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Blow-Up and Boundedness in Quasilinear Parabolic-Elliptic Chemotaxis System with Nonlinear Signal Production

Blow-Up and Boundedness in Quasilinear Parabolic-Elliptic Chemotaxis System with Nonlinear Signal Production
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Year:    2023

Author:    Ruxi Cao, Zhongping Li

Journal of Partial Differential Equations, Vol. 36 (2023), Iss. 3 : pp. 262–285

Abstract

In this paper, we consider the quasilinear chemotaxis system of parabolic-elliptic type $$\begin{cases} u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(f(u)\nabla v), & x\in \Omega,\ t>0, \\ 0=\Delta v-\mu(t)+g(u), &  x\in \Omega, \ t>0 \end{cases}$$
under homogeneous Neumann boundary conditions in a smooth bounded domain  $\Omega\subset\mathbb{R}^n, \ n\geq1$. The nonlinear diffusivity $D(\xi)$ and chemosensitivity $f(\xi)$ as well as nonlinear signal production $g(\xi)$ are supposed to extend the prototypes $$D(\xi)=C_{0}(1+\xi)^{-m}, \ \ f(\xi)=K(1+\xi)^{k}, \ \ g(\xi)=L(1+\xi)^{l}, \ \ C_{0}>0,\xi\geq 0,K,k,L,l>0,m\in\mathbb{R}.$$ We proved that if $m+k+l>1+\frac{2}{n}$, then there exists nonnegative radially symmetric initial data $u_{0}$ such that the corresponding solutions blow up in finite time. However, the system admits a global bounded classical solution for arbitrary initial datum when $m+k+l<1+\frac{2}{n}$.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jpde.v36.n3.2

Journal of Partial Differential Equations, Vol. 36 (2023), Iss. 3 : pp. 262–285

Published online:    2023-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    24

Keywords:    Chemotaxis nonlinear diffusion blow-up boundedness nonlinear signal production.

Author Details

Ruxi Cao

Zhongping Li