A short remark on comparison estimates in a chemotaxis system with local sensing

Abstract

This paper deals with classical solutions $ (u, v) $ to chemotaxis stystems with local sensing. In Fujie-Jiang (Calc. Var. Partial Differential Equations, 2021), some auxiliary function $ w $ is introduced and the comparison estimate: $ d_1v\leq w $ ($ d_1>0 $) plays an important role in the analysis. Recently, in Fujie-Senba (Nonlinearity, 2022) this comparison estimate is refined: $ d_1 v\leq w \leq d_2v $ ($ d_1, d_2>0 $) for some restricted case. In this paper, it is pointed out that in light of the lower bound for $ v $, some part of the proof in Fujie-Senba (Nonlinearity, 2022) can be applied in more general setting.