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The Keller-Segel system with logistic growth and signal-dependent motility

  • *Corresponding author: Zhi-An Wang

    *Corresponding author: Zhi-An Wang
The research of H.Y. Jin was supported by the NSF of China (No. 11871226), Guangdong Basic and Applied Basic Research Foundation (No. 2020A1515010140), Guangzhou Science and Technology Program No.202002030363 and the Fundamental Research Funds for the Central Universities. The research of Z.A. Wang was supported by the Hong Kong RGC GRF grant 15303019 (Project ID P0030816)
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  • The paper is concerned with the following chemotaxis system with nonlinear motility functions

    $\begin{equation}\label{0-1}\begin{cases}u_t = \nabla \cdot (\gamma(v)\nabla u- u\chi(v)\nabla v)+\mu u(1-u), &x\in \Omega, ~~t>0, \\ 0 = \Delta v+ u-v, & x\in \Omega, ~~t>0, \\u(x, 0) = u_0(x), & x\in \Omega, \end{cases}~~~~(\ast)\end{equation}$

    subject to homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset \mathbb{R}^2 $ with smooth boundary, where the motility functions $ \gamma(v) $ and $ \chi(v) $ satisfy the following conditions

    ● $ (\gamma, \chi)\in [C^2[0, \infty)]^2 $ with $ \gamma(v)>0 $ and $ \frac{|\chi(v)|^2}{\gamma(v)} $ is bounded for all $ v\geq 0 $.

    By employing the method of energy estimates, we establish the existence of globally bounded solutions of ($\ast$) with $ \mu>0 $ for any $ u_0 \in W^{1, \infty}(\Omega) $ with $ u_0 \geq (\not\equiv) 0 $. Then based on a Lyapunov function, we show that all solutions $ (u, v) $ of ($\ast$) will exponentially converge to the unique constant steady state $ (1, 1) $ provided $ \mu>\frac{K_0}{16} $ with $ K_0 = \max\limits_{0\leq v \leq \infty}\frac{|\chi(v)|^2}{\gamma(v)} $.

    Mathematics Subject Classification: Primary: 35A01, 35B40, 35K57, 35Q92; Secondary: 92C17.


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