Remarks on two connected papers about Keller-Segel systems with nonlinear production

TL;DR

This paper critically reviews two recent studies on nonlinear Keller-Segel chemotaxis models, clarifying their results, addressing gaps, and providing a more comprehensive understanding of solution boundedness in these complex systems.

Contribution

It offers a detailed analysis and clarification of previous results, filling gaps and refining assumptions in the study of nonlinear Keller-Segel chemotaxis models.

Findings

Identified and addressed a proof gap in Tao et al.'s work

Refined the assumptions in Frassu and Viglialoro's model

Provided a more complete understanding of solution boundedness in nonlinear chemotaxis systems

Abstract

These notes aim to provide a deeper insight on the specifics of two articles dealing with chemotaxis models with nonlinear production. More precisely, we are referring to the papers "Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal production" by X. Tao, S. Zhou and M. Ding [J. Math. Anal. Appl. 474:1 (2019) 733-747] and "Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation" by S. Frassu and G. Viglialoro [Acta Appl. Math. 171:1 (2021), 19]. These works, independently published in these last years, present results leaving open room for further improvement. Indeed, in the first a gap in the proof of the main claim appears, whereas the cornerstone assumption in the second is not sharp. In these pages we give a more complete picture to the relative underlying comprehension.