Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion

TL;DR

This paper proves the existence of locally bounded global solutions for a chemotaxis system with singular sensitivity and nonlinear diffusion, under certain conditions on the diffusion exponent, in bounded domains.

Contribution

It establishes the existence of solutions to a chemotaxis model with singular sensitivity and nonlinear diffusion, extending previous results to broader parameter ranges.

Findings

Existence of locally bounded global solutions under specified conditions.

Solutions exist for diffusion exponent m > 1 + N/4.

Results apply to smooth bounded domains with regular initial data.

Abstract

We show the existence of locally bounded global solutions to the chemotaxis system \[ u_t = \nabla\cdot(D(u)\nabla u) - \nabla\cdot(\frac{u}{v} \nabla v) \] \[ v_t = \Delta v - uv \] with homogeneous Neumann boundary conditions and suitably regular positive initial data in smooth bounded domains $\Omega \subset \mathbb{R}^N$, $N\geq2$, for $D(u)\geq \delta u^{m-1}$ with some $\delta>0$, provided that $m>1+\frac N4$.