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Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice
Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms
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 C.P. 04510 Cd. Mx. (Mexico) |
2. | Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca' Foscari Venezia Mestre, Campus Scientifico, Via Torino 155, 30170 Venezia Mestre (Italy) |
$ p $ |
$ u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t > 0, $ |
$ \varepsilon>0 $ |
$ p>1 $ |
$ F(u) = \frac{1}{2\theta}|1-u^2|^\theta, $ |
$ \theta>1 $ |
$ \theta\in \mathbb{R} $ |
$ u = \pm1 $ |
$ \pm 1 $ |
$ \varepsilon $ |
$ \theta = p $ |
$ \theta>p $ |
$ \theta<p $ |
References:
[1] |
M. Allen and J. W. Cahn,
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[2] |
J. Benedikt, P. Girg, L. Kotrla and P. Takáč,
Origin of the $p$-Laplacian and A. Missbach, Electron. J. Differ. Eq., 2018 (2018), 1-17.
|
[3] |
F. Bethuel and D. Smets,
Slow motion for equal depth multiple-well gradient systems: The degenerate case, Discrete Contin. Dyn. Syst., 33 (2013), 67-87.
doi: 10.3934/dcds.2013.33.67. |
[4] |
L. Bronsard and R. V. Kohn,
On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43 (1990), 983-997.
doi: 10.1002/cpa.3160430804. |
[5] |
L. Bronsard and R. V. Kohn,
Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237.
doi: 10.1016/0022-0396(91)90147-2. |
[6] |
J. Carr and R. L. Pego,
Metastable patterns in solutions of $u_t = \epsilon^2u_xx-f(u)$, Comm. Pure Appl. Math., 42 (1989), 523-576.
doi: 10.1002/cpa.3160420502. |
[7] |
J. Carr and R. L. Pego,
Invariant manifolds for metastable patterns in $u_t = \epsilon^2u_xx-f(u)$, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 133-160.
doi: 10.1017/S0308210500031425. |
[8] |
M. S. Chang, S. C. Lee and C.-C. Yen,
Minimizers and gamma-convergence of energy functionals derived from $p$-Laplacian equation, Taiwanese J. Math., 13 (2009), 2021-2036.
doi: 10.11650/twjm/1500405655. |
[9] |
X. Chen,
Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.
doi: 10.1016/0022-0396(92)90146-E. |
[10] |
X. Chen,
Generation, propagation, and annihilation of metastable patterns, J. Differential Equations, 206 (2004), 399-437.
doi: 10.1016/j.jde.2004.05.017. |
[11] |
P. de Mottoni and M. Schatzman,
Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.
doi: 10.1090/S0002-9947-1995-1672406-7. |
[12] |
E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[13] |
S. Dipierro, A. Farina and E. Valdinoci,
Density estimates for degenerate double-well potentials, SIAM J. Math. Anal., 50 (2018), 6333-6347.
doi: 10.1137/17M114933X. |
[14] |
S. Dipierro, A. Pinamonti and E. Valdinoci,
Rigidity results for elliptic boundary value problems: Stable solutions for quasilinear equations with Neumann or Robin boundary conditions, Int. Math. Res. Not. IMRN, 2020 (2020), 1366-1384.
doi: 10.1093/imrn/rny055. |
[15] |
L. Evans, H. M. Soner and P. E. Souganidis,
Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.
doi: 10.1002/cpa.3160450903. |
[16] |
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. |
[17] |
R. Folino,
Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems, Electron. J. Differ. Eq., 2019 (2019), 1-21.
|
[18] |
R. Folino, C. A. Hernández Melo, L. F. López Ríos and R. G. Plaza, Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity,, Z. Angew. Math. Phys., 71 (2020), Paper No. 132, 25 pp.
doi: 10.1007/s00033-020-01362-0. |
[19] |
R. Folino, C. Lattanzio and C. Mascia,
Slow dynamics for the hyperbolic Cahn-Hilliard equation in one-space dimension, Math. Methods Appl. Sci., 42 (2019), 2492-2512.
doi: 10.1002/mma.5525. |
[20] |
R. Folino, C. Lattanzio, C. Mascia and M. Strani, Metastability for nonlinear convection-diffusion equations,, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 35, 20 pp.
doi: 10.1007/s00030-017-0459-5. |
[21] |
R. Folino, R. G. Plaza and M. Strani, Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion,, J. Math. Anal. Appl., 493 (2021), 124455, 29pp.
doi: 10.1016/j.jmaa.2020.124455. |
[22] |
G. Fusco and J. K. Hale,
Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations, 1 (1989), 75-94.
doi: 10.1007/BF01048791. |
[23] |
C. P. Grant,
Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34.
doi: 10.1137/S0036141092226053. |
[24] |
E. J. Hurtado and M. Sônego, On the energy functionals derived from a non-homogeneous $p$-Laplacian equation: $\Gamma$-convergence, local minimizers and stable transition layers,, J. Math. Anal. Appl., 483 (2020), 123634, 13pp.
doi: 10.1016/j.jmaa.2019.123634. |
[25] |
J. G. L. Laforgue and R. E. O'Malley Jr.,
Shock layer movement for Burgers' equation, SIAM J. Appl. Math., 55 (1995), 332-347.
doi: 10.1137/S003613999326928X. |
[26] |
K. A. Lee, A. Petrosyan and J. L. Vázquez,
Large-time geometric properties of solutions of the evolution $p$-Laplacian equation, J. Differential Equations, 229 (2006), 389-411.
doi: 10.1016/j.jde.2005.07.028. |
[27] |
P. Lindqvist, Notes on the Stationary $p$-Laplace Equation, SpringerBriefs in Mathematics, Springer, Cham, 2019.
doi: 10.1007/978-3-030-14501-9. |
[28] |
J. L. Lions, Quelques Méthodes de Résolution des Problèemes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. |
[29] |
B. Lou,
Singular limit of a $p$-Laplacian reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 110 (2003), 377-383.
doi: 10.1023/A:1021083015108. |
[30] |
C. Mascia and M. Strani,
Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws, SIAM J. Math. Anal., 45 (2013), 3084-3113.
doi: 10.1137/120875119. |
[31] |
F. Otto and M. G. Reznikoff,
Slow motion of gradient flows, J. Differential Equations, 237 (2007), 372-420.
doi: 10.1016/j.jde.2007.03.007. |
[32] |
A. Petrosyan and E. Valdinoci,
Density estimates for a degenerate/singular phase-transition model, SIAM J. Math. Anal., 36 (2005), 1057-1079.
doi: 10.1137/S0036141003437678. |
[33] |
L. G. Reyna and M. J. Ward,
On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math., 48 (1995), 79-120.
doi: 10.1002/cpa.3160480202. |
[34] |
M. Strani,
On the metastable behavior of solutions to a class of parabolic systems, Asymptot. Anal., 90 (2014), 325-344.
doi: 10.3233/ASY-141250. |
[35] |
S. Takeuchi and Y. Yamada,
Asymptotic properties of a reaction-diffusion equation with degenerate $p$-Laplacian, Nonlinear Anal., 42 (2000), 41-61.
doi: 10.1016/S0362-546X(98)00329-0. |
show all references
References:
[1] |
M. Allen and J. W. Cahn,
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[2] |
J. Benedikt, P. Girg, L. Kotrla and P. Takáč,
Origin of the $p$-Laplacian and A. Missbach, Electron. J. Differ. Eq., 2018 (2018), 1-17.
|
[3] |
F. Bethuel and D. Smets,
Slow motion for equal depth multiple-well gradient systems: The degenerate case, Discrete Contin. Dyn. Syst., 33 (2013), 67-87.
doi: 10.3934/dcds.2013.33.67. |
[4] |
L. Bronsard and R. V. Kohn,
On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43 (1990), 983-997.
doi: 10.1002/cpa.3160430804. |
[5] |
L. Bronsard and R. V. Kohn,
Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237.
doi: 10.1016/0022-0396(91)90147-2. |
[6] |
J. Carr and R. L. Pego,
Metastable patterns in solutions of $u_t = \epsilon^2u_xx-f(u)$, Comm. Pure Appl. Math., 42 (1989), 523-576.
doi: 10.1002/cpa.3160420502. |
[7] |
J. Carr and R. L. Pego,
Invariant manifolds for metastable patterns in $u_t = \epsilon^2u_xx-f(u)$, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 133-160.
doi: 10.1017/S0308210500031425. |
[8] |
M. S. Chang, S. C. Lee and C.-C. Yen,
Minimizers and gamma-convergence of energy functionals derived from $p$-Laplacian equation, Taiwanese J. Math., 13 (2009), 2021-2036.
doi: 10.11650/twjm/1500405655. |
[9] |
X. Chen,
Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.
doi: 10.1016/0022-0396(92)90146-E. |
[10] |
X. Chen,
Generation, propagation, and annihilation of metastable patterns, J. Differential Equations, 206 (2004), 399-437.
doi: 10.1016/j.jde.2004.05.017. |
[11] |
P. de Mottoni and M. Schatzman,
Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.
doi: 10.1090/S0002-9947-1995-1672406-7. |
[12] |
E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[13] |
S. Dipierro, A. Farina and E. Valdinoci,
Density estimates for degenerate double-well potentials, SIAM J. Math. Anal., 50 (2018), 6333-6347.
doi: 10.1137/17M114933X. |
[14] |
S. Dipierro, A. Pinamonti and E. Valdinoci,
Rigidity results for elliptic boundary value problems: Stable solutions for quasilinear equations with Neumann or Robin boundary conditions, Int. Math. Res. Not. IMRN, 2020 (2020), 1366-1384.
doi: 10.1093/imrn/rny055. |
[15] |
L. Evans, H. M. Soner and P. E. Souganidis,
Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.
doi: 10.1002/cpa.3160450903. |
[16] |
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. |
[17] |
R. Folino,
Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems, Electron. J. Differ. Eq., 2019 (2019), 1-21.
|
[18] |
R. Folino, C. A. Hernández Melo, L. F. López Ríos and R. G. Plaza, Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity,, Z. Angew. Math. Phys., 71 (2020), Paper No. 132, 25 pp.
doi: 10.1007/s00033-020-01362-0. |
[19] |
R. Folino, C. Lattanzio and C. Mascia,
Slow dynamics for the hyperbolic Cahn-Hilliard equation in one-space dimension, Math. Methods Appl. Sci., 42 (2019), 2492-2512.
doi: 10.1002/mma.5525. |
[20] |
R. Folino, C. Lattanzio, C. Mascia and M. Strani, Metastability for nonlinear convection-diffusion equations,, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 35, 20 pp.
doi: 10.1007/s00030-017-0459-5. |
[21] |
R. Folino, R. G. Plaza and M. Strani, Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion,, J. Math. Anal. Appl., 493 (2021), 124455, 29pp.
doi: 10.1016/j.jmaa.2020.124455. |
[22] |
G. Fusco and J. K. Hale,
Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations, 1 (1989), 75-94.
doi: 10.1007/BF01048791. |
[23] |
C. P. Grant,
Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34.
doi: 10.1137/S0036141092226053. |
[24] |
E. J. Hurtado and M. Sônego, On the energy functionals derived from a non-homogeneous $p$-Laplacian equation: $\Gamma$-convergence, local minimizers and stable transition layers,, J. Math. Anal. Appl., 483 (2020), 123634, 13pp.
doi: 10.1016/j.jmaa.2019.123634. |
[25] |
J. G. L. Laforgue and R. E. O'Malley Jr.,
Shock layer movement for Burgers' equation, SIAM J. Appl. Math., 55 (1995), 332-347.
doi: 10.1137/S003613999326928X. |
[26] |
K. A. Lee, A. Petrosyan and J. L. Vázquez,
Large-time geometric properties of solutions of the evolution $p$-Laplacian equation, J. Differential Equations, 229 (2006), 389-411.
doi: 10.1016/j.jde.2005.07.028. |
[27] |
P. Lindqvist, Notes on the Stationary $p$-Laplace Equation, SpringerBriefs in Mathematics, Springer, Cham, 2019.
doi: 10.1007/978-3-030-14501-9. |
[28] |
J. L. Lions, Quelques Méthodes de Résolution des Problèemes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. |
[29] |
B. Lou,
Singular limit of a $p$-Laplacian reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 110 (2003), 377-383.
doi: 10.1023/A:1021083015108. |
[30] |
C. Mascia and M. Strani,
Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws, SIAM J. Math. Anal., 45 (2013), 3084-3113.
doi: 10.1137/120875119. |
[31] |
F. Otto and M. G. Reznikoff,
Slow motion of gradient flows, J. Differential Equations, 237 (2007), 372-420.
doi: 10.1016/j.jde.2007.03.007. |
[32] |
A. Petrosyan and E. Valdinoci,
Density estimates for a degenerate/singular phase-transition model, SIAM J. Math. Anal., 36 (2005), 1057-1079.
doi: 10.1137/S0036141003437678. |
[33] |
L. G. Reyna and M. J. Ward,
On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math., 48 (1995), 79-120.
doi: 10.1002/cpa.3160480202. |
[34] |
M. Strani,
On the metastable behavior of solutions to a class of parabolic systems, Asymptot. Anal., 90 (2014), 325-344.
doi: 10.3233/ASY-141250. |
[35] |
S. Takeuchi and Y. Yamada,
Asymptotic properties of a reaction-diffusion equation with degenerate $p$-Laplacian, Nonlinear Anal., 42 (2000), 41-61.
doi: 10.1016/S0362-546X(98)00329-0. |



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