August  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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

[8]

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

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[13]

M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem,, Comm. Partial Differential Equations, 17 (1992), 1695. doi: 10.1080/03605309208820900. Google Scholar

[14]

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

[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. 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 (2007), 1542. doi: 10.1137/060649574. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[29]

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

[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. Google Scholar

[31]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[40]

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

[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. Google Scholar

[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. Google Scholar

[43]

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

[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. Google Scholar

[45]

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

[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. Google Scholar

[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. Google Scholar

[48]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

M. del Pino, Layers with nonsmooth interface in a semilinear elliptic problem,, Comm. Partial Differential Equations, 17 (1992), 1695. doi: 10.1080/03605309208820900. Google Scholar

[14]

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

[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. 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 (2007), 1542. doi: 10.1137/060649574. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[29]

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

[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. Google Scholar

[31]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[40]

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

[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. Google Scholar

[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. Google Scholar

[43]

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

[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. Google Scholar

[45]

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

[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. Google Scholar

[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. Google Scholar

[48]

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

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