# American Institute of Mathematical Sciences

June  2017, 22(4): 1635-1644. doi: 10.3934/dcdsb.2017078

## Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity

 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author

Received  May 2016 Revised  September 2016 Published  February 2017

Fund Project: Supported by the National Natural Science Foundation of China (11171048) and the Fundamental Research Funds for the Central Universities (DUT16LK24).

This paper deals with the global boundedness of solutions to a fully parabolic Keller-Segel system $u_t=Δ u-\nabla (u^α \nabla v)$, $v_t=Δ v-v+u$ under non-flux boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^{n}$. The case of $α≥ \max\{1,\frac{2}{n}\}$ with $n≥1$ was considered in a previous paper of the authors [Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. B, 21 (2016), 1317-1327]. In the present paper we prove for the other case $α∈(\frac{2}{3},1)$ that if $\|u_0\|_{L^\frac{nα}{2}(Ω)}$ and $\|\nabla v_0\|_{L^{nα}(Ω)}$ are small enough with $n≥q3$, then the solutions are globally bounded with both $u$ and $v$ decaying to the same constant steady state $\bar{u}_0=\frac{1}{|Ω|}∈t_Ω u_0(x) dx$ exponentially in the $L^∞$-norm as $t? ∞$. Moreover, the above conclusions still hold for all $α≥q2$ and $n≥q1$, provided $\|u_0\|_{L^{nα-n}(Ω)}$ and $\|\nabla v_0\|_{L^{∞}(Ω)}$ sufficiently small.

Citation: Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078
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