  In this paper, we consider the exogenous chemotaxis system with physical
mixed zero-flux and Dirichlet boundary conditions in one dimension. Since the
Dirichlet boundary condition can not contribute necessary estimates for the
cross-diffusion structure in the system, the global-in-time existence and
asymptotic behavior of solutions remain open up to date. In this paper, we
overcome this difficulty by employing the technique of taking anti-derivative
so that the Dirichlet boundary condition can be fully used, and show that the
system admits global strong solutions which exponentially stabilize to the
unique stationary solution as time tends to infinity against some suitable
small perturbations. To the best of our knowledge, this is the first result
obtained on the global well-posedness and asymptotic behavior of solutions to
the exogenous chemotaxis system with physical boundary conditions.
