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 $\alpha \in (1-c_1,2]$, 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 $1-c_2<\alpha \le 2$ 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.

References in corpus (5)

• The fractional Keller-Segel model
• A maximum principle for the Muskat problem for fluids with different densities
• On a generalized doubly parabolic Keller-Segel system in one spatial dimension
• On a nonlocal analog of the Kuramoto-Sivashinsky equation
• On the effect of boundaries in two-phase porous flow

Cited by in corpus (7)

• On a generalized doubly parabolic Keller-Segel system in one spatial dimension
• On a drift-diffusion system for semiconductor devices
• Critical Keller-Segel meets Burgers on ${\mathbb{S}}^1$: large-time smooth solutions
• The Optimal Temporal Decay Estimates for the Fractional Power Dissipative Equation in Negative Besov Spaces
• Suppression of blow up by a logistic source in $2$D Keller-Segel system with fractional dissipation
• Blowup of solutions for nonlinear nonlocal heat equations
• Around a singular solution of a nonlocal nonlinear heat equation