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

In this paper we study the Cauchy problem for the semilinear fractional power dissipative equation u_t+(−Δ)^αu=F(u) for the initial data u₀ in critical Besov spaces with , where α>0, F(u)=P(D)u^b+1 with P(D) being a homogeneous pseudo-differential operator of order d0,2α) 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 .