Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity

This paper is concerned with the parabolic-elliptic Keller-Segel system with signal-dependent sensitivity $\chi(v)$,\begin{align*}\begin{cases}u_t=\Delta u - \nabla \cdot ( u \nabla \chi(v))&\mathrm{in}\ \Omega\times(0,\infty), \\0=\Delta v -v+u&\mathrm{in}\ \Omega\times(0,\infty),\end{cases}\end{align*}under homogeneous Neumann boundary condition in a smoothly bounded domain$\Omega \subset \mathbb{R}^2$with nonnegative initial data $u_0 \in C^{0}(\overline{\Omega})$, $\not\equiv 0$.  In the special case $\chi(v)=\chi_0 \log v\, (\chi_0>0)$,global existence and boundedness of the solution to the system were proved under some smallness condition on $\chi_0$ by Biler (1999) and Fujie, Winkler and Yokota (2015).In the present work, global existence and boundedness in the system will be established for general sensitivity $\chi$ satisfying $\chi'>0$ and$\chi'(s) \to 0 $ as $s\to \infty$.In particular, this establishes global existence and boundedness in the case $\chi(v)=\chi_0\log v$ with large $\chi_0>0$.Moreover, although the methods in the previous results are effective for only few specific cases, the present method can be applied to more general cases requiring only the essential conditions. Actually, our condition is necessary, since there are many radial blow-up solutions in the case $\inf_{s>0} \chi^\prime (s) >0$.