Delayed loss of stability in singularly perturbed finite-dimensional gradient flows
Giovanni Scilla, Francesco Solombrino · Asymptotic Analysis · 2018
In this paper we study the singular vanishing-viscosity limit of a gradient flow in a finite dimensional Euclidean 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 [Formula: see text] of the limit. For our class of functionals, [Formula: see text] 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 [Formula: see text] where strict local minimality with respect to the driving energy gets lost. Moreover, we show that, in a right neighborhood of [Formula: see text], rescaled solutions of the singularly perturbed problem converge to heteroclinic solutions of the gradient flow. Our results complement the previous ones by Zanini [ Discrete Contin. Dyn. Syst. Ser. A 18 ( 2007 ), 657–675], 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.