# American Institute of Mathematical Sciences

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

 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.
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

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##### 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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