Strong solutions to the nonlinear heat equation in homogeneous Besov spaces

Abstract

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 = ∞ and the initial data φ satisfies that λ 2 b φ ( λ x ) = φ ( x ) for any λ > 0 , our result leads to the existence of global self-similar solutions to the problem.