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

Abstract We study local and global existence and uniqueness of solutions to the drift-diffusion equation with fractional dissipation ( − Δ ) θ / 2 . In the preceding works for some associated equations, the cases θ = 1 and θ 1 are known as critical and supercritical respectively. In the critical and supercritical cases, we may not apply the L p -theory for semilinear equations of parabolic type used in the subcritical case 1 θ ≤ 2 . We discuss local existence with large data and global existence with small data in the Besov space B p , q n p − θ ( R n ) , 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.