Nonlinear stability of diffusive contact wave for a chemotaxis model

• Published in: Journal of Differential Equations Vol. 308; pp. 286 - 326
• Main Author: Zeng, Yanni
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 25.01.2022
• Subjects: Asymptotic behavior; Chemotaxis; Diffusive contact wave; Hyperbolic-parabolic balance laws; Logarithmic singularity; Logistic growth; 35B35; Logarithmic singularity; 35B45; Logistic growth; 35A01; Asymptotic behavior; 35B40; Diffusive contact wave; Chemotaxis; 35M31; 35Q92; Hyperbolic-parabolic balance laws
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2021.11.008

We consider a 2×2 system of hyperbolic-parabolic balance laws. Our system is the converted form under inverse Hopf-Cole transformation of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity, non-diffusive chemical signal and density-dependent production/consumption rate. We study Cauchy problem when the Cauchy data are near a diffusive contact wave. The contact wave connects two different end-states as x→±∞, reflecting the situation when the logarithmic singularity plays an intrinsic role in the original chemotaxis model. We establish global existence of solution and study time asymptotic behavior of the solution. Consequently, we obtain nonlinear stability of the diffusive contact wave. Our result shows a significant difference when comparing our model to Euler equations with damping. In our case, there exists a secondary wave in the asymptotic ansatz. Therefore, the solution to Cauchy problem converges to the diffusive contact wave slower than in the case of Euler equations with damping. Besides its own physical relevance, our model is a prototype of a general system of hyperbolic-parabolic balance laws. Our results shed light on the future study of nonlinear stability of elementary waves for a general system.