 2020

Nonlinear Analysis

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Let (M, d, µ) be a metric measure space with upper and lower densities:where β, β ⋆ are two positive constants which are less than or equal to the Hausdorff dimension of M. Assume that pt(·, ·) is a heat kernel on M satisfying Gaussian upper estimates and L is the generator of the semigroup associated with pt(·, ·). In this paper, via a method independent of Fourier transform, we establish the decay estimates for the kernels of the fractional heat semigroup {e −tL α }t>0 and the operators {L θ/2 e −tL α }t>0, respectively. By these estimates, we obtain the regularity for the Cauchy problem of the fractional dissipative equation associated with L on (M, d, µ). Moreover, based on the geometric-measure-theoretic analysis of a new L p -type capacity defined in M × (0, ∞), we also characterize a nonnegative Randon measureis the weak solution of the fractional diffusion equation (∂t + L α )u(t, x) = 0 in M × (0, ∞) subject to u(0, x) = f (x) in M.

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A tracing of the fractional temperature field

Abstract: Abstract. This note is devoted to a study of L q -tracing of the fractional temperature field u(t, x) -the weak solution of the fractional heat equation

Cited by 17 publications

References 26 publications

Embeddings of Function Spaces via the Caffarelli-Silvestre Extension, Capacities and Wolff potentials

Embeddings of Function Spaces via the Caffarelli-Silvestre Extension, Capacities and Wolff potentials

Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian

Fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations

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