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

Raffaele Folino

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 $\varepsilon\ll 1$ is the diffusion coefficient, we show that in a time scale of order $\varepsilon^{-k}$ 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 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 and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen-Cahn equation on the interval 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 is the diffusion coefficient, we show that in a time scale of order nothing happens and the solution maintains the same number of transitions of its initial datum . The novelty consists mainly in the role of the initial velocity , 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.

Paper Structure

This paper contains 4 sections, 12 theorems, 138 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Slow evolution
  3. Numerical explorations
  4. Existence and uniqueness

Key Result

Theorem 1.2

We consider the initial boundary value problem Cauchy problem with $F$ and $g$ satisfying g strettamente positiva-hp-F-3. We suppose that the initial data $u^\varepsilon_0, u^\varepsilon_1$ satisfy u^eps tende a v and energia iniziale< for some $k>0$. Then for any $m>0$

Figures (4)

  • Figure 1: Initial data: $u_0(x)=\cos(\frac{\pi}{2} x)/10$, $u_1(x)=0$. The values of constants are: $\varepsilon=0.01, \tau=0.8$.
  • Figure 2: Initial data: $u_0(x)=0$, $u_1(x)=\cos(\frac{\pi}{2} x)$. The values of constants are: $\varepsilon=0.1, \tau=0.8$.
  • Figure 3: Initial data: $u_0(x)=\tanh\left(\frac{x}{\sqrt2\varepsilon}\right)$, $u_1(x)=-x$. The values of constants are: $\varepsilon=0.2, \tau=0.6$.
  • Figure 4: Initial data: $u_0(x)$ with two transitions and $u_1(x)=-x$. The values of constants are: $\varepsilon=0.01, \tau=0.9$.

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • proof : Proof of Theorem \ref{['metastability-thm']}
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 12 more