Asymptotics for a nonlinear integral equation with a generalized heat kernel

This paper is concerned with a nonlinear integral equation (P) u(x, t) = integral (N)(R) G(x - y, t) phi(y)dy + integral(t)(0) integral(N)(R) G(x - y, t - s)f(y, s : u)dyds, Where N >= 1, phi is an element of L-infinity (R-N) boolean AND L-1(R-N, (1+vertical bar x vertical bar(K))dx for some K >= 0. Hence, G = G(x, t) is phi 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 -> infinity.