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

summary:In this paper, we consider solutions to the following chemotaxis system with general sensitivity \[ \left\{ \begin{array}{l} \tau u_t = \Delta u - \nabla \cdot (u \nabla \chi (v)) \quad \mbox{ in } \Omega \times (0,\infty), \\ \eta v_t = \Delta v - v + u \quad \mbox{ in } \Omega \times (0,\infty), \\ \displaystyle \frac{\partial u}{\partial \nu} = \frac{\partial u}{\partial \nu} = 0 \quad \mbox{ on } \partial \Omega \times (0,\infty). \end{array} \right. \] 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 $\mathbf{R}^n$ ($n \geq 2$). 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 $n \geq 2$. 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