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Global solutions of the heat equation with a nonlinear boundary condition

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Abstract

We consider the heat equation with a nonlinear boundary condition,

$$(P) \left\{\begin{array}{ll} \partial_t u = \Delta u, & x \in \Omega, \quad t > 0, \\ \partial_\nu u=u^p, & x \in \partial \Omega,\quad t > 0,\\ u (x,0) = \phi (x),& x\in\Omega, \end{array}\right.$$

where \({\Omega = \{x = (x^{\prime},x_N) \in {\bf R}^{N} : x_N > 0\}, N \ge 2, \partial_t = \partial{/}\partial t , \partial_\nu = -\partial{/}\partial x_{N}}\), p > 1 + 1/N, and (N − 2)p < N. In this paper we give a complete classification of the large time behaviors of the nonnegative global solutions of (P).

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Authors and Affiliations

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

    Kazuhiro Ishige & Tatsuki Kawakami

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  1. Kazuhiro Ishige
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  2. Tatsuki Kawakami
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Correspondence to Kazuhiro Ishige.

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Communicated by Y.Giga.

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Ishige, K., Kawakami, T. Global solutions of the heat equation with a nonlinear boundary condition. Calc. Var. 39, 429–457 (2010). https://doi.org/10.1007/s00526-010-0316-4

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