Czechoslovak Mathematical Journal, Vol. 74, No. 1, pp. 127-151, 2024
Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source
Xiangdong Zhao
Received November 24, 2022. Published online October 13, 2023.
Abstract: We study the chemotaxis system with singular sensitivity and logistic-type source: $u_t=\Delta u-\chi\nabla\cdot(u \nabla v/ v) +ru-\mu u^k$, $0=\Delta v-v+u$ under the non-flux boundary conditions in a smooth bounded domain $\Omega\subset\mathbb{R}^n$, $\chi,r,\mu>0$, $k>1$ and $n\ge1$. It is shown with $k\in(1,2)$ that the system possesses a global generalized solution for $n\ge2$ which is bounded when $\chi>0$ is suitably small related to $r>0$ and the initial datum is properly small, and a global bounded classical solution for $n=1$.
Keywords: chemotaxis; singular sensitivity; global solvability
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Affiliations: Xiangdong Zhao, School of Mathematics, Liaoning Normal University, No.850 Huanghe Road Shahekou District, Dalian 116029, P. R. China e-mail: zhaoxd1223@163.com
