Boundedness and stability of a 2-D parabolic-elliptic system arising in biological transport networks
Jose A. Carrillo, Bin Li, Li Xie
TL;DR
This work analyzes a 2-D parabolic-elliptic system modeling biological transport networks under Dirichlet boundary conditions. It proves that for $\gamma \ge 1$ and sufficiently large diffusion $\kappa$, time-dependent classical solutions are globally bounded and unique, and the stationary solution is unique and semi-trivial with exponential global stability toward $(0, p_\infty^*)$, where $-\Delta p_\infty^* = S$. The methodology hinges on a robust energy-dissipation framework together with higher-order elliptic regularity to achieve uniform-in-time bounds and to control the challenging cubic nonlinearity and potential elliptic singularities. The results also reveal a bifurcation-type behavior with respect to $\kappa$, clarifying the role of diffusion in BTN network formation and long-time dynamics. Overall, the paper provides a rigorous foundation for infinite-time regularity, uniqueness, and exponential stabilization in this biologically relevant PDE model.
Abstract
This paper is concerned with the Dirichlet initial-boundary value problem of a 2-D parabolic-elliptic system proposed to model the formation of biological transport networks. Even if global weak solutions for this system are known to exist, how to improve the regularity of weak solutions is a challenging problem due to the peculiar cubic nonlinearity and the possible elliptic singularity of the system. Global-in-time existence of classical solutions has recently been established showing that finite time singularities cannot emerge in this problem. However, whether or not singularities in infinite time can be precluded was still pending. In this work, we show that classical solutions of the initial-boundary value problem are uniformly bounded in time as long as $γ\geq1$ and $κ$ is suitably large, closing this gap in the literature. Moreover, uniqueness of classical solutions is also achieved based on the uniform-in-time bounds. Furthermore, it is shown that the corresponding stationary problem possesses a unique classical stationary solution which is semi-trivial, and that is globally exponentially stable, that is, all solutions of the time dependent problem converge exponentially fast to the semi-trivial steady state for $κ$ large enough.