Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Effect of nonlinear diffusion on a lower bound for the blow-up time in a fully parabolic chemotaxis system (1902.09787v1)

Published 26 Feb 2019 in math.AP

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 $\Omega$.

Summary

We haven't generated a summary for this paper yet.