Global solutions for a supercritical drift-diffusion equation

TL;DR

This paper establishes conditions for the global existence and smoothness of solutions to a supercritical drift-diffusion equation with logistic term, extending classical models in mathematical biology.

Contribution

It generalizes the classical Keller-Segel model by proving global solutions for fractional diffusion orders in the supercritical regime, with explicit parameter dependence.

Findings

Existence of global weak solutions for fractional order ppa in (1-c_1, 2]

Global smooth solutions for ppa in (1-c_2, 2] with 0<c_2<c_1

Diffusion in the supercritical regime when ppa<1

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 $\alpha \in (1-c_1, 2]$, where $c_1>0$ is an explicit constant depending on the physical parameters present in the problem (chemosensitivity and strength of logistic damping). Furthermore, in the range $1-c_2<\alpha\leq 2$ with $0<c_2<c_1$, the solution is globally smooth. Let us emphasize that when $\alpha<1$, the diffusion is in the supercritical regime.