Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Zhao, Xiangdong

Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 127-151.

Résumé

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 $n\ge 1$. It is shown with $k\in (1,2)$ that the system possesses a global generalized solution for $n\ge 2$ 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 $n=1$.

Détail

DOI : 10.21136/CMJ.2023.0544-22

Classification : 35B45, 35K55, 92C17
Keywords: chemotaxis; singular sensitivity; global solvability