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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 |
References:
[1] |
P. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. |
[2] |
P. Faĭf and U. Grinli, Interior transition layers for elliptic boundary value problems with a small parameter, Uspehi Mat. Nauk, 29 (1974), 103-131. |
[3] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[4] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, in "Mathematical Analysis and Applications, Part A," 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[5] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, 2001. |
[6] |
J. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405.
doi: 10.1007/BF03167908. |
[7] |
R. V. Kohn and P. Sternberg, Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 69-84. |
[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), 259-274. |
[9] |
A. Malchiodi, W.-M. Ni and J. Wei, Boundary-clustered interfaces for the Allen-Cahn equation, Pacific J. Math., 229 (2007), 447-468.
doi: 10.2140/pjm.2007.229.447. |
[10] |
H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454.
doi: 10.2977/prims/1195188180. |
[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), 645-673. |
[12] |
K. Nakashima, Stable transition layers in a balanced bistable equation, Diff. Integral Eqns., 13 (2000), 1025-1038. |
[13] |
K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Diff. Eqns., 191 (2003), 234-276.
doi: 10.1016/S0022-0396(02)00181-X. |
[14] |
A. S. do Nascimento, Local minimizers induced by spatial inhomogeneity with inner transition layer, J. Diff. Eqns., 133 (1997), 203-223.
doi: 10.1006/jdeq.1996.3206. |
[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), 391-432. |
[16] |
M. del Pino, M. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal., 38 (2006/07), 1542-1564.
doi: 10.1137/060649574. |
[17] |
K. Sakamoto, Construction and stability analysis of transition layer solutions in reaction-diffusion systems, Tohoku Math. J. (2), 42 (1990), 17-44.
doi: 10.2748/tmj/1178227692. |
[18] |
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000.
doi: 10.1512/iumj.1972.21.21079. |
show all references
References:
[1] |
P. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. |
[2] |
P. Faĭf and U. Grinli, Interior transition layers for elliptic boundary value problems with a small parameter, Uspehi Mat. Nauk, 29 (1974), 103-131. |
[3] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[4] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, in "Mathematical Analysis and Applications, Part A," 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[5] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, 2001. |
[6] |
J. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405.
doi: 10.1007/BF03167908. |
[7] |
R. V. Kohn and P. Sternberg, Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 69-84. |
[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), 259-274. |
[9] |
A. Malchiodi, W.-M. Ni and J. Wei, Boundary-clustered interfaces for the Allen-Cahn equation, Pacific J. Math., 229 (2007), 447-468.
doi: 10.2140/pjm.2007.229.447. |
[10] |
H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454.
doi: 10.2977/prims/1195188180. |
[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), 645-673. |
[12] |
K. Nakashima, Stable transition layers in a balanced bistable equation, Diff. Integral Eqns., 13 (2000), 1025-1038. |
[13] |
K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Diff. Eqns., 191 (2003), 234-276.
doi: 10.1016/S0022-0396(02)00181-X. |
[14] |
A. S. do Nascimento, Local minimizers induced by spatial inhomogeneity with inner transition layer, J. Diff. Eqns., 133 (1997), 203-223.
doi: 10.1006/jdeq.1996.3206. |
[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), 391-432. |
[16] |
M. del Pino, M. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal., 38 (2006/07), 1542-1564.
doi: 10.1137/060649574. |
[17] |
K. Sakamoto, Construction and stability analysis of transition layer solutions in reaction-diffusion systems, Tohoku Math. J. (2), 42 (1990), 17-44.
doi: 10.2748/tmj/1178227692. |
[18] |
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000.
doi: 10.1512/iumj.1972.21.21079. |
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