2024/10/29 by José M. Arrieta, Arrieta, José M., Juliana Fernandes +3
Engineering · Mathematics · #35B40 #35B44 #35K55 #37B35 #37G35 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #Dynamics and Control of Mechanical Systems #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2410.22146
openalex publication_date 2024/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a scalar parabolic partial differential equation on the interval with nonlinear boundary conditions that are asymptotically sublinear. As the parameter crosses critical values (e.g. the Steklov eigenvalues), it is known that there are large equilibria that arise through a bifurcation from infinity (i.e., such equilibria converge, after rescaling, to the Steklov eigenfunctions). We provide a compactification approach to the study of such unbounded bifurcation curves of equilibria, their stability, and heteroclinic orbits. In particular, we construct an induced semiflow at infinity such that the Steklov eigenfunctions are equilibria. Moreover, we prove the existence of infinite-time blow-up solutions that converge, after rescaling, to certain eigenfunctions that are equilibria of the induced semiflow at infinity.