Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball

Abstract

We consider the initial-boundary value problem \begin{equation}\qquad\left\{ \begin{array}{lll} \frac{\partial}{\partial t}u = \Delta u-V(|x|)u & \mbox{in}& \Omega_L\times(0,\infty), \mu u+(1-\mu)\frac{\partial}{\partial\nu}u = 0 & \mbox{on}& \partial\Omega_L\times(0,\infty), u(\cdot,0) = \phi(\cdot)\in L^p(\Omega_L), & p \ge 1,& \end{array}\right. \tag{P}\end{equation} where $\Omega_L = \{ |x| \in \mathbf{R}^{N} : |x > L\}, N \geq 2, L > 0, 0 \leq \mu \leq 1, v$ is the outer unit normal vector to $\partial \Omega_L$, and $V$ is a nonnegative smooth function such that $V(r) = O(r^{-2})$ as $r \to \infty$. In this paper, we study the decay rates of the derivatives $\bigtriangledown^j_x u$ of the solution $u$ to $(P)$ as $t \to \infty$.