On bifurcation from infinity: a compactification approach

TL;DR

This paper introduces a compactification method to analyze bifurcations from infinity in scalar parabolic PDEs with nonlinear boundary conditions, revealing new insights into equilibrium behavior and blow-up solutions.

Contribution

It develops a novel compactification approach to study unbounded bifurcation curves, stability, and heteroclinic orbits, connecting large equilibria to eigenfunctions.

Findings

Existence of bifurcation from infinity at critical parameter values.

Construction of an induced semiflow at infinity with eigenfunctions as equilibria.

Proof of infinite-time blow-up solutions converging to eigenfunctions.

Abstract

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.