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Volume 36, Issue 3
Blow-Up and Boundedness in Quasilinear Parabolic-Elliptic Chemotaxis System with Nonlinear Signal Production

Ruxi Cao & Zhongping Li

DOI: 10.4208/jpde.v36.n3.2

J. Part. Diff. Eq., 36 (2023), pp. 262-285.

Published online: 2023-08

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  • 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 $m+k+l<1+\frac{2}{n}$.
  • Keywords

Chemotaxis, nonlinear diffusion, blow-up, boundedness, nonlinear signal production.

  • AMS Subject Headings

35K55, 32O92, 35B35, 92C17

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{JPDE-36-262, author = {Cao , Ruxi and Li , Zhongping}, title = {Blow-Up and Boundedness in Quasilinear Parabolic-Elliptic Chemotaxis System with Nonlinear Signal Production}, journal = {Journal of Partial Differential Equations}, year = {2023}, volume = {36}, number = {3}, pages = {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 $m+k+l<1+\frac{2}{n}$.
}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v36.n3.2}, url = {http://global-sci.org/intro/article_detail/jpde/21888.html} }
TY - JOUR T1 - Blow-Up and Boundedness in Quasilinear Parabolic-Elliptic Chemotaxis System with Nonlinear Signal Production AU - Cao , Ruxi AU - Li , Zhongping JO - Journal of Partial Differential Equations VL - 3 SP - 262 EP - 285 PY - 2023 DA - 2023/08 SN - 36 DO - http://doi.org/10.4208/jpde.v36.n3.2 UR - https://global-sci.org/intro/article_detail/jpde/21888.html KW - Chemotaxis, nonlinear diffusion, blow-up, boundedness, nonlinear signal production. AB -

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 $m+k+l<1+\frac{2}{n}$.
Cao , Ruxi and Li , Zhongping. (2023). Blow-Up and Boundedness in Quasilinear Parabolic-Elliptic Chemotaxis System with Nonlinear Signal Production. Journal of Partial Differential Equations. 36 (3). 262-285. doi:10.4208/jpde.v36.n3.2
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