Refined asymptotic profiles for a semilinear heat equation

We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation, ∂ _tu = Δu+F(x,t,u) in R ^N × (0, ∞) u(x, 0) = φ(x) in R ^N, (P) where F ∈ C(R ^N × [0, ∞) × R) and φ ∈ L ¹ (R ^N, (1 + {pipe}x{pipe}) ^K dx) with K ≥ 0. Assume that u is a solution of (P) satisfying {pipe}F(x,t,u(x,t)){pipe} ≤ C(1+t) ^-A {pipe}u(x,t){pipe}, (x,t) ∈ R ^N × (0,∞) 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.