American Institute of Mathematical Sciences

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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

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