We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system $\partial_t u = \Delta u - \nabla \cdot (u \nabla \mathcal{ K}_u), \quad -\Delta \mathcal{ K}_u = u \quad \textup{ in}\;\; \mathbb{ R}^d,\; d = 3,4,$ and derive the final blowup profile $ u(r,T) \sim c_d \frac{ |\log r|^\frac{ d-2}{ d}}{ r^2} \quad \textup{ as}\;\; r \to 0, \;\; c_d > 0.$ To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner, Constantin, Kadanoff, Schenkel, and Venkataramani (Nonlinearity, 1999).
Categories: math.AP