Long time asymptotics of sub-threshold solutions of a semilinear Cauchy problem

Abstract

We show that the solution of the semilinear heat equation u t = uxx +u p (with x ∈ R, p > 3, and nonnegative Cauchy data) behaves for large t like the solution of the corresponding linear problem plus a small correction of order t −1/2−c ,w herec := 1/2, if p 4, and c = (p −3)/2, if 3< p < 4. The result is known in special cases like small initial data. We prove it here for positive sub-threshold initial data satisfying some assumptions. Part of our results are contained in the recent work (10), but the motivation of this paper is to provide a new method leading to somewhat more general space-time estimates. 1. Main result We consider the long time asymptotical behaviour of the solution of the classi- cal semilinear diffusion equation and the associated Cauchy problem for the unknown function u = u(x,t), x ∈ R, t 0: