  In this paper, we are concerned with the asymptotic behavior of solutions to
the Cauchy problem (or initial-boundary value problem) of one-dimensional
Keller-Segel model. For the Cauchy problem, we prove that the solutions
time-asymptotically converge to the nonlinear diffusion wave whose profile is
self-similar solution to the corresponding parabolic equation, which is derived
by Darcy's law, as in [11, 28]. For the initial-boundary value problem, we
consider two cases: Dirichlet boundary condition and null Neumann boundary
condition on (u, \rho). In the case of Dirichlet boundary condition, similar to
the Cauchy problem, the asymptotic profile is still the self-similar solution
of the corresponding parabolic equation, which is derived by Darcy's law, thus
we only need to deal with boundary effect. In the case of null-Neumann boundary
condition, the global existence and asymptotic behavior of solutions near
constant steady states are established. The proof is based on the elementary
energy method and some delicate analysis of the corresponding asymptotic
profiles.
