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

Hao Yu; Wei Wang; Sining Zheng

doi:10.3934/dcdsb.2017078

Article Contents

2017, Volume 22, Issue 4: 1635-1644. Doi: 10.3934/dcdsb.2017078

This issue Previous Article Stability of traveling waves for autocatalytic reaction systems with strong decay Next Article Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces

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
* Corresponding author

Received: April 30, 2016

Revised: August 31, 2016

Published: August 2017

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

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Abstract

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.

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Mathematics Subject Classification: Primary:35K55, 35B35, 35B40;Secondary:92C17.

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