Strong solutions to the nonlinear heat equation in homogeneous Besov spaces

In this paper we study the Cauchy problem of the nonlinear heat equation in homogeneous Besov spaces B-p,r(s)(R-n) with s < 0.The nonlinear estimate is established by means of the Littlewood-Paley trichotomy and is used to prove the global well-posedness of solutions for small initial data in the homogeneous Besov space B-p,r(s) (R-n) with s = n/p - 2/b < 0. In particular, when r = infinity and the initial data phi satisfies that lambda 2/b phi(lambda x) = phi(x) for any lambda > 0, our result leads to the existence of global self-similar solutions to the problem. (c) 2006 Elsevier Ltd. All rights reserved.