Large time behavior of solutions of the heat equation with inverse square potential

Ishige, Kazuhiro; Mukai, Asato

doi:10.3934/dcds.2018176

Cited by 8 publications

(8 citation statements)

References 27 publications

(53 reference statements)

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“…where d = N − 2α 1 (λ) > 2 corresponds to the spatial dimension. See [1,3,11,13] for the analysis of the radial heat equation. We first consider (2.2) with a nonnegative and nontrivial initial value w 0 (r), Lemma 2.1.…”

Section: Radial Heat Equationmentioning

confidence: 99%

On the evolution equation with a dynamic Hardy-type potential

et al. 2021

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Motivated by the celebrated paper of Baras and Goldstein (1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, non-existence and uniqueness of solutions, and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.

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Section: Radial Heat Equationmentioning

confidence: 99%

On the evolution equation with a dynamic Hardy-type potential

et al. 2021

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“…Nonnegative Schrödinger operators and their heat semigroups appear in various fields and have been studied intensively by many authors since the pioneering work due to Simon [29] (see e.g., [3]- [6], [9]- [14], [17]- [28], [31]- [33] and references therein). The inverse square potential is a typical one appearing in the study of the Schrödinger operators and it also arises in the linearized analysis for nonlinear diffusion equations, in particular, in solid-fuel ignition phenomena which can be modeled by ∂ t u = ∆u + e u .…”

Section: Introductionmentioning

confidence: 99%

“…(See Section 2.2.) In this paper, combing the arguments in [9]- [12] and [14], we study the large time behavior of H(u(t)) in the cases (S), (S * ) and (C) and reveal the relationship between the large time behavior of H(u(t)) and the corresponding harmonic functions. We remark that L V is not necessarily subcritical.…”

Section: Introductionmentioning

confidence: 99%

“…Our arguments in this paper are based on [14], where the precise description of the large time behavior of the solution of (1.1) was discussed under condition (V). Applying the arguments in [14], we modify the arguments in [9]- [12] and study the large time behavior of the hot spots. We study the following subjects when the hots spots tend to the space infinity as t → ∞:…”

Section: Introductionmentioning

confidence: 99%

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Hot spots of solutions to the heat equation with inverse square potential

Ishige

Kabeya

Mukai

2018

Applicable Analysis

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We investigate the large time behavior of the hot spots of the solution to the Cauchy problemwhere ϕ ∈ L 2 (R N , e |x| 2 /4 dx) and V = V (r) decays quadratically as r → ∞. In this paper, based on the arguments in [K. Ishige and A. Mukai, preprint (arXiv:1709.00809)], we classify the large time behavior of the hot spots of u and reveal the relationship between the behavior of the hot spots and the harmonic functions for −∆ + V .

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“…for compact sets K ⊂ R N \ {0} and t > 0 (see [17] and [19]). We study the behavior of derivatives of v k,i by using the radially symmetry of v k,i , and obtain upper decay estimates of ∥∇ α e −tH ∥ (L p,σ →L q,θ ) .…”

mentioning

confidence: 99%

Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II

Ishige¹,

Tateishi²

2022

DCDS

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<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ H: = -\Delta+V $\end{document}</tex-math></inline-formula> be a nonnegative Schrödinger operator on <inline-formula><tex-math id="M2">\begin{document}$ L^2({\bf R}^N) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ N\ge 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ V $\end{document}</tex-math></inline-formula> is a radially symmetric inverse square potential. Let <inline-formula><tex-math id="M5">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula> be the operator norm of <inline-formula><tex-math id="M6">\begin{document}$ \nabla^\alpha e^{-tH} $\end{document}</tex-math></inline-formula> from the Lorentz space <inline-formula><tex-math id="M7">\begin{document}$ L^{p, \sigma}({\bf R}^N) $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M8">\begin{document}$ L^{q, \theta}({\bf R}^N) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M9">\begin{document}$ \alpha\in\{0, 1, 2, \dots\} $\end{document}</tex-math></inline-formula>. We establish both of upper and lower decay estimates of <inline-formula><tex-math id="M10">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula> and study sharp decay estimates of <inline-formula><tex-math id="M11">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula>. Furthermore, we characterize the Laplace operator <inline-formula><tex-math id="M12">\begin{document}$ -\Delta $\end{document}</tex-math></inline-formula> from the view point of the decay of <inline-formula><tex-math id="M13">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula>.</p>

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Large time behavior of solutions of the heat equation with inverse square potential

Cited by 8 publications

References 27 publications

On the evolution equation with a dynamic Hardy-type potential

On the evolution equation with a dynamic Hardy-type potential

Hot spots of solutions to the heat equation with inverse square potential

Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II

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