FRACTIONAL KELLER-SEGEL EQUATION: GLOBAL WELL-POSEDNESS AND FINITE TIME BLOW-UP

This article studies the aggregation diffusion equation partial derivative(t)rho = Delta(alpha/2) rho + lambda div((K*rho)rho), where Delta(alpha/2) denotes the fractional Laplacian and K= x/vertical bar x vertical bar(beta) is an attractive kernel. This equation is a generalization of the classical Keller-Segel equation, which arises in the modelling of the motion of cells. In the diffusion dominated case beta < alpha, we prove global well-posedness for an L-k(1) initial condition, and in the fair competition case beta = alpha for an L-k(1) boolean AND LlnL initial condition with small mass. In the aggregation dominated case beta > alpha, we prove global or local well-posedness for an L-p initial condition, depending on some smallness condition on the L-p norm of the initial condition. We also prove that finite time blow-up of even solutions occurs under some initial mass concentration criteria.