Motion of interfaces for a damped hyperbolic Allen–Cahn equation

Raffaele Folino; Corrado Lattanzio; Corrado Mascia

doi:10.3934/cpaa.2020205

Article Contents

2020, Volume 19, Issue 9: 4507-4543. Doi: 10.3934/cpaa.2020205

This issue Previous Article Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing Next Article Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential

Motion of interfaces for a damped hyperbolic Allen–Cahn equation

1.
Departamento de Matemáticas y Mecánica, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Circuito Escolar s/n, Ciudad Universitaria, 04510, Cd. de México, Mexico
2.
Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università degli Studi dell'Aquila, Via Vetoio, 67100, L'Aquila, Italy
3.
Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro, 2, 00185, Roma, Italy

^* Corresponding author
^* Corresponding author

Received: November 30, 2019

Revised: January 31, 2020

Published: June 2020

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Abstract

This paper concerns with the motion of the interface for a damped hyperbolic Allen–Cahn equation, in a bounded domain of $ \mathbb{R}^n $, for $ n = 2 $ or $ n = 3 $. In particular, we focus the attention on radially symmetric solutions and extend to the hyperbolic framework some well-known results of the classic parabolic case: it is shown that, under appropriate assumptions on the initial data and on the boundary conditions, the interface moves by mean curvature as the diffusion coefficient goes to $ 0 $.

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Mathematics Subject Classification: Primary: 35L70; Secondary: 35B25, 35K57.

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Figure 1. Solution for $ \tau = 1 $, $ \varepsilon = 0.02 $ and different values of $ t $. Top left: $ t = 0 $, top right: $ t = 250 $, bottom left: $ t = 400 $, bottom right: $ t = 450 $

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