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)^α u = F (u) for the initial data u₀ in critical Besov spaces over(B, ̇)_{2, r}^σ with σ {delta equal to} frac(n, 2) - frac(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 > - frac(n, 2).