Poincaré compactification for semiflows of reaction-diffusion equations with large diffusion and convection heating at the boundary
Leonardo Pires
TL;DR
This work investigates the Poincaré compactification of the limiting planar semiflow arising from a reaction-diffusion equation with large diffusion coupled to an ODE via a boundary heating condition. By reducing the PDE-ODE system to a two-dimensional system on an invariant manifold, the authors perform a Poincaré compactification on the Poincaré disk and prove that the compactified vector fields converge in the $C^1$-norm as $\varepsilon \to 0$, establishing a robust link between the perturbed and unperturbed dynamics. The analysis relies on a rigorous invariant-manifold framework, spectral decompositions, and careful control of nonlinear terms (made globally Lipschitz via truncation) under dissipativity and degree assumptions. An explicit example with polynomial nonlinearities demonstrates the limiting planar dynamics and highlights potential changes in equilibria at infinity upon compactification, including the possible loss of Morse–Smale structure due to newly introduced equilibria at the boundary. The results provide a rigorous bridge between high-dimensional PDE-ODE behavior and low-dimensional compactified dynamics, with implications for understanding heating-boundary effects in diffusion-dominated regimes.
Abstract
In this paper, we study the Poincaré compactification of the limiting planar semiflow of a coupled PDE-ODE system composed by a reaction-diffusion equation with large diffusion coupled with an ODE by a boundary condition in a heating transition region. The nonlinear sources are dissipative polynomials. We guarantee conditions to apply the Invariant Manifold Theorem in order to reduce the dimension of the PDE and we prove that the compactified vector fields are close in the $C^1$-norm.