Global solutions for a supercritical drift–diffusion equation

Abstract

We study the global existence of solutions to a one-dimensional drift–diffusion equation with logistic term, generalizing the classical parabolic–elliptic Keller–Segel aggregation equation arising in mathematical biology. In particular, we prove that there exists a global weak solution, if the order of the fractional diffusion α∈(1−c1,2], where c1>0 is an explicit constant depending on the physical parameters present in the problem (chemosensitivity and strength of logistic damping). Furthermore, in the range 1−c2<α≤2 with 0<c2<c1, the solution is globally smooth. Let us emphasize that when α<1, the diffusion is in the supercritical regime.