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

## TL;DR

This paper proves the global boundedness of solutions to a complex chemotaxis system with nonlinear diffusion, sensitivity, and logistic growth, under certain parameter conditions, ensuring well-behaved long-term dynamics.

## Contribution

It establishes conditions for global bounded solutions in a fully parabolic chemotaxis model with nonlinear diffusion and sensitivity, extending previous results to more general settings.

## Key findings

- Existence of global bounded classical solutions for large enough .
- Boundedness depends on the logistic damping coefficient .
- Results hold for a broad class of initial data.

## Abstract

In this paper we study the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_{ t}=\nabla \cdot ((u+1)^{m-1} \nabla u-(u+1)^\alpha \chi(v)\nabla v) + ku-\mu u^2 & x\in \Omega, t>0, \\ v_{t} = \Delta v-vu & x\in \Omega, t>0,\\ \end{cases} \end{equation*} $\Omega$ being a bounded and smooth domain of $\mathbb{R}^n$, $n\geq 1$, and where $m,k \in \mathbb{R}$, $\mu>0$ and $\alpha < \frac{m+1}{2}$. For any $v\geq 0$ the chemotactic sensitivity function is assumed to behave as the prototype $\chi(v) = \frac{\chi_0}{(1+av)^2}$, with $a\geq 0$ and $\chi_0>0$. We prove that for nonnegative and sufficiently regular initial data $u(x,0)$ and $v(x,0),$ the corresponding initial-boundary value problem admits a global bounded classical solution provided $\mu$ is large enough.