Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4610334 | Journal of Differential Equations | 2015 | 28 Pages |
Abstract
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.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Yuusuke Sugiyama, Masakazu Yamamoto, Keiichi Kato,