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Keywords:
chemotaxis; singular sensitivity; global solvability
chemotaxis; singular sensitivity; global solvability
Summary:
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 $n=1$.
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