Solvability of the Cauchy problem for fractional semilinear parabolic equations in critical and doubly critical cases

Let 00. We are concerned with the Cauchy problem for the fractional semilinear parabolic equation ∂tu+(-Δ)θ/2u=f(u)inRN×(0,T),u(x,0)=u0(x)≥0inRN.Here, f∈C[0,∞) denotes a rather general growing nonlinearity and u0 may be unbounded. We study local in time solvability in the so-called critical and doubly critical cases. In particular, when f(u)=u1+θ/Nlog(u+e)a, we obtain a sharp integrability condition on u0 which explicitly determines local in time existence/nonexistence of a nonnegative solution.