Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition
type: Item
instance of: scholarly article
title: Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition
zbMATH Open document ID: 1096.35055
author: Kazuhiro Ishige
published in: Journal of Differential Equations
publication date: 2005-06-01T00:00:00Z
review text: This paper concerns the very precise movement as \(t \to \infty\) of hot spots \(H(t)\) of the solution \(u = u(x,t)\) of the Cauchy-Neumann problem for the heat equation in the exterior domain of a ball in \(\mathbb R^N\): \[ \partial_t u= \Delta u \text{ in } \Omega \times (0,\infty), \partial_\nu u= 0 \text{ on } \partial \Omega \times (0,\infty), \text{ and } u(x,0) = \phi(x) \text{ in } \Omega, \] where \(\Omega = \{ x \in \mathbb R^N\;: | x| > L\}\) for \(L > 0,\) \(\nu =\nu(x)\) is the outer unit normal vector to \(\partial\Omega\) at \(x \in \partial\Omega\), and where initial data \(\phi = \phi(x)\) satisfies \[ \int_\Omega \phi(x)\,dx > 0\;\text{ and } \int_\Omega | \phi(x)| ^2 e^{\lambda| x| ^2} dx < \infty \text{ for some } \lambda > 0. \] \(H(t)\) is defined by \[ H(t) = \{ x \in \overline{\Omega} : u(x,t) = \max_{y \in \overline{\Omega}} u(y,t) \}. \] The author finds that the point \(A(t)\) given by \[ A(t) = \int_\Omega xu(x,t)\left(1+\frac {L^N}{N-1} | x| ^{-N}\right) dx \Big/ \int_\Omega u(x,t) \,dx\quad \text{ for } t > 0 \] is stationary in \(t\), while the center of mass of \(u(\cdot,t)\) is not necessarily stationary. Let \(A_\phi\) be defined by replacing \(u(x,t)\) with \(\phi(x)\) in the definition of \(A(t)\). This paper gives the following interesting results: (1) If \(A_\phi\) is in \(\mathbb R^N \setminus \overline{\Omega}\), then after a finite time \(H(t)\) is contained in \(\partial\Omega\). (2) Assume that \(A_\phi \not=0.\) Put \(x_\infty = L A_\phi/| A_\phi| \) if \(A_\phi \in \mathbb R^N \setminus \overline{\Omega}\) and \(x_\infty = A_\phi\) if \(A_\phi \in \overline{\Omega}.\) Then \(\lim_{t \to \infty} \sup\{ | x_\infty-y| : y \in H(t) \} = 0.\) In order to prove these results, the author studies the asymptotic behavior of the radial solution \(v_k\) of the Cauchy-Neumann problem: \[ \partial_t v_k= \Delta v_k -\frac {\omega_k}{| x| ^2} v_k \text{ in } \Omega \times (0,\infty), \partial_\nu v_k= 0 \text{ on } \partial \Omega \t…
reviewed by: Shigeru Sakaguchi
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 2173050
zbMATH Keywords: heat equation
zbMATH Keywords: hot spots
zbMATH Keywords: Cauchy-Neumann problem
zbMATH Keywords: exterior domain
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: eigenvalues of the Laplace-Beltrami operator
MaRDI profile type: Publication
full work available at URL: j.jde.2004.11.002
OpenAlex ID: W1995318548
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DOI: 10.1016/J.JDE.2004.11.002
description: scientific article; zbMATH DE number 2173050