Asymptotics for a nonlinear integral equation with a generalized heat kernel

Abstract

This paper is concerned with a nonlinear integral equation $$(P)u(x,t)={\int }_{{\mathbf{R}}^N}G(x-y,t)\phi (y)dy+{\int }_0^t{\int }_{{\mathbf{R}}^N}G(x-y,t-s)f(y,s:u)dyds,$$ where $N\ge 1$, $\phi \in L^{\infty }({\mathbf{R}}^N)\cap L^1({\mathbf{R}}^N,(1+|x{|}^K)dx)$ 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 $t\to \infty$.

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