Global well-posedness of logarithmic Keller-Segel type systems

TL;DR

This paper proves the global existence of classical solutions for a class of logarithmic Keller-Segel systems in two or more dimensions, extending previous results by enlarging parameter ranges and introducing new initial data conditions.

Contribution

It extends the known parameter range for chemotactic sensitivity and introduces new small initial data conditions, applicable to both biological and urban crime models.

Findings

Global classical solutions are established under broader conditions.

The range of chemotactic sensitivity $hi$ is enlarged.

New small initial data conditions are proposed for global existence.

Abstract

We consider a class of logarithmic Keller-Segel type systems modeling the spatio-temporal behavior of either chemotactic cells or criminal activities in spatial dimensions two and higher. Under certain assumptions on parameter values and given functions, the existence of classical solutions is established globally in time, provided that initial data are sufficiently regular. In particular, we enlarge the range of chemotatic sensitivity $\chi$, compared to known results, in case that spatial dimensions are between two and eight. In addition, we provide new type of small initial data to obtain global classical solution, which is also applicable to the urban crime model. We discuss long-time asymptotic behaviors of solutions as well.