# Effect of nonlinear diffusion on a lower bound for the blow-up time in a fully parabolic chemotaxis system

## TL;DR

This paper extends previous results on lower bounds for blow-up time in a chemotaxis system by considering nonlinear diffusion and non-convex domains, using sharp inequalities for improved estimates.

## Contribution

It generalizes earlier work to include nonlinear diffusion effects and non-convex domains, providing a more comprehensive lower bound for blow-up time.

## Key findings

- Derived a new lower bound for blow-up time in the chemotaxis system.
- Showed the effect of nonlinear diffusion on blow-up behavior.
- Removed the convexity restriction on the domain in the analysis.

## Abstract

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system \begin{equation*} \begin{cases} u_t=\nabla \cdot [(u+\alpha)^{m_1-1} \nabla u-\chi u(u+\alpha)^{m_2-2} \nabla v] & {\rm in} \; \Omega \times (0,T), \\[1mm] v_t=\Delta v-v+u & {\rm in} \; \Omega \times (0,T) \end{cases} \end{equation*} under Neumann boundary conditions and initial conditions, where $\Omega$ is a general bounded domain in $\mathbb{R}^n$ with smooth boundary, $\alpha>0$, $\chi>0$, $m_1, m_2 \in \mathbb{R}$ and $T>0$. Recently, Anderson-Deng (2017) gave a lower bound for the blow-up time in the case that $m_1=1$ and $\Omega$ is a convex bounded domain. The purpose of this paper is to generalize the result in Anderson-Deng (2017) to the case that $m_1 \neq 1$ and $\Omega$ is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo-Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of $\Omega$.