Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition
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: \[ \partial_t v_k= \Delta v_k -\frac {\omega_k}{| x| ^2} v_k \text{ in } \Omega \times (0,\infty), \partial_\nu v_k= 0 \text{ on } \partial \Omega \times (0,\infty), \text{ and } v_k(x,0) = \phi(x) \text{ in } \Omega, \] where \(k \in \mathbb N \cup \{0\},\) \(\phi\) is a radial function in \(L^2(\Omega,\rho\, dx)\) with \(\rho(x) = \exp(| x| ^2/4)\), and where \(\{\omega_k\} \) are the eigenvalues of the Laplace-Beltrami operator on the \((N-1)\)-dimensional unit sphere \(\mathbb S^{N-1}\). Moreover, as in \textit{N. Mizoguchi, H. Ninomiya} and \textit{E. Yanagida} [J. Math. Anal. Appl. 218, 495--518 (1998; Zbl 0914.35064)], he studies a rescaled function \(w_k\) of \(v_k\) defined by \[ w_k(y,s) = (1+t)^{\frac {N+k}2} v_k(x,t),\;y = (1+t)^{-\frac 12}x,\;s = \log(1+t). \]
eigenvalues of the Laplace-Beltrami operator
scientific article; zbMATH DE number 2173050