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

Abstract

The aim of this paper is to study the metastable properties of the solutions to a hyperbolic relaxation of the classic Cahn‐Hilliard equation in one‐space dimension, subject to either Neumann or Dirichlet boundary conditions. To perform this goal, we make use of an “energy approach," already proposed for various evolution PDEs, including the Allen‐Cahn and the Cahn‐Hilliard equations. In particular, we shall prove that certain solutions maintain a N‐transition layer structure for a very long time, thus proving their metastable dynamics. More precisely, we will show that, for an exponentially long time, such solutions are very close to piecewise constant functions assuming only the minimal points of the potential, with a finitely number of transition layers, which move with an exponentially small velocity.

Metastable dynamics for hyperbolic variations of the Allen–Cahn equation

“…which permits to prove that some solutions to (1.1)-(1.2) maintain the same structure of the initial datum for the time T ε as ε → 0 + , for details see [6,17,19]. We stress again that the key point of the energy approach is the estimate (2.2), which implies (2.7).…”

“…represents the minimum energy to have a transition between −1 and +1 [6,17,19]. An example of initial data satisfying (2.8) can be found in [19].…”

“…In this paper, we assume that (1.14) is satisfied and then we study the metastable dynamics of the solutions when the mass is conserved. It is worth to stress that, by using the energy functional (1.15) and adapting the procedure of [19], one can prove the exponentially slow motion of the solutions also without the assumption (1.14) (see Section 2.1). On the contrary, the strictly positiveness of the damping coefficient g (1.4) is crucial because it guarantees the dissipative character of equation (1.1).…”

On the speed rate of convergence of solutions to conservation laws with nonlinear diffusions

“…(where P ε is a nonlinear differential operator that is singular with respect to the parameter ε) approaches his stable (metastable) steady state in an exponentially long time interval, usually of the order O(exp (c/ε)). There is a huge literature investigating such phenomenon for different evolution PDEs and by means of different techniques: far from being exhaustive see, among others, [2,9,12,13,24,30] for Allen-Cahn and Cahn-Hilliard like equations, [20,27,29,32] for viscous shock problems, and the references therein.…”

Stability properties and dynamics of solutions to viscous conservation laws with mean curvature operator

“…Among others, we include viscous shock problems (see, for example [16,17,20,25] for viscous conservation laws, and [3,27,30] for Burgers type equations), phase transition problems described by the Allen-Cahn equation, with the fundamental contributions [4,10] and the most recent references [22,28], and the Cahn-Hilliard equation studied in [2] and [24]. We finally quote some recent papers on metastability for hyperbolic versions of both the Allen-Cahn and the Cahn-Hilliard equation [6,7,8].…”