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

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.