American Institute of Mathematical Sciences

Advanced
April  2012, 32(4): 1391-1420. doi: 10.3934/dcds.2012.32.1391

Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains

Fang Li 1, and  Kimie Nakashima 2,

1. 

Center for Partial Differential Equations, East China Normal University, 500 Dongchuan Road, Shanghai, 200241, China
2. 

Tokyo University of Marine Science and Technology, 4-5-7 Konan, Minato-ku, Tokyo 108-8477

Received  October 2010 Revised  August 2011 Published  October 2011

In this paper, we study a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. By upper and lower solution method, we obtain a sufficient condition for a hypersurface $S$ in the domain $\Omega$ to support stable transition layers, and a necessary condition for $S$ in $\Omega$ to support transition layers, not necessarily stable. In addition, sharp estimates on depths of transition layers have also been derived.
Keywords: transition layers, spatial inhomogeneity., Allen-Cahn equation.
Mathematics Subject Classification: Primary: 35J25, 35B25; Secondary: 35B3.
Citation: 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
References:
[1]

P. Fife, "Dynamics of Internal Layers and Diffusive Interfaces,", CBMS-NSF Regional Conference Series in Applied Mathematics, 53 (1988).   Google Scholar

[2]

P. Faĭf and U. Grinli, Interior transition layers for elliptic boundary value problems with a small parameter,, Uspehi Mat. Nauk, 29 (1974), 103.   Google Scholar

[3]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[4]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, in, (1981), 369.   Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar

[6]

J. Hale and K. Sakamoto, Existence and stability of transition layers,, Japan J. Appl. Math., 5 (1988), 367.  doi: 10.1007/BF03167908.  Google Scholar

[7]

R. V. Kohn and P. Sternberg, Local minimisers and singular perturbations,, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 69.   Google Scholar

[8]

F. Li, K. Nakashima and W.-M. Ni, Stability from the point of view of diffusion, relaxation and spatial inhomogeneity,, Discrete Contin. Dyn. Syst., 20 (2008), 259.   Google Scholar

[9]

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

[10]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations,, Publ. Res. Inst. Math. Sci., 15 (1979), 401.  doi: 10.2977/prims/1195188180.  Google Scholar

[11]

H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1984), 645.   Google Scholar

[12]

K. Nakashima, Stable transition layers in a balanced bistable equation,, Diff. Integral Eqns., 13 (2000), 1025.   Google Scholar

[13]

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

[14]

A. S. do Nascimento, Local minimizers induced by spatial inhomogeneity with inner transition layer,, J. Diff. Eqns., 133 (1997), 203.  doi: 10.1006/jdeq.1996.3206.  Google Scholar

[15]

N. N. Nefedov and K. Sakamoto, Multi-dimensional stationary internal layers for spatially inhomogeneous reaction-diffusion equations with balanced nonlinearity,, Hiroshima Math. J., 33 (2003), 391.   Google Scholar

[16]

M. del Pino, M. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model,, SIAM J. Math. Anal., 38 (2006), 1542.  doi: 10.1137/060649574.  Google Scholar

[17]

K. Sakamoto, Construction and stability analysis of transition layer solutions in reaction-diffusion systems,, Tohoku Math. J. (2), 42 (1990), 17.  doi: 10.2748/tmj/1178227692.  Google Scholar

[18]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 21 (1971), 979.  doi: 10.1512/iumj.1972.21.21079.  Google Scholar

show all references

References:
[1]

P. Fife, "Dynamics of Internal Layers and Diffusive Interfaces,", CBMS-NSF Regional Conference Series in Applied Mathematics, 53 (1988).   Google Scholar

[2]

P. Faĭf and U. Grinli, Interior transition layers for elliptic boundary value problems with a small parameter,, Uspehi Mat. Nauk, 29 (1974), 103.   Google Scholar

[3]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[4]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, in, (1981), 369.   Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar

[6]

J. Hale and K. Sakamoto, Existence and stability of transition layers,, Japan J. Appl. Math., 5 (1988), 367.  doi: 10.1007/BF03167908.  Google Scholar

[7]

R. V. Kohn and P. Sternberg, Local minimisers and singular perturbations,, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 69.   Google Scholar

[8]

F. Li, K. Nakashima and W.-M. Ni, Stability from the point of view of diffusion, relaxation and spatial inhomogeneity,, Discrete Contin. Dyn. Syst., 20 (2008), 259.   Google Scholar

[9]

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

[10]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations,, Publ. Res. Inst. Math. Sci., 15 (1979), 401.  doi: 10.2977/prims/1195188180.  Google Scholar

[11]

H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1984), 645.   Google Scholar

[12]

K. Nakashima, Stable transition layers in a balanced bistable equation,, Diff. Integral Eqns., 13 (2000), 1025.   Google Scholar

[13]

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

[14]

A. S. do Nascimento, Local minimizers induced by spatial inhomogeneity with inner transition layer,, J. Diff. Eqns., 133 (1997), 203.  doi: 10.1006/jdeq.1996.3206.  Google Scholar

[15]

N. N. Nefedov and K. Sakamoto, Multi-dimensional stationary internal layers for spatially inhomogeneous reaction-diffusion equations with balanced nonlinearity,, Hiroshima Math. J., 33 (2003), 391.   Google Scholar

[16]

M. del Pino, M. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model,, SIAM J. Math. Anal., 38 (2006), 1542.  doi: 10.1137/060649574.  Google Scholar

[17]

K. Sakamoto, Construction and stability analysis of transition layer solutions in reaction-diffusion systems,, Tohoku Math. J. (2), 42 (1990), 17.  doi: 10.2748/tmj/1178227692.  Google Scholar

[18]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 21 (1971), 979.  doi: 10.1512/iumj.1972.21.21079.  Google Scholar

[1]

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
[2]

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
[3]

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
[4]

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
[5]

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
[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]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

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
[16]

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
[17]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[18]

Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159
[19]

Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823
[20]

Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences