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Heat equation with a singular potential on the boundary and the Kato inequality

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Abstract

This paper is concerned with the heat equation in the half-space ℝ N+ with the singular potential function on the boundary,

$\left\{ \begin{gathered} \frac{\partial } {{\partial t}}u - \Delta u = 0\operatorname{in} \mathbb{R}_ + ^N \times (0,T), \hfill \\ \frac{\partial } {{\partial x_N }}u + \frac{\omega } {{|x|}}u = 0on\partial \mathbb{R}_ + ^N \times (0,T), \hfill \\ u(x,0) = u_0 (x) \geqslant ()0in\mathbb{R}_ + ^N , \hfill \\ \end{gathered} \right. $
((P))

where N ≥ 3, ω > 0, 0 < T ≤ ∞, and u 0 ∈ C 0(ℝ N+ ). We prove the existence of a threshold number ω N for the existence and the nonexistence of positive solutions of (P), which is characterized as the best constant of the Kato inequality in ℝ N+ .

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Authors and Affiliations

  1. Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan

    Kazuhiro Ishige

  2. Faculty of Symbiotic Systems Science, Fukushima University, Kanayagawa, Fukushima, 960-1269, Japan

    Michinori Ishiwata

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  1. Kazuhiro Ishige
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  2. Michinori Ishiwata
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Correspondence to Kazuhiro Ishige.

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Ishige, K., Ishiwata, M. Heat equation with a singular potential on the boundary and the Kato inequality. JAMA 118, 161–176 (2012). https://doi.org/10.1007/s11854-012-0032-4

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  • DOI: https://doi.org/10.1007/s11854-012-0032-4

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