Global existence, uniqueness and $L^{\infty}$-bound of weak solutions of fractional time-space Keller-Segel system

TL;DR

This paper establishes the global existence, uniqueness, and boundedness of weak solutions for a fractional Keller-Segel system with logistic terms, using advanced PDE techniques and fractional inequalities.

Contribution

It introduces new results on the existence, boundedness, and blow-up criteria of weak solutions for a fractional space-time Keller-Segel model with logistic source terms.

Findings

Global existence and $L^{ abla}$-bound of solutions proven.

Uniqueness established under strong damping conditions.

Blow-up criterion linked to $L^{h}$-norms of solutions.

Abstract

This paper studies the properties of weak solutions to a class of space-time fractional parabolic-elliptic Keller-Segel equations with logistic source terms in $\mathbb{R}^{n}$, $n\geq 2$. The global existence and $L^{\infty}$-bound of weak solutions are established. We mainly divide the damping coefficient into two cases: (i) $b>1-\frac{\alpha}{n}$, for any initial value and birth rate; (ii) $0<b\leq 1-\frac{\alpha}{n}$, for small initial value and small birth rate. The existence result is obtained by verifying the existence of a solution to the constructed regularization equation and incorporate the generalized compactness criterion of time fractional partial differential equation. At the same time, we get the $L^{\infty}$-bound of weak solutions by establishing the fractional differential inequality and using the Moser iterative method. Furthermore, we prove the uniqueness of weak solutions by using the hyper-contractive estimates when the damping coefficient is strong. Finally, we also propose a blow-up criterion for weak solutions, that is, if a weak solution blows up in finite time, then for all $h>q$, the $L^{h}$-norms of the weak solution blow up at the same time.