Asymptotic stability of strong rarefaction waves to a parabolic-hyperbolic system arising from chemotaxis
Year: 2026
Type: preprint
Abstract: We are interested in the asymptotic behavior of solutions toward strong rarefaction waves for a parabolic-hyperbolic system arising from chemotaxis. Suppose that the Riemann problem to the corresponding inviscid system admits rarefaction waves. We show that if the initial data is a small perturbation of an approximate rarefaction wave, then the Cauchy problem has a unique global solution that converges to the rarefaction wave asymptotically in time. The stability results hold regardless of the wave strengths. The waves can be either a single rarefaction wave or a composite of two rarefaction waves. In the composite case, we separate the initial centers of the two approximate profiles to make their interaction error term small while retaining convergence to the prescribed Riemann wave. The proofs are based on the energy method, where the key observations are the monotonicity of the approximate rarefaction waves with respect to both space and time, and the appropriate replacement of spatial derivative of the hyperbolic component with the parabolic component in proper forms.
Source: arXiv (Cornell University)
Authors: Liu, Sitong, Li, Jingyu
Institution: Northeast Normal University
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G3064071896
OpenAlex ID: W7163194965