On behavior of solutions to a chemotaxis system with a nonlinear sensitivity function

Senba, Takasi, and Fujie, Kentarou. "On behavior of solutions to a chemotaxis system with a nonlinear sensitivity function." Proceedings of Equadiff 14. Bratislava: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, 2017. 45-52. <http://eudml.org/doc/294893>.

@inProceedings{Senba2017,
abstract = {In this paper, we consider solutions to the following chemotaxis system with general sensitivity \[ \left\lbrace \begin\{array\}\{l\} \tau u\_t = \Delta u - \nabla \cdot (u \nabla \chi (v)) \quad \mbox\{ in \} \Omega \times (0,\infty ), \\ \eta v\_t = \Delta v - v + u \quad \mbox\{ in \} \Omega \times (0,\infty ), \\ \displaystyle \frac\{\partial u\}\{\partial \nu \} = \frac\{\partial u\}\{\partial \nu \} = 0 \quad \mbox\{ on \} \partial \Omega \times (0,\infty ). \end\{array\} \right. \] Here, $\tau $ and $\eta $ are positive constants, $\chi $ is a smooth function on $(0,\infty )$ satisfying $\chi ^\prime (\cdot ) >0$ and $\Omega $ is a bounded domain of $\mathbf \{R\}^n$ ($n \ge 2$). It is well known that the chemotaxis system with direct sensitivity ($\chi (v) = \chi _0 v$, $\chi _0>0$) has blowup solutions in the case where $n \ge 2$. On the other hand, in the case where $\chi (v) = \chi _0 \log v$ with $0 < \chi _0 \ll 1$, any solution to the system exists globally in time and is bounded. We present a sufficient condition for the boundedness of solutions to the system and some related systems.},
author = {Senba, Takasi, Fujie, Kentarou},
booktitle = {Proceedings of Equadiff 14},
keywords = {Chemotaxis system, nonlinear sensitivity, time-global existence},
location = {Bratislava},
pages = {45-52},
publisher = {Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing},
title = {On behavior of solutions to a chemotaxis system with a nonlinear sensitivity function},
url = {http://eudml.org/doc/294893},
year = {2017},
}

TY - CLSWK
AU - Senba, Takasi
AU - Fujie, Kentarou
TI - On behavior of solutions to a chemotaxis system with a nonlinear sensitivity function
T2 - Proceedings of Equadiff 14
PY - 2017
CY - Bratislava
PB - Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing
SP - 45
EP - 52
AB - In this paper, we consider solutions to the following chemotaxis system with general sensitivity \[ \left\lbrace \begin{array}{l} \tau u_t = \Delta u - \nabla \cdot (u \nabla \chi (v)) \quad \mbox{ in } \Omega \times (0,\infty ), \\ \eta v_t = \Delta v - v + u \quad \mbox{ in } \Omega \times (0,\infty ), \\ \displaystyle \frac{\partial u}{\partial \nu } = \frac{\partial u}{\partial \nu } = 0 \quad \mbox{ on } \partial \Omega \times (0,\infty ). \end{array} \right. \] Here, $\tau $ and $\eta $ are positive constants, $\chi $ is a smooth function on $(0,\infty )$ satisfying $\chi ^\prime (\cdot ) >0$ and $\Omega $ is a bounded domain of $\mathbf {R}^n$ ($n \ge 2$). It is well known that the chemotaxis system with direct sensitivity ($\chi (v) = \chi _0 v$, $\chi _0>0$) has blowup solutions in the case where $n \ge 2$. On the other hand, in the case where $\chi (v) = \chi _0 \log v$ with $0 < \chi _0 \ll 1$, any solution to the system exists globally in time and is bounded. We present a sufficient condition for the boundedness of solutions to the system and some related systems.
KW - Chemotaxis system, nonlinear sensitivity, time-global existence
UR - http://eudml.org/doc/294893
ER -