On convergence in smooth gradient systems with branching of equilibria

TL;DR

This paper extends Zelenyak's 1968 methodology to N-dimensional parabolic equations, demonstrating exponential convergence to degenerate equilibria in both autonomous and perturbed settings within Hilbert spaces.

Contribution

It generalizes Zelenyak's approach to higher-dimensional parabolic systems, providing a framework for exponential convergence analysis.

Findings

Exponential convergence to degenerate equilibria established

Applicable to autonomous and perturbed equations in Hilbert spaces

Methodology extends to second- and higher-order parabolic equations

Abstract

T.I. Zelenyak's ideas and method of 1968 are shown to apply to N-dimensional second- and higher-order parabolic equtions and ensure a fast exponential convergence to degenerate equilibria. The methotology applies to autonomous and perturbed equations in Hilbert spaces.