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On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence
Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains
1. | College of Mathematics and Computational Sciences, Shenzhen University, Nanhai Ave 3688, Shenzhen 518060 |
2. | College of Mathematics and Computer Sciences, Hunan Normal University, Changsha 410081, China |
References:
[1] |
N. D. Alikakos and P. W. Bates, On the singular limit in a phase field model of phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 141-178. |
[2] |
N. D. Alikakos, P. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805.
doi: 10.1090/S0002-9947-99-02134-0. |
[3] |
N. D. Alikakos, P. 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. |
[4] |
N. D. Alikakos, X. Chen and G. Fusco, Motion of a droplet by surface tension along the boundray, Calc. Var. Partial Differential Equations, 11 (2000), 233-305.
doi: 10.1007/s005260000052. |
[5] |
N. D. Alikakos and H. C. Simpson, A variational approach for a class of singular perturbation problems and applications, Proc. Roy. Soc. Edinburgh Sect. A, 107 (1987), 27-42.
doi: 10.1017/S0308210500029334. |
[6] |
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. |
[7] |
S. Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem, J. Differential Equations, 67 (1987), 212-242.
doi: 10.1016/0022-0396(87)90147-1. |
[8] |
L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces, Math. Res. Lett., 3 (1996), 41-50. |
[9] |
I. Chavel, "Eigenvalues in Riemannian Geometry,'' Pure and Applied Mathematics, 115, Academic Press, Inc., Orlando, FL, 1984. |
[10] |
I. Chavel, "Riemannian Geometry - A Modern Introduction,'' Cambridge Tracts in Math. 108, Cambridge Univ. Press, Cambridge, 1993. |
[11] |
E. N. Dancer and S. Yan, Multi-layer solutions for an elliptic problem, J. Differential Equations, 194 (2003), 382-405.
doi: 10.1016/S0022-0396(03)00176-1. |
[12] |
E. N. Dancer and S. Yan, Construction of various types of solutions for an elliptic problem, Calc. Var. Partial Differential Equations, 20 (2004), 93-118.
doi: 10.1007/s00526-003-0229-6. |
[13] |
M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem, Comm. Partial Differential Equations, 17 (1992), 1695-1708.
doi: 10.1080/03605309208820900. |
[14] |
M. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837.
doi: 10.1090/S0002-9947-1995-1303116-3. |
[15] |
M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.
doi: 10.1002/cpa.20135. |
[16] |
M. del Pino, M. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model,, SIAM J. Math. Anal., 38 (): 1542.
doi: 10.1137/060649574. |
[17] |
M. del Pino, M. 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. |
[18] |
M. del Pino, M. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces, Discrete Contin. Dynam. Systems-A, 28 (2010), 975-1006.
doi: 10.3934/dcds.2010.28.975. |
[19] |
M. del Pino, M. Kowalczyk, J. 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. |
[20] |
A. S. do Nascimento, Stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in $N$-dimensional domains, J. Differential Equations, 190 (2003), 16-38.
doi: 10.1016/S0022-0396(02)00147-X. |
[21] |
Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation, J. Differential Equations, 249 (2010), 215-239.
doi: 10.1016/j.jde.2010.03.024. |
[22] |
P. C. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations, J. Math. Anal. Appl., 54 (1976), 497-521.
doi: 10.1016/0022-247X(76)90218-3. |
[23] |
P. Fife and M. W. Greenlee, Interior transition Layers of elliptic boundary value problem with a small parameter, Russian Math. Survey, 29 (1974), 103-131.
doi: 10.1070/RM1974v029n04ABEH001291. |
[24] |
G. Flores and P. Padilla, Higher energy solutions in the theory of phase transitions: a variational approach, J. Differential Equations, 169 (2001), 190-207.
doi: 10.1006/jdeq.2000.3898. |
[25] |
C. E. Garza-Hume and P. Padilla, Closed geodesic on oval surfaces and pattern formation, Comm. Anal. Geom., 11 (2003), 223-233. |
[26] |
J. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405.
doi: 10.1007/BF03167908. |
[27] |
R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Royal Soc. Edinburgh A, 111 (1989), 69-84.
doi: 10.1017/S0308210500025026. |
[28] |
M. Kowalczyk, On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions, Annali di Matematica Pura ed Applicata, 184 (2005), 17-52.
doi: 10.1007/s10231-003-0088-y. |
[29] |
F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains, Discrete Contin. Dynam. Systems-A, 32 (2012), 1391-1420.
doi: 10.3934/dcds.2012.32.1391. |
[30] |
P. Li and S. Yau, On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88 (1983), 309-318.
doi: 10.1007/BF01213210. |
[31] |
F. Mahmoudi, A. Malchiodi and J. Wei, Transition layer for the heterogeneous Allen-Cahn equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 609-631.
doi: 10.1016/j.anihpc.2007.03.008. |
[32] |
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. |
[33] |
A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation, Journal of Fixed Point Theory and Applications, 1 (2007), 305-336.
doi: 10.1007/s11784-007-0016-7. |
[34] |
S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math., 1 (1949), 242-256.
doi: 10.4153/CJM-1949-021-5. |
[35] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[36] |
K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Differential Equations, 191 (2003), 234-276.
doi: 10.1016/S0022-0396(02)00181-X. |
[37] |
K. Nakashima and K. Tanaka, Clustering layers and boundary layers in spatially inhomogeneous phase transition problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 107-143.
doi: 10.1016/S0294-1449(02)00008-2. |
[38] |
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770.
doi: 10.1137/0518124. |
[39] |
F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Diff. Geom., 64 (2003), 359-423. |
[40] |
P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Comm. Pure Appl. Math., 51 (1998), 551-579.
doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6. |
[41] |
P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I, Comm Pure Appl. Math., 56 (2003), 1078-1134.
doi: 10.1002/cpa.10087. |
[42] |
P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II, Calc. Var. Partial Differential Equations, 21 (2004), 157-207.
doi: 10.1007/s00526-003-0251-8. |
[43] |
K. Sakamoto, Construction and stability an alysis of transition layer solutions in reaction-diffusion systems, Tohoku Math. J., 42 (1990), 17-44.
doi: 10.2748/tmj/1178227692. |
[44] |
K. Sakamoto, Infinitely many fine modes bifurcating from radially symmetric internal layers, Asymptot. Anal., 42 (2005), 55-104. |
[45] |
P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.
doi: 10.1007/s002050050081. |
[46] |
J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model, Asymptot. Anal., 69 (2010), 175-218. |
show all references
References:
[1] |
N. D. Alikakos and P. W. Bates, On the singular limit in a phase field model of phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 141-178. |
[2] |
N. D. Alikakos, P. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805.
doi: 10.1090/S0002-9947-99-02134-0. |
[3] |
N. D. Alikakos, P. 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. |
[4] |
N. D. Alikakos, X. Chen and G. Fusco, Motion of a droplet by surface tension along the boundray, Calc. Var. Partial Differential Equations, 11 (2000), 233-305.
doi: 10.1007/s005260000052. |
[5] |
N. D. Alikakos and H. C. Simpson, A variational approach for a class of singular perturbation problems and applications, Proc. Roy. Soc. Edinburgh Sect. A, 107 (1987), 27-42.
doi: 10.1017/S0308210500029334. |
[6] |
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. |
[7] |
S. Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem, J. Differential Equations, 67 (1987), 212-242.
doi: 10.1016/0022-0396(87)90147-1. |
[8] |
L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces, Math. Res. Lett., 3 (1996), 41-50. |
[9] |
I. Chavel, "Eigenvalues in Riemannian Geometry,'' Pure and Applied Mathematics, 115, Academic Press, Inc., Orlando, FL, 1984. |
[10] |
I. Chavel, "Riemannian Geometry - A Modern Introduction,'' Cambridge Tracts in Math. 108, Cambridge Univ. Press, Cambridge, 1993. |
[11] |
E. N. Dancer and S. Yan, Multi-layer solutions for an elliptic problem, J. Differential Equations, 194 (2003), 382-405.
doi: 10.1016/S0022-0396(03)00176-1. |
[12] |
E. N. Dancer and S. Yan, Construction of various types of solutions for an elliptic problem, Calc. Var. Partial Differential Equations, 20 (2004), 93-118.
doi: 10.1007/s00526-003-0229-6. |
[13] |
M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem, Comm. Partial Differential Equations, 17 (1992), 1695-1708.
doi: 10.1080/03605309208820900. |
[14] |
M. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837.
doi: 10.1090/S0002-9947-1995-1303116-3. |
[15] |
M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.
doi: 10.1002/cpa.20135. |
[16] |
M. del Pino, M. Kowalczyk and J. Wei, Resonance and interior layers in an inhomogeneous phase transition model,, SIAM J. Math. Anal., 38 (): 1542.
doi: 10.1137/060649574. |
[17] |
M. del Pino, M. 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. |
[18] |
M. del Pino, M. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces, Discrete Contin. Dynam. Systems-A, 28 (2010), 975-1006.
doi: 10.3934/dcds.2010.28.975. |
[19] |
M. del Pino, M. Kowalczyk, J. 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. |
[20] |
A. S. do Nascimento, Stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in $N$-dimensional domains, J. Differential Equations, 190 (2003), 16-38.
doi: 10.1016/S0022-0396(02)00147-X. |
[21] |
Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation, J. Differential Equations, 249 (2010), 215-239.
doi: 10.1016/j.jde.2010.03.024. |
[22] |
P. C. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations, J. Math. Anal. Appl., 54 (1976), 497-521.
doi: 10.1016/0022-247X(76)90218-3. |
[23] |
P. Fife and M. W. Greenlee, Interior transition Layers of elliptic boundary value problem with a small parameter, Russian Math. Survey, 29 (1974), 103-131.
doi: 10.1070/RM1974v029n04ABEH001291. |
[24] |
G. Flores and P. Padilla, Higher energy solutions in the theory of phase transitions: a variational approach, J. Differential Equations, 169 (2001), 190-207.
doi: 10.1006/jdeq.2000.3898. |
[25] |
C. E. Garza-Hume and P. Padilla, Closed geodesic on oval surfaces and pattern formation, Comm. Anal. Geom., 11 (2003), 223-233. |
[26] |
J. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math., 5 (1988), 367-405.
doi: 10.1007/BF03167908. |
[27] |
R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Royal Soc. Edinburgh A, 111 (1989), 69-84.
doi: 10.1017/S0308210500025026. |
[28] |
M. Kowalczyk, On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions, Annali di Matematica Pura ed Applicata, 184 (2005), 17-52.
doi: 10.1007/s10231-003-0088-y. |
[29] |
F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains, Discrete Contin. Dynam. Systems-A, 32 (2012), 1391-1420.
doi: 10.3934/dcds.2012.32.1391. |
[30] |
P. Li and S. Yau, On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88 (1983), 309-318.
doi: 10.1007/BF01213210. |
[31] |
F. Mahmoudi, A. Malchiodi and J. Wei, Transition layer for the heterogeneous Allen-Cahn equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 609-631.
doi: 10.1016/j.anihpc.2007.03.008. |
[32] |
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. |
[33] |
A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation, Journal of Fixed Point Theory and Applications, 1 (2007), 305-336.
doi: 10.1007/s11784-007-0016-7. |
[34] |
S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math., 1 (1949), 242-256.
doi: 10.4153/CJM-1949-021-5. |
[35] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[36] |
K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Differential Equations, 191 (2003), 234-276.
doi: 10.1016/S0022-0396(02)00181-X. |
[37] |
K. Nakashima and K. Tanaka, Clustering layers and boundary layers in spatially inhomogeneous phase transition problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 107-143.
doi: 10.1016/S0294-1449(02)00008-2. |
[38] |
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770.
doi: 10.1137/0518124. |
[39] |
F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Diff. Geom., 64 (2003), 359-423. |
[40] |
P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Comm. Pure Appl. Math., 51 (1998), 551-579.
doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6. |
[41] |
P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I, Comm Pure Appl. Math., 56 (2003), 1078-1134.
doi: 10.1002/cpa.10087. |
[42] |
P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II, Calc. Var. Partial Differential Equations, 21 (2004), 157-207.
doi: 10.1007/s00526-003-0251-8. |
[43] |
K. Sakamoto, Construction and stability an alysis of transition layer solutions in reaction-diffusion systems, Tohoku Math. J., 42 (1990), 17-44.
doi: 10.2748/tmj/1178227692. |
[44] |
K. Sakamoto, Infinitely many fine modes bifurcating from radially symmetric internal layers, Asymptot. Anal., 42 (2005), 55-104. |
[45] |
P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.
doi: 10.1007/s002050050081. |
[46] |
J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model, Asymptot. Anal., 69 (2010), 175-218. |
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