  In this paper, we study a free boundary problem for a class of
parabolic-elliptic type chemotaxis model in high dimensional symmetry domain
\Omega. By using the contraction mapping principle and operator semigroup
approach, we establish the existence of the solution for such kind of
chemotaxis system in the domain \Omega with free boundary condition. Besides,
we get the explicit formula for the free boundary and show the chemotactic
collapse for the solution of the system.
