Slow motion for a hyperbolic variation of Allen–Cahn equation in one space dimension

Abstract

The aim of this paper is to prove that, for specific initial data (u 0 , u 1 ) and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen-Cahn equation on the interval [a, b] shares the well-known dynamical metastability valid for the classical parabolic case. In particular, using the "energy approach" proposed by Bronsard and Kohn [8], if ε 1 is the diffusion coefficient, we show that in a time scale of order ε −k nothing happens and the solution maintains the same number of transitions of its initial datum u 0 . The novelty consists mainly in the role of the initial velocity u 1 , which may create or eliminate transitions in later times. Numerical experiments are also provided in the particular case of the Allen-Cahn equation with relaxation.

Slow dynamics for the hyperbolic Cahn‐Hilliard equation in one‐space dimension

“…The results on metastability for the Allen‐Cahn Equation can be extended to the hyperbolic Allen‐Cahn equation where is strictly positive (see other studies). In particular, in the past study, using the dynamical approach, the authors prove existence and persistence of metastable states for an exponentially long time.…”

“…where g ∶ R → R is strictly positive (see other studies [26][27][28] ). In particular, in the past study, 28 using the dynamical approach, the authors prove existence and persistence of metastable states for an exponentially long time.…”

“…On the other hand, the energy approach is applied to (1.18) in the other research. 27 In this case, the energy functional is…”

“…If u is a solution of with homogeneous Neumann boundary conditions, then E ε [ u , u t ] is a non‐increasing function of t , namely, In order to have slow motion of the solutions, the initial velocity u 1 = u t (·,0) has to be very small. Precisely, in the previous work, it is proved that if the initial profile u 0 has a transition layer structure of order k and the L 2 –norm of the u 1 is bounded by , then the solution maintains the same transition layer structure of its initial datum for a time T ε ≥ ε − k . These results are extended to the case of systems (when u is a vector‐valued function) in the other paper, where imposing stronger conditions on the initial data, persistence of metastable states for an exponentially long time is obtained.…”

“…In particular, in the past study, using the dynamical approach, the authors prove existence and persistence of metastable states for an exponentially long time. On the other hand, the energy approach is applied to in the other research . In this case, the energy functional is where P ε is defined in .…”