Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption

Abstract

Assuming that $0<χ<\sqrt{\frac{2}n}$, $κ\ge 0$ and $μ>\frac{n-2}{n}$, we prove global existence of classical solutions to a chemotaxis system slightly generalizing \[ \begin{split} u_t &= Δu - χ\nabla\cdot ( \frac{u}{v} \nabla v ) + κu -μu^2\\ v_t &= Δv - u v \end{split} \] in a bounded domain $Ω\subset \mathbb{R}^n$, with homogeneous Neumann boundary conditions and for widely arbitrary positive initial data. In the spatially one-dimensional setting, we prove global existence and, moreover, boundedness of the solution for any $χ>0$, $μ>0$, $κ\ge 0$.