Delayed loss of stability in singularly perturbed finite-dimensional gradient flows

TL;DR

This paper investigates the delayed loss of stability in finite-dimensional gradient flows under singular perturbations, explicitly calculating the critical time for stability loss and analyzing the convergence of solutions near this point.

Contribution

It introduces a class of energy functionals and initial conditions allowing explicit calculation of the stability loss time, extending previous results by relaxing transversality assumptions.

Findings

Explicit formula for the first discontinuity time $t^*$

$t^*$ coincides with the blow-up time of the linearized system

Rescaled solutions converge to heteroclinic solutions near $t^*$

Abstract

In this paper we study the singular vanishing-viscosity limit of a gradient flow in a finite dimensional Hilbert space, focusing on the so-called delayed loss of stability of stationary solutions. We find a class of time-dependent energy functionals and initial conditions for which we can explicitly calculate the first discontinuity time $t^*$ of the limit. For our class of functionals, $t^*$ coincides with the blow-up time of the solutions of the linearized system around the equilibrium, and is in particular strictly greater than the time $t_c$ where strict local minimality with respect to the driving energy gets lost. Moreover, we show that, in a right neighborhood of $t^*$, rescaled solutions of the singularly perturbed problem converge to heteroclinic solutions of the gradient flow. Our results complement the previous ones by Zanini, where the situation we consider was excluded by assuming the so-called transversality conditions, and the limit evolution consisted of strict local minimizers of the energy up to a negligible set of times.