• Previous Article
    Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle
  • CPAA Home
  • This Issue
  • Next Article
    Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions
May  2020, 19(5): 2575-2616. doi: 10.3934/cpaa.2020113

Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation

1. 

Department of Mathematics, South China Agricultural University, Guangzhou 510642, China,

2. 

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

3. 

School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

* Corresponding author

Received  December 2018 Revised  November 2019 Published  March 2020

Fund Project: Jun Yang is supported by National Natural Science Foundation of China (Grant No. 11771167 and No. 11831009)

We consider the nonlinear problem of inhomogeneous Allen-Cahn equation
$ \epsilon^2\Delta u+V(y)\,(1-u^2)\,u = 0\quad \mbox{in}\ \Omega, \qquad \frac {\partial u}{\partial \nu} = 0\quad \mbox{on}\ \partial \Omega, $
where
$ \Omega $
is a bounded domain in
$ \mathbb R^2 $
with smooth boundary,
$ \epsilon $
is a small positive parameter,
$ \nu $
denotes the unit outward normal of
$ \partial \Omega $
,
$ V $
is a positive smooth function on
$ \bar\Omega $
. Let
$ \Gamma\subset\Omega $
be a smooth curve dividing
$ \Omega $
into two disjoint regions and intersecting orthogonally with
$ \partial\Omega $
at exactly two points
$ P_1 $
and
$ P_2 $
. Moreover, by considering
$ {\mathbb R}^2 $
as a Riemannian manifold with the metric
$ g = V(y)\,({\mathrm d}{y}_1^2+{\mathrm d}{y}_2^2) $
, we assume that: the curve
$ \Gamma $
is a non-degenerate geodesic in the Riemannian manifold
$ ({\mathbb R}^2, g) $
, the Ricci curvature of the Riemannian manifold
$ ({\mathbb R}^2, g) $
along the normal
$ \mathbf{n} $
of
$ \Gamma $
is positive at
$ \Gamma $
, the generalized mean curvature of the submanifold
$ \partial\Omega $
in
$ ({\mathbb R}^2, g) $
vanishes at
$ P_1 $
and
$ P_2 $
. Then for any given integer
$ N\geq 2 $
, we construct a solution exhibiting
$ N $
-phase transition layers near
$ \Gamma $
(the zero set of the solution has
$ N $
components, which are curves connecting
$ \partial\Omega $
and directed along the direction of
$ \Gamma $
) with mutual distance
$ O(\epsilon|\log \epsilon|) $
, provided that
$ \epsilon $
stays away from a discrete set of values to avoid the resonance of the problem. Asymptotic locations of these layers are governed by a Toda system.
Citation: Suting Wei, Jun Yang. Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2575-2616. doi: 10.3934/cpaa.2020113
References:
[1]

N. D. AlikakosP. W. Bates and G. Fusco, Solutions to the nonautonomous bistable equation with specified Morse index, I. Existence, Trans. Amer. Math. Soc., 340 (1993), 641-654.  doi: 10.2307/2154670.  Google Scholar

[2]

N. D. AlikakosX. Chen and G. Fusco, Motion of a droplet by surface tension along the boundray, Calc. Var. Partial Differ. Equ., 11 (2000), 233-305.  doi: 10.1007/s005260000052.  Google Scholar

[3]

S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metall., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[4]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces, Math. Res. Lett., 3 (1996), 41-50.  doi: 10.4310/MRL.1996.v3.n1.a4.  Google Scholar

[5]

M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem, Commun. Partial Differ. Equ., 17 (1992), 1695-1708.  doi: 10.1080/03605309208820900.  Google Scholar

[6]

M. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837.  doi: 10.2307/2155064.  Google Scholar

[7]

M. del PinoM. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Commun. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135.  Google Scholar

[8]

M. del PinoM. Kowalczyk and J. Wei, The Toda system and clustering interface in the Allen-Cahn equation, Arch. Ration. Mech. Anal., 190 (2008), 141-187.  doi: 10.1007/s00205-008-0143-3.  Google Scholar

[9]

M. del PinoM. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces, Discrete Contin. Dyn. Syst., 28 (2010), 975-1006.  doi: 10.3934/dcds.2010.28.975.  Google Scholar

[10]

M. del PinoM. KowalczykJ. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature, Geom. Funct. Anal., 20 (2010), 918-957.  doi: 10.1007/s00039-010-0083-6.  Google Scholar

[11]

M. P. do Carmo, Differential Geometry of Curves and Surfaces, Translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N. J., 1976.  Google Scholar

[12]

Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation, J. Differ. Equ., 249 (2010), 215-239.  doi: 10.1016/j.jde.2010.03.024.  Google Scholar

[13]

Z. Du and L. Wang, Interface foliation for an inhomogeneous Allen-Cahn equation in Riemannian manifolds, Calc. Var. Partial Differ. Equ., 47 (2013), 343-381.  doi: 10.1007/s00526-012-0521-4.  Google Scholar

[14]

X. FanB. Xu and J. Yang, Phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation, J. Differ. Equ., 266 (2019), 5821-5866.  doi: 10.1016/j.jde.2018.10.051.  Google Scholar

[15]

G. Flores and P. Padilla, Higher energy solutions in the theory of phase transitions: a variational approach, J. Differ. Equ., 169 (2001), 190-207.  doi: 10.1006/jdeq.2000.3898.  Google Scholar

[16]

C. E. Garza-Hume and P. Padilla, Closed geodesic on oval surfaces and pattern formation, Comm. Anal. Geom., 11 (2003), 223-233.  doi: 10.4310/CAG.2003.v11.n2.a3.  Google Scholar

[17]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. R. Soc. Edinb. Sect. A Math., 111 (1989), 69-84.  doi: 10.1017/S0308210500025026.  Google Scholar

[18]

M. Kowalczyk, On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions, Ann. Mat. Pura Appl., 184 (2005), 17-52.  doi: 10.1007/s10231-003-0088-y.  Google Scholar

[19]

F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains, Discrete Contin. Dyn. Syst.-A, 32 (2012), 1391-1420.  doi: 10.3934/dcds.2012.32.1391.  Google Scholar

[20]

A. MalchiodiW. M. Ni and J. Wei, Boundary clustered interfaces for the Allen-Cahn equation, Pac. J. Math., 229 (2007), 447-468.  doi: 10.2140/pjm.2007.229.447.  Google Scholar

[21]

A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation, J. Fixed Point Theory Appl., 1 (2007), 305-336.  doi: 10.1007/s11784-007-0016-7.  Google Scholar

[22]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), 123-142.  doi: 10.1007/BF00251230.  Google Scholar

[23]

F. Morgan, Manifolds with Density, Notices Amer. Math. Soc., 52 (2005), 853-858.   Google Scholar

[24]

K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Differ. Equ., 191 (2003), 234-276.  doi: 10.1016/S0022-0396(02)00181-X.  Google Scholar

[25]

K. Nakashima and K. Tanaka, Clustering layers and boundary layers in spatially inhomogeneous phase transition problems, Ann. Inst. Henri Poincaré-Anal. Non Linéaire, 20 (2003), 107-143.  doi: 10.1016/S0294-1449(02)00008-2.  Google Scholar

[26]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differ. Geom., 64 (2003), 359-423.  doi: 10.4310/jdg/1090426999.  Google Scholar

[27]

P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Commun. Pure Appl. Math., 51 (1998), 551-579.  doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6.  Google Scholar

[28]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I, Commun. Pure Appl. Math., 56 (2003), 1078-1134.  doi: 10.1002/cpa.10087.  Google Scholar

[29]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II, Calc. Var. Partial Differ. Equ., 21 (2004), 157-207.  doi: 10.1007/s00526-003-0251-8.  Google Scholar

[30]

K. Sakamoto, Existence and stability of three-dimensional boundary-interior layers for the Allen-Cahn equation, Taiwan. J. Math., 9 (2005), 331-358.  doi: 10.11650/twjm/1500407844.  Google Scholar

[31]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Ration. Mech. Anal., 141 (1998), 375-400.  doi: 10.1007/s002050050081.  Google Scholar

[32]

F. TangS. Wei and J. Yang, Phase transition layers for Fife-Greenlee problem on smooth bounded domain, Discrete Contin. Dyn. Syst.-A, 38 (2018), 1527-1552.  doi: 10.3934/dcds.2018063.  Google Scholar

[33]

J. Wei and J. Yang, Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain, Discrete Contin. Dyn. Syst.-A, 22 (2008), 465-508. doi: 10.3934/dcds.2008.22.465.  Google Scholar

[34]

J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model, Asymptotic Anal., 69 (2010), 175-218.  doi: 10.3233/ASY-2010-0999.  Google Scholar

[35]

S. Wei, B. Xu and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains, Calc. Var. Partial Differ. Equ., 57 (2018), Article: 87. doi: 10.1007/s00526-018-1347-5.  Google Scholar

[36]

S. Wei and J. Yang, Connectivity of boundaries by clustering phase transition layers of Fife-Greenlee problem on smooth bounded domain, J. Differ. Equ., to appear. doi: 10.1016/j.jde.2020.01.014.  Google Scholar

[37]

J. Yang and X. Yang, Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains, Commun. Pure Appl. Anal., 12 (2013), 303-340.  doi: 10.3934/cpaa.2013.12.303.  Google Scholar

show all references

References:
[1]

N. D. AlikakosP. W. Bates and G. Fusco, Solutions to the nonautonomous bistable equation with specified Morse index, I. Existence, Trans. Amer. Math. Soc., 340 (1993), 641-654.  doi: 10.2307/2154670.  Google Scholar

[2]

N. D. AlikakosX. Chen and G. Fusco, Motion of a droplet by surface tension along the boundray, Calc. Var. Partial Differ. Equ., 11 (2000), 233-305.  doi: 10.1007/s005260000052.  Google Scholar

[3]

S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metall., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[4]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces, Math. Res. Lett., 3 (1996), 41-50.  doi: 10.4310/MRL.1996.v3.n1.a4.  Google Scholar

[5]

M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem, Commun. Partial Differ. Equ., 17 (1992), 1695-1708.  doi: 10.1080/03605309208820900.  Google Scholar

[6]

M. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837.  doi: 10.2307/2155064.  Google Scholar

[7]

M. del PinoM. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Commun. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135.  Google Scholar

[8]

M. del PinoM. Kowalczyk and J. Wei, The Toda system and clustering interface in the Allen-Cahn equation, Arch. Ration. Mech. Anal., 190 (2008), 141-187.  doi: 10.1007/s00205-008-0143-3.  Google Scholar

[9]

M. del PinoM. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces, Discrete Contin. Dyn. Syst., 28 (2010), 975-1006.  doi: 10.3934/dcds.2010.28.975.  Google Scholar

[10]

M. del PinoM. KowalczykJ. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature, Geom. Funct. Anal., 20 (2010), 918-957.  doi: 10.1007/s00039-010-0083-6.  Google Scholar

[11]

M. P. do Carmo, Differential Geometry of Curves and Surfaces, Translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N. J., 1976.  Google Scholar

[12]

Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation, J. Differ. Equ., 249 (2010), 215-239.  doi: 10.1016/j.jde.2010.03.024.  Google Scholar

[13]

Z. Du and L. Wang, Interface foliation for an inhomogeneous Allen-Cahn equation in Riemannian manifolds, Calc. Var. Partial Differ. Equ., 47 (2013), 343-381.  doi: 10.1007/s00526-012-0521-4.  Google Scholar

[14]

X. FanB. Xu and J. Yang, Phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation, J. Differ. Equ., 266 (2019), 5821-5866.  doi: 10.1016/j.jde.2018.10.051.  Google Scholar

[15]

G. Flores and P. Padilla, Higher energy solutions in the theory of phase transitions: a variational approach, J. Differ. Equ., 169 (2001), 190-207.  doi: 10.1006/jdeq.2000.3898.  Google Scholar

[16]

C. E. Garza-Hume and P. Padilla, Closed geodesic on oval surfaces and pattern formation, Comm. Anal. Geom., 11 (2003), 223-233.  doi: 10.4310/CAG.2003.v11.n2.a3.  Google Scholar

[17]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. R. Soc. Edinb. Sect. A Math., 111 (1989), 69-84.  doi: 10.1017/S0308210500025026.  Google Scholar

[18]

M. Kowalczyk, On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions, Ann. Mat. Pura Appl., 184 (2005), 17-52.  doi: 10.1007/s10231-003-0088-y.  Google Scholar

[19]

F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains, Discrete Contin. Dyn. Syst.-A, 32 (2012), 1391-1420.  doi: 10.3934/dcds.2012.32.1391.  Google Scholar

[20]

A. MalchiodiW. M. Ni and J. Wei, Boundary clustered interfaces for the Allen-Cahn equation, Pac. J. Math., 229 (2007), 447-468.  doi: 10.2140/pjm.2007.229.447.  Google Scholar

[21]

A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation, J. Fixed Point Theory Appl., 1 (2007), 305-336.  doi: 10.1007/s11784-007-0016-7.  Google Scholar

[22]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), 123-142.  doi: 10.1007/BF00251230.  Google Scholar

[23]

F. Morgan, Manifolds with Density, Notices Amer. Math. Soc., 52 (2005), 853-858.   Google Scholar

[24]

K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Differ. Equ., 191 (2003), 234-276.  doi: 10.1016/S0022-0396(02)00181-X.  Google Scholar

[25]

K. Nakashima and K. Tanaka, Clustering layers and boundary layers in spatially inhomogeneous phase transition problems, Ann. Inst. Henri Poincaré-Anal. Non Linéaire, 20 (2003), 107-143.  doi: 10.1016/S0294-1449(02)00008-2.  Google Scholar

[26]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differ. Geom., 64 (2003), 359-423.  doi: 10.4310/jdg/1090426999.  Google Scholar

[27]

P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Commun. Pure Appl. Math., 51 (1998), 551-579.  doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6.  Google Scholar

[28]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I, Commun. Pure Appl. Math., 56 (2003), 1078-1134.  doi: 10.1002/cpa.10087.  Google Scholar

[29]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II, Calc. Var. Partial Differ. Equ., 21 (2004), 157-207.  doi: 10.1007/s00526-003-0251-8.  Google Scholar

[30]

K. Sakamoto, Existence and stability of three-dimensional boundary-interior layers for the Allen-Cahn equation, Taiwan. J. Math., 9 (2005), 331-358.  doi: 10.11650/twjm/1500407844.  Google Scholar

[31]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Ration. Mech. Anal., 141 (1998), 375-400.  doi: 10.1007/s002050050081.  Google Scholar

[32]

F. TangS. Wei and J. Yang, Phase transition layers for Fife-Greenlee problem on smooth bounded domain, Discrete Contin. Dyn. Syst.-A, 38 (2018), 1527-1552.  doi: 10.3934/dcds.2018063.  Google Scholar

[33]

J. Wei and J. Yang, Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain, Discrete Contin. Dyn. Syst.-A, 22 (2008), 465-508. doi: 10.3934/dcds.2008.22.465.  Google Scholar

[34]

J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model, Asymptotic Anal., 69 (2010), 175-218.  doi: 10.3233/ASY-2010-0999.  Google Scholar

[35]

S. Wei, B. Xu and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains, Calc. Var. Partial Differ. Equ., 57 (2018), Article: 87. doi: 10.1007/s00526-018-1347-5.  Google Scholar

[36]

S. Wei and J. Yang, Connectivity of boundaries by clustering phase transition layers of Fife-Greenlee problem on smooth bounded domain, J. Differ. Equ., to appear. doi: 10.1016/j.jde.2020.01.014.  Google Scholar

[37]

J. Yang and X. Yang, Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains, Commun. Pure Appl. Anal., 12 (2013), 303-340.  doi: 10.3934/cpaa.2013.12.303.  Google Scholar

Figure 1.  $\begin{array}{*{20}{l}} {{ Curves}{\rm{: }}{C_1} = \left( {{y_1},{\varphi _1}\left( {{y_1}} \right)} \right),\;\;\;{\kern 1pt} {C_2} = \left( {{y_1},{\varphi _2}\left( {{y_1}} \right)} \right), - {\delta _0} < {y_1} < {\delta _0},}\\ {{ Points}{\rm{: }}{P_1} = (0,\varphi (0)),{P_2} = \left( {0,{\varphi _2}(0)} \right).} \end{array}$
[1]

Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure & Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303

[2]

Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407

[3]

Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391

[4]

Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024

[5]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319

[6]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703

[7]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301

[8]

Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099

[9]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127

[10]

Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639

[11]

Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947

[12]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015

[13]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679

[14]

Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577

[15]

Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009

[16]

Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024

[17]

Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077

[18]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020025

[19]

Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4907-4925. doi: 10.3934/dcds.2020205

[20]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (72)
  • HTML views (99)
  • Cited by (0)

Other articles
by authors

[Back to Top]