# Mathematische Nachrichten

## Boundedness in a fully parabolic chemotaxis‐consumption system with nonlinear diffusion and sensitivity, and logistic source

### Abstract

In this paper we study the zero‐flux chemotaxis‐system

Ω being a convex smooth and bounded domain of ${\mathbb{R}}^{n}$, $n\ge 1$, and where $m,k\in \mathbb{R}$, $\mu >0$ and $\alpha <\frac{m+1}{2}$. For any $v\ge 0$ the chemotactic sensitivity function is assumed to behave as the prototype $\chi \left(v\right)=\frac{{\chi }_{0}}{{\left(1+av\right)}^{2}}$, with $a\ge 0$ and ${\chi }_{0}>0$. We prove that for nonnegative and sufficiently regular initial data $u\left(x,0\right)$ and $v\left(x,0\right)$, the corresponding initial‐boundary value problem admits a unique globally bounded classical solution provided μ is large enough.

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