This paper is concerned with nonlinear stability of rarefaction wave to the Cauchy problem for a blood flow model, which describes the motion of blood through axi-symmetric compliant vessels. Inspired by the stability analysis of classical $p$-system, we show the solution of this typical model tends time-asymptotically toward the rarefaction wave under some suitably small conditions and there are more difficulties in the proof due to the appearance of strong nonlinear terms including second-order derivative of $v$ with respect to the spatial variable $x$. The main result is proved by employing the elementary $L^2$ energy methods. This is the first result about nonlinear stability of some nontrivial profiles (i.e., non-constant function patterns) for the blood flow model.

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, ρ). 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.