Refined asymptotic profiles for a semilinear heat equation

Abstract

We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation,

$$\partial_t u=\Delta u+F(x,t,u) \quad{\rm in} \;{\bf R}^N\times(0,\infty), \quad u(x,0)=\varphi(x)\quad{\rm in} \;{\bf R}^N,\quad\quad ({\rm P})$$

where ${F\in C({\bf R}^N\times[0,\infty)\times{\bf R})}$ and ${\varphi\in L^1({\bf R}^N, (1+|x|)^K\,dx)}$ with K ≥ 0. Assume that u is a solution of (P) satisfying

$$|F(x,t,u(x,t))|\le C(1+t)^{-A}|u(x,t)|,\quad (x,t)\in{\bf R}^N\times(0,\infty),$$

for some constants C > 0 and A > 1. Then it is well known that the solution u behaves like the heat kernel. In this paper we give the ([K] + 2)th order asymptotic expansion of the solution u, and reveal the relationship between the asymptotic profile of the solution u and the nonlinear term F. Here [K] is the integer satisfying K − 1 < [K] ≤ K.