A priori Bound of Solutions to a Class of Elliptic/Parabolic Cross Diffusion Systems of $m$ Equations on Two/Three Dimensional Domains. Existence on thin domains

TL;DR

This paper investigates bounds and existence of solutions for elliptic and parabolic cross-diffusion systems with multiple equations on low-dimensional domains, highlighting the critical role of domain thinness in three-dimensional cases.

Contribution

It provides new bounds, existence results, and counter-examples for cross-diffusion systems, especially emphasizing the necessity of domain thinness in three dimensions.

Findings

Established bounds for solutions on 2D/3D domains.

Proved existence and global existence under certain conditions.

Demonstrated the critical role of domain thinness in 3D cases.

Abstract

We establish several bounds for solutions to elliptic/parabolic cross-diffusion systems of $m$ equations ($m\ge2$) on 2d/3d domains $\Og$. We settle the existence and global existence problems in these cases and also provide new counter-examples for the case of general dimensions. Most importantly, we prove that when $m=N=3$, the thinness of $\Og$ in $\RR^3$ is sufficient and necessary. When $m$ is arbitrary and $N=3$, we establish global existence results for nonlinear cross-diffusion systems (the case of the scalar semilinear equation was considered in the literature but the classical methods do not seem to be applicable here).