2011, 30(3): 965-994. doi: 10.3934/dcds.2011.30.965

Coexistence phenomenon of concentration and transition of an inhomogeneous phase transition model on surfaces

1. 

College of Mathematics and Computational Sciences, Shenzhen University, Nanhai Ave 3688, Shenzhen 518060, China

Received  December 2009 Revised  January 2011 Published  January 2011

We consider the following singularly perturbed elliptic problem

ε2Δ ũ + (ũ –a(ŷ))(1- ũ2)=0 in M

&ytilde; where M is a two dimensional smooth compact Riemannian manifold associated with metric ğ, ε is a small parameter. The inhomogeneous term -1 < a(ŷ) < 1 takes maximum value b with 0 < b < 1. Assume that Γ = { ŷ ∈ M : a(ŷ) = 0} is a closed, smooth curve that Γ separate M into two disjoint components M+ and M- and also ∂a/∂v > 0 on Γ, where v is the normal of Γ pointing to the interior of M-. Moreover the maximum value loop Γ = { ŷ ∈ M : a(ŷ) = b} is a closed, smooth geodesic contained in M in such a way and Γ separate M- into two disjoint components. We will show the existence of solution possessing both transition and concentration phenomenon, i.e.

uε → + 1 in M-\ Γδ, uε → -1 in M+, uε → 1 – C along Γ as ε → 0,

where Γδ is a small neighborhood of Γ and C is a fixed positive constant.

Citation: Jun Yang. Coexistence phenomenon of concentration and transition of an inhomogeneous phase transition model on surfaces. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 965-994. doi: 10.3934/dcds.2011.30.965
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.

[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. doi: 10.1090/S0002-9947-99-02134-0.

[3]

N. D. Alikakos, X. Chen and G. Fusco, Motion of a droplet by surface tension along the boundray,, Cal. Var. PDE, 11 (2000), 233. doi: 10.1007/s005260000052.

[4]

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.

[5]

S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta. Metall., 27 (1979), 1084. doi: 10.1016/0001-6160(79)90196-2.

[6]

S. Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem,, J. Differential Equations, 67 (1987), 212. doi: 10.1016/0022-0396(87)90147-1.

[7]

H. Berestycki and P.-L.¡¡Lions, Existence of a ground state in nonlinear equations of the Klein-Gordon type,, in, (1978), 35.

[8]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces,, Math. Res. Lett., 3 (1996), 41.

[9]

E. N. Dancer and J. Wei, On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem,, Proc. Roy. Soc. Edinburgh A, 127 (1997), 691.

[10]

E. N. Dancer and J. Wei, On the location of spikes of solutions with sharp layers for a singularly perburbed semilinear Dirichlet problem,, J. Diff. Eqns., 157 (1999), 82. doi: 10.1006/jdeq.1998.3619.

[11]

E. N. Dancer and S. Yan, Multi-layer solutions for an elliptic problem,, J. Diff. Eqns., 194 (2003), 382. 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. 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. doi: 10.1080/03605309208820900.

[14]

M. del Pino, Radially symmetric internal layers in a semilinear elliptic system,, Trans. Amer. Math. Soc., 347 (1995), 4807. doi: 10.2307/2155064.

[15]

M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 70 (2007), 113. 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 (2007), 1542. doi: 10.1137/060649574.

[17]

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. doi: 10.1016/S0022-0396(02)00147-X.

[18]

Y. Du, The heterogeneous Allen-Cahn equation in a ball: Solutions with layers and spikes,, J. Differential Equations, 244 (2008), 117. doi: 10.1016/j.jde.2007.10.017.

[19]

Y. Du and Z. M. Guo, Boudary layer and spike layer solutions for a bistable elliptic problem with generalized boundary conditions,, J. Diff. Eqns., 221 (2006), 102. doi: 10.1016/j.jde.2005.08.006.

[20]

Y. Du, Z. M. Guo and F. Zhou, Boundary blow-up solutions with interior layers and spikes in a bistable problem,, Discrete and Continuous Dynamical Systems, 19 (2007), 271. doi: 10.3934/dcds.2007.19.271.

[21]

Y. Du and K. Nakashima, Morse index of layered solutions to the heterogeneous Allen-Cahn equation,, J. of Differential Equations, 238 (2007), 87. doi: 10.1016/j.jde.2007.03.024.

[22]

Y. Du and S. Yan, Boundary blow-up solutions with a spike layer,, J. Diff. Eqns., 205 (2004), 156. doi: 10.1016/j.jde.2004.06.010.

[23]

P. C. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations,, J. Math. Anal. Appl., 54 (1976), 497. doi: 10.1016/0022-247X(76)90218-3.

[24]

P. Fife and M. W. Greenlee, Interior transition Layers of elliptic boundary value problem with a small parameter,, Russian Math. Survey, 29 (1974), 103. doi: 10.1070/RM1974v029n04ABEH001291.

[25]

G. Flores and P. Padilla, Higher energy solutions in the theory of phase transitions: a variational approach,, J. Diff. Eqns., 169 (2001), 190. doi: 10.1006/jdeq.2000.3898.

[26]

C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems,, J. Diff. Eqns., 158 (1999), 1. doi: 10.1016/S0022-0396(99)80016-3.

[27]

J. Jang, On spike solutions of singularly perturbed semilinearly Dirichlet problems,, J. Diff. Eqns., 114 (1994), 370. doi: 10.1006/jdeq.1994.1154.

[28]

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

[29]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations,, Proc. Royal Soc. Edinburgh A, 111 (1989), 69.

[30]

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. doi: 10.1007/s10231-003-0088-y.

[31]

B. M. Levitan and I. S. Sargsjan, "Sturm-Liouville and Dirac Operator,", Mathematics and its application (Soviet Series), (1991).

[32]

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. doi: 10.1016/j.anihpc.2007.03.008.

[33]

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.

[34]

A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation,, Journal of Fixed Point Theory and Applications, 1 (2007), 305. doi: 10.1007/s11784-007-0016-7.

[35]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rat. Mech. Anal., 98 (1987), 357. doi: 10.1007/BF00251230.

[36]

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.

[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. doi: 10.1016/S0294-1449(02)00008-2.

[38]

W. M. Ni, I. Takagi and J. Wei, On the location and profile of intermediate solutions to a singularly perturbed semilinear Dirichlet problem,, Duke Math. J., 94 (1998), 597. doi: 10.1215/S0012-7094-98-09424-8.

[39]

Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. doi: 10.1137/0518124.

[40]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions,, J. Diff. Geom., 64 (2003), 359.

[41]

P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions,, Comm. Pure Appl. Math., 51 (1998), 551. doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6.

[42]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I,, Comm Pure Appl. Math., 56 (2003), 1078. doi: 10.1002/cpa.10087.

[43]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II,, Calc. Var. Partial Differential Equations, 21 (2004), 157.

[44]

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

[45]

K. Sakamoto, Infinitely many fine modes bifurcating from radially symmetric internal layers,, Asymptot. Anal., 42 (2005), 55.

[46]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. doi: 10.1007/s002050050081.

[47]

J. Wei and J. Yang, Solutions with transition layer and spike in an inhomogeneous phase transition model,, J. Differential Equations, 246 (2009), 3642. doi: 10.1016/j.jde.2008.12.021.

[48]

J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model,, Asymptotic Analysis, 69 (2010), 175.

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.

[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. doi: 10.1090/S0002-9947-99-02134-0.

[3]

N. D. Alikakos, X. Chen and G. Fusco, Motion of a droplet by surface tension along the boundray,, Cal. Var. PDE, 11 (2000), 233. doi: 10.1007/s005260000052.

[4]

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.

[5]

S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta. Metall., 27 (1979), 1084. doi: 10.1016/0001-6160(79)90196-2.

[6]

S. Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem,, J. Differential Equations, 67 (1987), 212. doi: 10.1016/0022-0396(87)90147-1.

[7]

H. Berestycki and P.-L.¡¡Lions, Existence of a ground state in nonlinear equations of the Klein-Gordon type,, in, (1978), 35.

[8]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces,, Math. Res. Lett., 3 (1996), 41.

[9]

E. N. Dancer and J. Wei, On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem,, Proc. Roy. Soc. Edinburgh A, 127 (1997), 691.

[10]

E. N. Dancer and J. Wei, On the location of spikes of solutions with sharp layers for a singularly perburbed semilinear Dirichlet problem,, J. Diff. Eqns., 157 (1999), 82. doi: 10.1006/jdeq.1998.3619.

[11]

E. N. Dancer and S. Yan, Multi-layer solutions for an elliptic problem,, J. Diff. Eqns., 194 (2003), 382. 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. 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. doi: 10.1080/03605309208820900.

[14]

M. del Pino, Radially symmetric internal layers in a semilinear elliptic system,, Trans. Amer. Math. Soc., 347 (1995), 4807. doi: 10.2307/2155064.

[15]

M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 70 (2007), 113. 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 (2007), 1542. doi: 10.1137/060649574.

[17]

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. doi: 10.1016/S0022-0396(02)00147-X.

[18]

Y. Du, The heterogeneous Allen-Cahn equation in a ball: Solutions with layers and spikes,, J. Differential Equations, 244 (2008), 117. doi: 10.1016/j.jde.2007.10.017.

[19]

Y. Du and Z. M. Guo, Boudary layer and spike layer solutions for a bistable elliptic problem with generalized boundary conditions,, J. Diff. Eqns., 221 (2006), 102. doi: 10.1016/j.jde.2005.08.006.

[20]

Y. Du, Z. M. Guo and F. Zhou, Boundary blow-up solutions with interior layers and spikes in a bistable problem,, Discrete and Continuous Dynamical Systems, 19 (2007), 271. doi: 10.3934/dcds.2007.19.271.

[21]

Y. Du and K. Nakashima, Morse index of layered solutions to the heterogeneous Allen-Cahn equation,, J. of Differential Equations, 238 (2007), 87. doi: 10.1016/j.jde.2007.03.024.

[22]

Y. Du and S. Yan, Boundary blow-up solutions with a spike layer,, J. Diff. Eqns., 205 (2004), 156. doi: 10.1016/j.jde.2004.06.010.

[23]

P. C. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations,, J. Math. Anal. Appl., 54 (1976), 497. doi: 10.1016/0022-247X(76)90218-3.

[24]

P. Fife and M. W. Greenlee, Interior transition Layers of elliptic boundary value problem with a small parameter,, Russian Math. Survey, 29 (1974), 103. doi: 10.1070/RM1974v029n04ABEH001291.

[25]

G. Flores and P. Padilla, Higher energy solutions in the theory of phase transitions: a variational approach,, J. Diff. Eqns., 169 (2001), 190. doi: 10.1006/jdeq.2000.3898.

[26]

C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems,, J. Diff. Eqns., 158 (1999), 1. doi: 10.1016/S0022-0396(99)80016-3.

[27]

J. Jang, On spike solutions of singularly perturbed semilinearly Dirichlet problems,, J. Diff. Eqns., 114 (1994), 370. doi: 10.1006/jdeq.1994.1154.

[28]

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

[29]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations,, Proc. Royal Soc. Edinburgh A, 111 (1989), 69.

[30]

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. doi: 10.1007/s10231-003-0088-y.

[31]

B. M. Levitan and I. S. Sargsjan, "Sturm-Liouville and Dirac Operator,", Mathematics and its application (Soviet Series), (1991).

[32]

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. doi: 10.1016/j.anihpc.2007.03.008.

[33]

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.

[34]

A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation,, Journal of Fixed Point Theory and Applications, 1 (2007), 305. doi: 10.1007/s11784-007-0016-7.

[35]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rat. Mech. Anal., 98 (1987), 357. doi: 10.1007/BF00251230.

[36]

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.

[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. doi: 10.1016/S0294-1449(02)00008-2.

[38]

W. M. Ni, I. Takagi and J. Wei, On the location and profile of intermediate solutions to a singularly perturbed semilinear Dirichlet problem,, Duke Math. J., 94 (1998), 597. doi: 10.1215/S0012-7094-98-09424-8.

[39]

Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. doi: 10.1137/0518124.

[40]

F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions,, J. Diff. Geom., 64 (2003), 359.

[41]

P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions,, Comm. Pure Appl. Math., 51 (1998), 551. doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6.

[42]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I,, Comm Pure Appl. Math., 56 (2003), 1078. doi: 10.1002/cpa.10087.

[43]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II,, Calc. Var. Partial Differential Equations, 21 (2004), 157.

[44]

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

[45]

K. Sakamoto, Infinitely many fine modes bifurcating from radially symmetric internal layers,, Asymptot. Anal., 42 (2005), 55.

[46]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. doi: 10.1007/s002050050081.

[47]

J. Wei and J. Yang, Solutions with transition layer and spike in an inhomogeneous phase transition model,, J. Differential Equations, 246 (2009), 3642. doi: 10.1016/j.jde.2008.12.021.

[48]

J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model,, Asymptotic Analysis, 69 (2010), 175.

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