Metastable dynamics of a hyperbolic variation of the Allen–Cahn equation with homo- geneous 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 M0 for the hyperbolic Allen–Cahn equation: if the initial datum is in a tubular neighborhood of M0 , 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.

Metastable dynamics for hyperbolic variations of the Allen-Cahn equation / Folino, Raffaele; Lattanzio, Corrado; Mascia, Corrado. - In: COMMUNICATIONS IN MATHEMATICAL SCIENCES. - ISSN 1539-6746. - STAMPA. - 15:7(2017), pp. 2055-2085.

Metastable dynamics for hyperbolic variations of the Allen-Cahn equation

Mascia, Corrado
2017

Abstract

Metastable dynamics of a hyperbolic variation of the Allen–Cahn equation with homo- geneous 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 M0 for the hyperbolic Allen–Cahn equation: if the initial datum is in a tubular neighborhood of M0 , 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.
File allegati a questo prodotto
File Dimensione Formato  
Folino_Metastable-dynamics_2017.pdf

solo gestori archivio

Tipologia: Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 501.47 kB
Formato Adobe PDF
501.47 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
Folino_preprint_Metastable-dynamics_2017.pdf

accesso aperto

Tipologia: Documento in Pre-print (manoscritto inviato all'editore, precedente alla peer review)
Licenza: Creative commons
Dimensione 635.46 kB
Formato Adobe PDF
635.46 kB Adobe PDF Visualizza/Apri PDF

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1072583
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 14
  • ???jsp.display-item.citation.isi??? 10
social impact