# American Institute of Mathematical Sciences

July  2019, 24(7): 2971-2988. doi: 10.3934/dcdsb.2018295

## $L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity

 1 School of Mathematical Sciences, Peking University, Beijing 100871, China 2 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author: Mengyao Ding

Received  November 2017 Revised  July 2018 Published  October 2018

Fund Project: The first author is supported by the National Natural Science Foundation of China (11571020, 11671021, 11171048).

This paper studies the parabolic-elliptic Keller-Segel system with supercritical sensitivity: $u_{t} = \nabla·(D(u)\nabla u)-\nabla ·(S(u)\nabla v)$, $0 = Δ v -v+u$ in $Ω× (0,T)$, where the bounded domain $Ω\subset\mathbb{R}^n$, $n≥2$, subject to the non-flux boundary conditions, $D(u)≥ a_0(u+1)^{-q}$, $0≤ S(u)≤ b_0u(u+1)^{α-q-1}$ with $q \in \mathbb{R}$, $α>\frac{2}{n}$, and $a_0, b_0>0$. It is proved that the problem possesses a unique globally bounded solution for $α>\frac{2}{n}$ whenever $\|u_0\|_{L^{\frac{nα}{2}}}$ is sufficiently small. In addition, we establish the large-time behavior of solutions when $q = 0$.

Citation: Mengyao Ding, Sining Zheng. $L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2971-2988. doi: 10.3934/dcdsb.2018295
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