Geometric analysis of a truncated Galerkin discretization of fast-slow PDEs with transcritical singularities

TL;DR

This paper investigates the geometric structure of a Galerkin discretization of fast-slow PDEs with transcritical bifurcations, providing insights into invariant manifolds and approximation of the infinite-dimensional dynamics.

Contribution

It introduces a geometric blow-up analysis for Galerkin approximations of PDEs with transcritical bifurcations, linking finite-dimensional and infinite-dimensional dynamics.

Findings

Invariant manifolds characterized via blow-up analysis.

Galerkin PDE approximations relate to classical reaction-diffusion problems.

Analysis of PDEs in blow-up charts as free boundary value problems.

Abstract

We consider a fast-slow partial differential equation (PDE) with reaction-diffusion dynamics in the fast variable and the slow variable driven by a differential operator on a bounded domain. Assuming a transcritical normal form for the reaction term and viewing the slow variable as a dynamic bifurcation parameter, we analyze the passage through the fast subsystem bifurcation point for the spectral Galerkin approximation of the PDE. We characterize the invariant manifolds for the finite-dimensional Galerkin ODEs using geometric desingularization via a blow-up analysis. In addition to the crucial approximation procedure, we also make the domain dynamic during the blow-up analysis. Finally, we elaborate in which sense our results approximate the infinite-dimensional problem. Within our analysis, we find that the PDEs appearing in entry and exit blow-up charts are quasi-linear free boundary value problems, while in the central/scaling chart we obtain a PDE, which is often encountered in classical reaction-diffusion problems exhibiting solutions with finite-time singularities.