Local and global solvability and blow up for the drift-diffusion equation with the fractional dissipation in the critical space

We study local and global existence and uniqueness of solutions to the drift-diffusion equation with fractional dissipation (−Δ)θ/2(−Δ)θ/2. In the preceding works for some associated equations, the cases θ=1θ=1 and θ<1θ<1 are known as critical and supercritical respectively. In the critical and supercritical cases, we may not apply the LpLp-theory for semilinear equations of parabolic type used in the subcritical case 1<θ≤21<θ≤2. We discuss local existence with large data and global existence with small data in the Besov space Bp,qnp−θ(Rn), which corresponds to the scaling invariant space of the equation. Furthermore we show that solutions can blow up in finite time if initial data is not small.