Poincar\'e compactification for semiflows of reaction-diffusion equations with large diffusion and convection heating at the boundary

TL;DR

This paper develops a Poincaré compactification method for analyzing the long-term behavior of a coupled PDE-ODE system with large diffusion and boundary heating, using invariant manifold theory.

Contribution

It introduces a novel approach to compactify and analyze the semiflow of reaction-diffusion equations with boundary coupling and large diffusion.

Findings

Successfully applies invariant manifold theorem to reduce PDE dimension.

Shows compactified vector fields are close in the $C^1$-norm.

Provides conditions for the method's applicability.

Abstract

In this paper, we study the Poincar\'e 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.