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

Abstract

We show the existence of locally bounded global solutions to the chemotaxis system \[ u_t = \nabla\cdot(D(u)\nabla u) - \nabla\cdot(\frac{u}{v} \nabla v) \] \[ v_t = Δv - uv \] with homogeneous Neumann boundary conditions and suitably regular positive initial data in smooth bounded domains $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$, for $D(u)\ge \delta u^{m-1}$ with some $\delta >0$, provided that $m>1+\frac{N}{4}$.