A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation

• Published in: Journal of functional analysis 2021-08, Vol.281 (4), p.109069, Article 109069
• Main Authors: Tao, Youshan, Winkler, Michael
• Format: Article
• Language: English
• Subjects: Asymptotic behavior; Cross-diffusion; Global existence; Thin-film equation
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2021.109069

This work is concerned with a two-component parabolic system accounting for a doubly cross-diffusive interaction mechanism which was predicted in Tsyganov et al. (2003) [48] as responsible for the occurrence of certain solitary propagating waves in so-called pursuit-evasion systems. This system formally possesses two basic entropy-like structures, but especially in the presence of large data the regularity features thereby implied seem insufficient to ensure global extensibility of local-in-time classical solutions provided by known results on classical solvability in general parabolic systems of not necessarily tridiagonal type. Attempting to nevertheless develop a basic theory of existence and qualitative behavior, the manuscript firstly constructs global solutions within a natural concept of weak solvability and for arbitrarily large data, and secondly derives a result on large-time stabilization toward homogeneous equilibria. A major challenge connected with this appears to consist in designing a suitable regularization which complies with the two requirements of asserting global solvability in the corresponding approximate systems on the one hand, and of retaining consistency with essential structural properties on the other. To adequately cope with this, a fourth-order regularization is pursued which, besides essentially respecting said entropy features, conforms to the fundamental sine qua non of positivity preservation by involving thin-film type degeneracies in the associated artificial diffusion operators. Here the use of embeddings enforces a restriction to spatially one-dimensional settings, in which an apparently novel refinement of Gagliardo-Nirenberg interpolation reveals a crucial L1 compactness feature of corresponding cross-diffusive fluxes.