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

2020

Preprint

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Section: Introductionmentioning

confidence: 99%

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

Ishige¹,

2020

Preprint

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“…for compact sets K ⊂ R N \ {0} and t > 0 (see [19] and [21]). 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 ) .…”

Section: Introductionmentioning

confidence: 99%

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

Ishige¹,

2020

Preprint

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Let H := −∆ + V be a nonnegative Schrödinger operator on L 2 (R N ), where N ≥ 2 and V is a radially symmetric inverse square potential. Let ∇ α e −tH (L p,σ →L q,θ ) be the operator norm of ∇ α e −tH from the Lorentz space L p,σ (R N ) to L q,θ (R N ), where α ∈ {0, 1, 2, . . . }. We establish both of upper and lower decay estimates of ∇ α e −tH (L p,σ →L q,θ ) and study sharp decay estimates of ∇ α e −tH (L p,σ →L q,θ ) . Furthermore, we characterize the Laplace operator −∆ from the view point of the decay of ∇ α e −tH

“…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,

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>