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September  2020, 19(9): 4507-4543. doi: 10.3934/cpaa.2020205

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

Received  December 2019 Revised  February 2020 Published  June 2020

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 $.

Citation: Raffaele Folino, Corrado Lattanzio, Corrado Mascia. Motion of interfaces for a damped hyperbolic Allen–Cahn equation. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4507-4543. doi: 10.3934/cpaa.2020205
References:
[1]

M. AlfaroD. Hilhorst and H. Matano, The singular limit of the Allen–Cahn equation and the FitzHugh–Nagumo system, J. Differ. Equ., 245 (2008), 505-565.  doi: 10.1016/j.jde.2008.01.014.  Google Scholar

[2]

S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085-1095.   Google Scholar

[3]

L. Bronsard and R. V. Kohn, On the slowness of phase boundary motion in one space dimension, Commun. Pure Appl. Math., 43 (1990), 983-997.  doi: 10.1002/cpa.3160430804.  Google Scholar

[4]

L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg–Landau dynamics, J. Differ. Equ., 90 (1991), 211-237.  doi: 10.1016/0022-0396(91)90147-2.  Google Scholar

[5]

J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t = \varepsilon^2u_xx-f(u)$, Commun. Pure Appl. Math., 42 (1989), 523-576.  doi: 10.1002/cpa.3160420502.  Google Scholar

[6]

C. Cattaneo, Sulla conduzione del calore, Atti del Semin. Mat. e Fis. Univ. Modena, 3 (1948), 83-101.   Google Scholar

[7]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differ. Equ., 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[8]

X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differ. Equ., 206 (2004), 399-437.  doi: 10.1016/j.jde.2004.05.017.  Google Scholar

[9]

Y. G. ChenY. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differ. Geom., 33 (1991), 749-786.   Google Scholar

[10]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.  doi: 10.2307/2154960.  Google Scholar

[11]

L. C. EvansH. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Commun. Pure Appl. Math., 45 (1992), 1097-1123.  doi: 10.1002/cpa.3160450903.  Google Scholar

[12]

L. C. Evans and J. Spruck, Motion of level set by mean curvature I, J. Differ. Geom., 33 (1991), 635-681.   Google Scholar

[13]

R. Folino, Slow motion for a hyperbolic variation of Allen–Cahn equation in one space dimension, J. Hyperbolic Differ. Equ., 14 (2017), 1-26.  doi: 10.1142/S0219891617500011.  Google Scholar

[14]

R. Folino, Slow motion for one-dimensional nonlinear damped hyperbolic Allen–Cahn systems, Electron. J. Differ. Equ., 2019 (2019), Art. 113, pp 21.  Google Scholar

[15]

R. FolinoC. Lattanzio and C. Mascia, Metastable dynamics for hyperbolic variations of the Allen–Cahn equation, Commun. Math. Sci., 15 (2017), 2055-2085.  doi: 10.4310/CMS.2017.v15.n7.a12.  Google Scholar

[16]

G. Fusco and J. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dyn. Differ. Equ., 1 (1989), 75-94.  doi: 10.1007/BF01048791.  Google Scholar

[17]

E. Giusti, Minimal Surfaces and Functions of Bounded Variations, Birkhäuser, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[18]

L. Herrera and D. Pavon, Hyperbolic theories of dissipation: Why and when do we need them, Physica A, 307 (2002), 121-130.   Google Scholar

[19]

D. Hilhorst and M. Nara, Singular limit of a damped wave equation with a bistable nonlinearity, SIAM J. Math. Anal., 46 (2014), 1701-1730.  doi: 10.1137/130921945.  Google Scholar

[20]

T. Hillen, Qualitative analysis of semilinear Cattaneo equations, Math. Models Meth. Appl. Sci., 8 (1998), 507-519.  doi: 10.1142/S0218202598000238.  Google Scholar

[21]

E. E. Holmes, Are diffusion models too simple? A comparison with telegraph models of invasion, Amer. Naturalist, 142 (1993), 779-795.   Google Scholar

[22]

D. D. Joseph and L. Preziosi, Heat waves, Rev. Modern Phys., 61 (1989), 41-73.  doi: 10.1103/RevModPhys.61.41.  Google Scholar

[23]

D. D. Joseph and L. Preziosi, Addendum to the paper: "Heat waves" [Rev. Modern Phys. 61 (1989), 41–73], Rev. Modern Phys., 62 (1990), 375-391.  doi: 10.1103/RevModPhys.62.375.  Google Scholar

[24]

C. LattanzioC. MasciaR. G. Plaza and C. Simeoni, Analytical and numerical investigation of traveling waves for the Allen–Cahn model with relaxation, Math. Models Meth. Appl. Sci., 26 (2016), 931-985.  doi: 10.1142/S0218202516500226.  Google Scholar

[25]

V. Mendez, S. Fedotov and W. Horsthemke, Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11443-4.  Google Scholar

[26]

J. RubinsteinP. Sternberg and J. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133.  doi: 10.1137/0149007.  Google Scholar

[27]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Ration. Mech. Anal., 101 (1988), 209-260.  doi: 10.1007/BF00253122.  Google Scholar

show all references

References:
[1]

M. AlfaroD. Hilhorst and H. Matano, The singular limit of the Allen–Cahn equation and the FitzHugh–Nagumo system, J. Differ. Equ., 245 (2008), 505-565.  doi: 10.1016/j.jde.2008.01.014.  Google Scholar

[2]

S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085-1095.   Google Scholar

[3]

L. Bronsard and R. V. Kohn, On the slowness of phase boundary motion in one space dimension, Commun. Pure Appl. Math., 43 (1990), 983-997.  doi: 10.1002/cpa.3160430804.  Google Scholar

[4]

L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg–Landau dynamics, J. Differ. Equ., 90 (1991), 211-237.  doi: 10.1016/0022-0396(91)90147-2.  Google Scholar

[5]

J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t = \varepsilon^2u_xx-f(u)$, Commun. Pure Appl. Math., 42 (1989), 523-576.  doi: 10.1002/cpa.3160420502.  Google Scholar

[6]

C. Cattaneo, Sulla conduzione del calore, Atti del Semin. Mat. e Fis. Univ. Modena, 3 (1948), 83-101.   Google Scholar

[7]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differ. Equ., 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[8]

X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differ. Equ., 206 (2004), 399-437.  doi: 10.1016/j.jde.2004.05.017.  Google Scholar

[9]

Y. G. ChenY. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differ. Geom., 33 (1991), 749-786.   Google Scholar

[10]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.  doi: 10.2307/2154960.  Google Scholar

[11]

L. C. EvansH. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Commun. Pure Appl. Math., 45 (1992), 1097-1123.  doi: 10.1002/cpa.3160450903.  Google Scholar

[12]

L. C. Evans and J. Spruck, Motion of level set by mean curvature I, J. Differ. Geom., 33 (1991), 635-681.   Google Scholar

[13]

R. Folino, Slow motion for a hyperbolic variation of Allen–Cahn equation in one space dimension, J. Hyperbolic Differ. Equ., 14 (2017), 1-26.  doi: 10.1142/S0219891617500011.  Google Scholar

[14]

R. Folino, Slow motion for one-dimensional nonlinear damped hyperbolic Allen–Cahn systems, Electron. J. Differ. Equ., 2019 (2019), Art. 113, pp 21.  Google Scholar

[15]

R. FolinoC. Lattanzio and C. Mascia, Metastable dynamics for hyperbolic variations of the Allen–Cahn equation, Commun. Math. Sci., 15 (2017), 2055-2085.  doi: 10.4310/CMS.2017.v15.n7.a12.  Google Scholar

[16]

G. Fusco and J. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dyn. Differ. Equ., 1 (1989), 75-94.  doi: 10.1007/BF01048791.  Google Scholar

[17]

E. Giusti, Minimal Surfaces and Functions of Bounded Variations, Birkhäuser, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[18]

L. Herrera and D. Pavon, Hyperbolic theories of dissipation: Why and when do we need them, Physica A, 307 (2002), 121-130.   Google Scholar

[19]

D. Hilhorst and M. Nara, Singular limit of a damped wave equation with a bistable nonlinearity, SIAM J. Math. Anal., 46 (2014), 1701-1730.  doi: 10.1137/130921945.  Google Scholar

[20]

T. Hillen, Qualitative analysis of semilinear Cattaneo equations, Math. Models Meth. Appl. Sci., 8 (1998), 507-519.  doi: 10.1142/S0218202598000238.  Google Scholar

[21]

E. E. Holmes, Are diffusion models too simple? A comparison with telegraph models of invasion, Amer. Naturalist, 142 (1993), 779-795.   Google Scholar

[22]

D. D. Joseph and L. Preziosi, Heat waves, Rev. Modern Phys., 61 (1989), 41-73.  doi: 10.1103/RevModPhys.61.41.  Google Scholar

[23]

D. D. Joseph and L. Preziosi, Addendum to the paper: "Heat waves" [Rev. Modern Phys. 61 (1989), 41–73], Rev. Modern Phys., 62 (1990), 375-391.  doi: 10.1103/RevModPhys.62.375.  Google Scholar

[24]

C. LattanzioC. MasciaR. G. Plaza and C. Simeoni, Analytical and numerical investigation of traveling waves for the Allen–Cahn model with relaxation, Math. Models Meth. Appl. Sci., 26 (2016), 931-985.  doi: 10.1142/S0218202516500226.  Google Scholar

[25]

V. Mendez, S. Fedotov and W. Horsthemke, Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11443-4.  Google Scholar

[26]

J. RubinsteinP. Sternberg and J. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133.  doi: 10.1137/0149007.  Google Scholar

[27]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Ration. Mech. Anal., 101 (1988), 209-260.  doi: 10.1007/BF00253122.  Google Scholar

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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