Global Existence of Weak Solutions to a Signal-dependent Keller-Segel Model for Local Sensing Chemotaxis

TL;DR

This paper proves the global existence of weak solutions for a degenerate, signal-dependent chemotaxis model, extending previous results by weakening assumptions and achieving higher regularity of solutions.

Contribution

It introduces a modified comparison approach and an approximation scheme to establish global weak solutions with higher regularity for a chemotaxis model with signal-dependent motility.

Findings

Established global existence of weak solutions in any spatial dimension.

Derived upper bounds for the signal-dependent motility function.

Achieved higher regularity of solutions compared to previous studies.

Abstract

This paper is devoted to global existence of weak solutions to the following degenerate kinetic model of chemotaxis \begin{equation} \begin{cases}\label{chemo0} u_t=\Delta (\gamma (v)u) \tau v_{t}=\Delta v-v+u \end{cases} \end{equation}in a smooth bounded domain with no-flux boundary conditions. The problem features a positive signal-dependent motility function $\gamma(\cdot)$ which may vanish as $v$ becomes unbounded. In this paper, we first modify the comparison approach developed recently in \cite{GM,GM2} to derive the upper bounds of $v$ under weakened assumptions on $\gamma(\cdot)$. 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.