Metastable dynamics for hyperbolic variations of the Allen–Cahn equation

Folino, Raffaele; Lattanzio, Corrado

doi:10.4310/cms.2017.v15.n7.a12

Cited by 19 publications

(19 citation statements)

References 55 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This approach can be refined to obtain persistence of the layered structure for time intervals of O(e C/ε ) and extended to the case when u is vector-valued. These topics, as well as the study of metastability for hyperbolic Allen-Cahn equation with the dynamical approach of Carr and Pego, are addressed in [12,13].…”

Section: Introductionmentioning

confidence: 99%

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

Folino

2017

J. Hyper. Differential Equations

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Folino

2017

J. Hyper. Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…We conclude by remarking that such peculiar feature is not shared by other more general forms of relaxation system, for instance τ ∂ tt u + g(t, x, u ; τ ) ∂ t u − µ ∂ xx u = f (u) , that is considered in [6], for example. Unless specific expression for the external field g are taken into account for physical reasons, the only approach to the numerical approximation of (3.6) seems to be the transcription into a first order system by putting ∂ t u = w , τ ∂ t w + g(t, x, u ; τ ) w − µ ∂ xx u = f (u) .…”

Section: 3mentioning

confidence: 87%

Kinetic schemes for assessing stability of traveling fronts for the Allen–Cahn equation with relaxation

Lattanzio

Plaza

Simeoni

2019

Applied Numerical Mathematics

Self Cite

View full text Add to dashboard Cite

This paper deals with the numerical (finite volume) approximation of reaction-diffusion systems with relaxation, among which the hyperbolic extension of the Allen-Cahn equation represents a notable prototype. Appropriate discretizations are constructed starting from the kinetic interpretation of the model as a particular case of reactive jump process. Numerical experiments * are provided for exemplifying the theoretical analysis (previously developed by the same authors) concerning the stability of traveling waves, and important evidence of the validity of those results beyond the formal hypotheses is numerically established.2010 Mathematics Subject Classification. 65M08, 35L60, 35A18.

show abstract

“…The results on metastability for the Allen‐Cahn Equation can be extended to the hyperbolic Allen‐Cahn equation

τ u_{t t} + g false(u false) u_{t} = ε^{2} u_{x x} - W^{'} false(u false),

where

g : double-struckR \to double-struckR

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. On the other hand, the energy approach is applied to in the other research .…”

Section: Introductionmentioning

confidence: 85%

“…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.…”

Section: Metastability For Evolution Pdesmentioning

confidence: 93%

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

Folino

Lattanzio

2019

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Metastable dynamics for hyperbolic variations of the Allen–Cahn equation

Cited by 19 publications

References 55 publications

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

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

Kinetic schemes for assessing stability of traveling fronts for the Allen–Cahn equation with relaxation

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

Contact Info

Product

Resources

About