The decay of the solutions for the heat equation with a potential Kazuhiro IshigeMichinori IshiwataTatsuki Kawakami 35B4035K15large time behaviorheat equationsemilinear heat equation 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 $p > 1$. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3771 10.1512/iumj.2009.58.3771 en Indiana Univ. Math. J. 58 (2009) 2673 - 2708 state-of-the-art mathematics http://iumj.org/access/