Communications in Mathematical Sciences

Volume 15 (2017)

Number 7

Metastable dynamics for hyperbolic variations of the Allen–Cahn equation

Pages: 2055 – 2085

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n7.a12

Authors

Raffaele Folino (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Coppito (L’Aquila), Italy)

Corrado Lattanzio (Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Coppito (L’Aquila), Italy)

Corrado Mascia (Dipartimento di Matematica, Sapienza Università di Roma, Italy)

Abstract

Metastable dynamics of a hyperbolic variation of the Allen–Cahn equation with homogeneous Neumann boundary conditions are considered. Using the “dynamical approach” proposed by Carr–Pego [J. Carr and R.L. Pego, Comm. Pure Appl. Math., 42:523–576, 1989] and Fusco–Hale [G. Fusco and J. Hale, J. Dynamics Diff. Eqs., 1:75–94, 1989] to study slow-evolution of solutions in the classic parabolic case, we prove existence and persistence of metastable patterns for an exponentially long time. In particular, we show the existence of an “approximately invariant” $N$-dimensional manifold $\mathcal{M}_0$ for the hyperbolic Allen–Cahn equation: if the initial datum is in a tubular neighborhood of $\mathcal{M}_0$, the solution remains in such neighborhood for an exponentially long time. Moreover, the solution has $N$ transition layers and the transition points move with exponentially small velocity. In addition, we determine the explicit form of a system of ordinary differential equations describing the motion of the transition layers and we analyze the differences with the corresponding motion valid for the parabolic case.

Keywords

Allen–Cahn equation, metastability, singular perturbations

2010 Mathematics Subject Classification

35B25, 35K57, 35L72

Received 30 October 2016

Accepted 5 July 2017

Published 16 October 2017