Blow-up analysis of fast-slow PDEs with loss of hyperbolicity

• DOI: 10.48550/arxiv.2007.09973
• Publication Date: 2020-01-01
• Authors: Maximilian Engel; Christian Kuehn
• Keywords: 34E15, 35K57, 37G10, 37L15, 37L65; FOS: Mathematics; Dynamical Systems (math.DS); Mathematics - Dynamical Systems; 0101 mathematics; 01 natural sciences

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