Abstract:

This paper investigates the following chemotactic model with logarithmic sensitivity in a one-dimensional bounded domain: $\begin{eqnarray*}\left\{\begin{array}{lll}\partial_{t}u=Du_{xx}+(u (\ln v)_{x})_{x}, \quad&x\in (0, 1), \quad t>0, \\partial_{t}v=\varepsilon v_{xx}+uv-\mu v, \quad&x\in (0, 1), \quad t>0.\end{array}\right.\end{eqnarray*} $ By using a Cole-Hopf type transformation, we transform the above singular repulsive chemotaxis model into a non-singular system of the form $\begin{eqnarray*}\left\{\begin{array}{lll}p_t = p_{xx} + (pq)_x,\quad & x\in (0,1),\quad t>0,\\q_t = \varepsilon q_{xx} + \varepsilon (q^2)_x + p_x,\quad & x\in (0,1),\quad t>0,\\\end{array}\right.\end{eqnarray*} $ Then under some mixed boundary conditions, we prove the global existence and exponential convergence of solutions to the initial-boundary value problem of the above system with regular initial data.

Key words: Chemotaxis, Logarithmic sensitivity, Mixed boundary conditions, Global existence, Exponential convergence