Well-posedness of the Cauchy problem for the fractional power dissipative equations

This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation ut+(−△)αu=F(u)ut+(−△)αu=F(u) for initial data in the Lebesgue space Lr(Rn)Lr(Rn) with either r≥rd≜nb/(2α−d)r≥rd≜nb/(2α−d) or the homogeneous Besov space Ḃp,∞−σ(Rn) with σ=(2α−d)/b−n/pσ=(2α−d)/b−n/p and 1≤p≤∞1≤p≤∞, where α>0,F(u)=f(u)α>0,F(u)=f(u) or Q(D)f(u)Q(D)f(u) with Q(D)Q(D) being a homogeneous pseudo-differential operator of order d∈[0,2α)d∈[0,2α) and f(u)f(u) is a function of uu which behaves like |u|bu|u|bu with b>0b>0.