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Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension

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  • This paper deals with the chemotaxis system $$ \left\{ \begin{array}{ll} u_t ={D} u_{xx}-\chi [u(\ln v)_x]_x, & x\in (0, 1), \ t>0,\\ v_t =\varepsilon v_{xx} +uv-\mu v, & x\in (0, 1), \ t>0, \end{array} \right. $$ under Neumann boundary condition, where $\chi<0$, $D>0$, $\varepsilon>0$ and $\mu>0$ are constants.
    It is shown that for any sufficiently smooth initial data $(u_0, v_0)$ fulfilling $u_0\ge 0$, $u_0 \not\equiv 0$ and $v_0>0$, the system possesses a unique global smooth solution that enjoys exponential convergence properties in $L^\infty(\Omega)$ as time goes to infinity, which depend on the sign of $\mu-\bar{u}_0$, where $\bar{u}_0 :=\int_0^1 u_0 dx$. Moreover, we prove that the constant pair $(\mu, (\frac{\mu}{\lambda})^{\frac{D}{\chi}})$ (where $\lambda>0$ is an arbitrary constant) is the only positive stationary solution. The biological implications of our results will be given in the paper.
    Mathematics Subject Classification: Primary: 35A01, 35B40, 35B44, 35K57; Secondary: 35Q92,92C17.

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