Metastable dynamics for hyperbolic variations of the Allen–Cahn equation

Raffaele Folino, Corrado Lattanzio, Corrado Mascia
2017 Communications in Mathematical Sciences  
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 [10] and Fusco-Hale [19] 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
more » ... tion: 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 differential equations describing the motion of the transition layers and we analyze the differences with the corresponding motion valid for the parabolic case.
doi:10.4310/cms.2017.v15.n7.a12 fatcat:ov3wuo2ujffjpidrg3y3g3zela