Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball. (English) Zbl 1128.35022

Ishige, Kazuhiro; Kabeya, Yoshitsugu

J. Math. Soc. Japan 59, No. 3, 861-898 (2007).

The paper is concerned with the linear initial-boundary value problem \[ u_t - \Delta u - V(| x| )u=0 \] in the exterior \(\Omega_L \subset \mathbb{R}^N\) of the ball with radius \(L\), with boundary conditions \(\mu u + (1-\mu) \partial_\nu u = 0\) and initial conditions \(u(\cdot,0) = \varphi \in L^p(\Omega_L)\). Here \(0 \leq \mu \leq 1\), and \(V\) is \(C^k\)-smooth and nonnegative such that \(V(r) \sim \omega r^{-2}\) as \(r \to \infty\), with \(\omega \geq 0\). The main results are estimates of the form \[ \| \nabla_x^ju(\cdot,t)\| _\infty \leq C t^{-\gamma} \| \varphi\| _p \] for derivatives up to order \(j \leq k+1\), as \(t \to \infty\), for space dimensions \(N \geq 3\) . Using eigenexpansions and suitable comparison theorems, the authors show that such estimates hold in the case where there is no even index \(q \leq k+1\) such that \(\mu = \frac{q}{q+L}\) and \(V(r) = \frac{\lambda_q}{r^2}\), where \(\lambda_q\) is the \(q\)th eigenvalue of the Laplace-Beltrami operator on the unit sphere. In this case, let \(\alpha(\omega)\) be the positive solution of \(\alpha(\alpha + N - 2) = \omega\) . Then \(\gamma = \frac{N}{2p} + \min (\frac{j}{2}, \frac{\alpha(\omega)}{2})\). The exponent cannot be improved if \(j > \alpha(\omega)\). A similar estimate is shown in the exceptional case where \(\mu = \frac{q}{q+L}\) and \(V(r) = \frac{\lambda_q}{r^2}\) for some even \(q> 0\).

Reviewer: Hans Engler (Bonn)

MSC:

35B40	Asymptotic behavior of solutions to PDEs
35B45	A priori estimates in context of PDEs
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

parabolic equation; decay estimates; exterior domain; eigenexpansions; eigenexpansions