Global existence of weak solutions to a signal-dependent Keller–Segel model for local sensing chemotaxis

Abstract

This paper is devoted to the global existence of weak solutions to the following degenerate kinetic model of chemotaxis (0.1)ut=Δ(γ(v)u)τvt=Δv−v+uin a smooth bounded domain with no-flux boundary conditions under the assumption 0≤τ<(sup(0,∞)γ)−1. The problem features a positive signal-dependent motility function γ(⋅) which may vanish as v becomes unbounded. In this paper, we first modify the comparison approach developed recently in Fujie and Jiang (2020); Fujie and Jiang (2021) to derive the upper bounds of v under the slightly weakened above assumptions on γ(⋅). Then by introducing a suitable approximation scheme which is compatible with the comparison method, we establish the global existence of weak solutions in any spatial dimension via compactness argument. Our weak solution has higher regularity than those obtained in previous literature (Burger et al., 2020; Desvillettes et al., 2019; Tao and Winkler, 2017).