Blow-up analysis of fast-slow PDEs with loss of hyperbolicity
Maximilian Engel, Christian Kuehn · arXiv (Cornell University) · 2020
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. In particular, we employ a spectral Galerkin approximation and characterize the invariant manifolds for the finite-dimensional Galerkin approximation for each finite truncation using geometric desingularization via a blow-up analysis. In addition to the crucial approximation procedure, a key step is to make the domain dynamic as well during the blow-up analysis. Finally, we prove that our results extend to the infinite-dimensional problem, showing the convergence of the finite-dimensional manifolds to infinite-dimensional Banach manifolds for different parameter regimes near the bifurcation point. 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. In summary, we establish a first full case of a geometric blow-up analysis for fast-slow PDEs with a non-hyperbolic point. Our methodological approach has the potential to deal with the loss of hyperbolicity for a wide variety of infinite-dimensional dynamical systems.