Convergence to diffusion waves for solutions of 1D Keller–Segel model

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