Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces

Abstract

In this paper we study the Cauchy problem for the semilinear fractional power dissipative equation u t + ( − Δ ) α u = F ( u ) for the initial data u 0 in critical Besov spaces B ˙ 2 , r σ with σ ≜ n 2 − 2 α − d b , where α > 0 , F ( u ) = P ( D ) u b + 1 with P ( D ) being a homogeneous pseudo-differential operator of order d ∈ [ 0 , 2 α ) and b > 0 being an integer. Making use of some estimates of the corresponding linear equation in the frame of mixed time–space spaces, the so-called “mono-norm method” which is different from the Kato's “double-norm method,” Fourier localization technique and Littlewood–Paley theory, we get the well-posedness result in the case σ > − n 2 .