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

Abstract

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 $\chi0$, $\varepsilon>0$ 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 $v_0>0$, thesystem possesses a unique global smooth solution that enjoysexponential convergence properties in $L^\infty(\Omega)$ as timegoes 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 constantpair $(\mu, (\frac{\mu}{\lambda})^{\frac{D}{\chi}})$ (where$\lambda>0$ is an arbitrary constant) is the only positivestationary solution. The biological implications of our results willbe given in the paper.

Locations

• Discrete and Continuous Dynamical Systems - B - View - PDF