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 and Fusco-Hale 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 M_0 for the hyperbolic Allen-Cahn equation: if the initial datum is in a tubular neighborhood of 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 dierential equations describing the motion of the transition layers and we analyze the dierences with the corresponding motion valid for the parabolic case.
Metastable dynamics for hyperbolic variations of Allen-Cahn equation / Folino, Raffaele; Lattanzio, Corrado; Mascia, Corrado. - In: COMMUNICATIONS IN MATHEMATICAL SCIENCES. - ISSN 1539-6746. - STAMPA. - ???:???(In corso di stampa), pp. ???-???.
Metastable dynamics for hyperbolic variations of Allen-Cahn equation
MASCIA, Corrado
In corso di stampa
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 and Fusco-Hale 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 M_0 for the hyperbolic Allen-Cahn equation: if the initial datum is in a tubular neighborhood of 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 dierential equations describing the motion of the transition layers and we analyze the dierences with the corresponding motion valid for the parabolic case.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.