  This paper is concerned with a nonlinear integral equation $$ (P)\qquad
u(x,t)=\int_{{\bf R}^N}G(x-y,t)\varphi(y)dy+\int_0^t\int_{{\bf
R}^N}G(x-y,t-s)f(y,s:u)dyds, \quad $$ where $N\ge 1$, $\varphi\in L^\infty({\bf
R}^N)\cap L^1({\bf R}^N,(1+|x|^K)dx)$ for some $K\ge 0$. Here $G=G(x,t)$ is a
generalization of the heat kernel. We are interested in the asymptotic
expansions of the solution of $(P)$ behaving like a multiple of the integral
kernel $G$ as $t\to\infty$.
