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 Ḃp,rs(Rn) with s<0s<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 Ḃp,rs(Rn) with s=n/p−2/b<0s=n/p−2/b<0. In particular, when r=∞r=∞ and the initial data φφ satisfies that λ2bφ(λx)=φ(x) for any λ>0λ>0, our result leads to the existence of global self-similar solutions to the problem.