Blow-up for a three dimensional Keller-Segel model with consumption of chemoattractant

We investigate blow-up properties for the initial-boundary value problem of a Keller-Segel model with consumption of chemoattractant when the spatial dimension is three. Through a kinetic reformulation of the Keller-Segel system, we first derive some higher-order estimates and obtain certain blow-up criteria for the local classical solutions. These blow-up criteria generalize the results in [4], [5] from the whole space R3 to the case of bounded smooth domain Ω⊂R3. Lower global blow-up estimate on ‖n‖L∞(Ω) is also obtained based on our higher-order estimates. Moreover, we prove local non-degeneracy for blow-up points.