article

Title: The decay of the solutions for the heat equation with a potential

Authors: Kazuhiro Ishige, Michinori Ishiwata and Tatsuki Kawakami

Issue: Volume 58 (2009), Issue 6, 2673-2708

Abstract:

$We study the large time behavior of the solutions for the Cauchy problem, \begin{gather*} \partial_{t} u = \Delta u + a(x,t)u \quad\mbox{in }\mathbb{R}^{N} \times (0,\infty),\\ u(x,0) = \varphi(x) \quad\mbox{in }\mathbb{R}^{N}, \end{gather*} where $\varphi \in L^{1}(\mathbb{R}^{N}, (1 + |x|^{K} )\, \mathrm{d}x)$ with $K \ge 0$ and $\| a(t) \|_{L^{\infty}(\mathbb{R}^{N})} = O(t^{-A})$ as $t \to \infty$ for some $A > 1$. In this paper we classify the decay rate of the solutions and give the precise estimates on the difference between the solutions and their asymptotic profiles. Furthermore, as an application, we discuss the large time behavior of the global solutions for the semilinear heat equation, \[ \partial_{t} u= \Delta u + \lambda |u|^{p-1} u, \] where $\lambda \in \mathbb{R}$ and $p > 1$.$