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December  2013, 8(4): 1009-1034. doi: 10.3934/nhm.2013.8.1009

Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I

Chaoqun Huang 1, and  Nung Kwan Yip 2,

1. 

Mathematics Department, University of Pittsburgh, Pittsburgh, PA 15260, United States
2. 

Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907

Received  February 2013 Revised  September 2013 Published  November 2013

We consider a singularly perturbed bistable reaction diffusion equation in a one-dimensional spatially degenerate inhomogeneous media. Degeneracy arises due to the choice of spatial inhomogeneity from some well-known class of normal forms or universal unfoldings. By means of a bilinear double well potential, we explicitly demonstrate the similarities and discrepancies between the bifurcation phenomena of the reaction diffusion equation and the limiting problem. The former is described by the location of the transition layer while the latter by the zeros of the spatial inhomogeneity function. Our result is the first which considers simultaneously the effects of singular perturbation, spatial inhomogeneity and bifurcation phenomena. (Part II [9] of this series analyzes the pitch-fork bifurcation for a general smooth double well potential where precise asymptotics and spectral analysis are needed.)
Keywords: Singular perturbation, universal unfolding, degenerate spatial inhomogeneity, imperfect bifurcation., normal form.
Mathematics Subject Classification: Primary: 35B25, 35B32, 35K57; Secondary: 34D15, 34E05, 34E1.
Citation: Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I. Networks & Heterogeneous Media, 2013, 8 (4) : 1009-1034. doi: 10.3934/nhm.2013.8.1009
References:
[1]

G. Alberti, Variational models for phase transitions, an approach via $\Gamma$-convergence, Calculus of Variations and Partial Differential Equations, (2000), 95-114.  Google Scholar

[2]

S. B Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem, Journal Differential Equations, 67 (1987), 212-242. doi: 10.1016/0022-0396(87)90147-1.  Google Scholar

[3]

S. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. I, Arch. Rational Mech. Anal., 59 (1975), 159-188.  Google Scholar

[4]

S. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. II, Arch. Rational Mech. Anal., 62 (1976), 209-235.  Google Scholar

[5]

P. C. Fife and W. M. Greenlee, Interior transition layers for elliptic boundary value problems with a small parameter, Russian Math. Surveys, 29 (1974), 103-131. doi: 10.1070/RM1974v029n04ABEH001291.  Google Scholar

[6]

M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Volume I, Applied Mathematical Sciences 51, Spring-Verlag, New York 1985.  Google Scholar

[7]

J. K. Hale and X. B. Lin, Multiple internal layer solutions generated by spatially oscillatory perturbations, J. Diff. Eqns., 154 (1999), 364-418. doi: 10.1006/jdeq.1998.3566.  Google Scholar

[8]

J. K. Hale and K. Sakamoto, Existence and stability of transition layers, Japan Journal Applied Math., 5 (1988), 367-405. doi: 10.1007/BF03167908.  Google Scholar

[9]

C. Q. Huang and N. K. Yip, Singular perturbation and bifurcation of diffused transition layers in degenerate inhomogeneous media, Part II, in Preprint, (2013). Google Scholar

[10]

J. E. Hutchinson and Y. Tonegawa, Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory, Calc. Var. Partial Differential Equations, 10 (2000), 49-84. doi: 10.1007/PL00013453.  Google Scholar

[11]

H. Ikeda, Singular perturbation approach to stability properties of traveling wave solutions of reaction-diffusion systems, Hiroshima Math. J., 19 (1989), 587-630.  Google Scholar

[12]

H. Ikeda, M. Mimura and Y. Nishirura, Global bifurcation phenomena of traveling wave solutions for some bistable reaction-diffusion systems, Nonlinear Analysis, Theory, Method and Application, 13 (1989), 507-526. doi: 10.1016/0362-546X(89)90061-8.  Google Scholar

[13]

H. Ikeda, Y. Nishiura and H. Suzuki, Stability of traveling waves and a relation between the Evans function and the SLEP equation, J. Reine Angew. Math., 475 (1996), 1-37.  Google Scholar

[14]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Royal Soc. Edinburgh Sect. A, 111 (1989), 69-84. doi: 10.1017/S0308210500025026.  Google Scholar

[15]

M. Kowalczyk, On the existence and morse index of solutions to Allen-Cahn equation in two dimensions, Annali di Mathematics Pura ed Applicata, 184 (2005), 17-52. doi: 10.1007/s10231-003-0088-y.  Google Scholar

[16]

H. Kukubu, Y. Nishirura and H. Oka, Heteroclinic and homoclinic bifurcation in bistable reaction-diffusion systems, Journal of Differential Equations, 86 (1990), 260-341. doi: 10.1016/0022-0396(90)90033-L.  Google Scholar

[17]

F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn in a multidimensional domains, Disc. Cont. Dyn. Sys., 32 (2012), 1391-1420. doi: 10.3934/dcds.2012.32.1391.  Google Scholar

[18]

L. Modica and S. Mortola, Il limite nella $\Gamma$-convergenze di una famiglia di funzionali ellittici, Boll. Un. Mat. Ital. A, 14 (1977), 426-529.  Google Scholar

[19]

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.  Google Scholar

[20]

A. S. do Nascimento, Stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in $N$-dimensional domains, J. Diff. Eqns., 190 (2003), 16-38. doi: 10.1016/S0022-0396(02)00147-X.  Google Scholar

[21]

N. N. Nefedov and K. Sakamoto, Multi-dimensional stationary internal layers for spatially inhomogeneous reaction-diffusion equation with balanced nonlinearity, Hiroshima Math. J., 33 (2003), 391-432.  Google Scholar

[22]

Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593. doi: 10.1137/0513037.  Google Scholar

[23]

Y. Nishiura, Singular limit approach to stability and bifurcation for bistable reaction diffusion systems, Rocky Mountain J. Math., 21 (1991), 727-767. doi: 10.1216/rmjm/1181072964.  Google Scholar

[24]

Y. Nishiura, H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion systems, SIAM J. Math. Anal., 18 (1987), 1726-1770. doi: 10.1137/0518124.  Google Scholar

[25]

Y. Nishiura, M. Mimura, H. Ikeda and H. Fujii, Singular limit analysis of stability of traveling wave solutions to bistable reaction-diffusion systems, SIAM J. Math. Anal., 21 (1990), 85-122. doi: 10.1137/0521006.  Google Scholar

[26]

F. Pacard and M. Ritore, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, Journal of Differential Geometry, 64 (2003), 359-423.  Google Scholar

[27]

H. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Communications on Pure and Applied Mathematics, 51 (1998), 551-579. doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6.  Google Scholar

[28]

M. del Pino, M. Kowalczyk and J. Wei, The Toda system and clustering interfaces in the Allen-Cahn equation, Arch. Rational Mech. Anal., 190 (2008), 141-187. doi: 10.1007/s00205-008-0143-3.  Google Scholar

[29]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. doi: 10.1007/BF00253122.  Google Scholar

show all references

References:
[1]

G. Alberti, Variational models for phase transitions, an approach via $\Gamma$-convergence, Calculus of Variations and Partial Differential Equations, (2000), 95-114.  Google Scholar

[2]

S. B Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem, Journal Differential Equations, 67 (1987), 212-242. doi: 10.1016/0022-0396(87)90147-1.  Google Scholar

[3]

S. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. I, Arch. Rational Mech. Anal., 59 (1975), 159-188.  Google Scholar

[4]

S. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. II, Arch. Rational Mech. Anal., 62 (1976), 209-235.  Google Scholar

[5]

P. C. Fife and W. M. Greenlee, Interior transition layers for elliptic boundary value problems with a small parameter, Russian Math. Surveys, 29 (1974), 103-131. doi: 10.1070/RM1974v029n04ABEH001291.  Google Scholar

[6]

M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Volume I, Applied Mathematical Sciences 51, Spring-Verlag, New York 1985.  Google Scholar

[7]

J. K. Hale and X. B. Lin, Multiple internal layer solutions generated by spatially oscillatory perturbations, J. Diff. Eqns., 154 (1999), 364-418. doi: 10.1006/jdeq.1998.3566.  Google Scholar

[8]

J. K. Hale and K. Sakamoto, Existence and stability of transition layers, Japan Journal Applied Math., 5 (1988), 367-405. doi: 10.1007/BF03167908.  Google Scholar

[9]

C. Q. Huang and N. K. Yip, Singular perturbation and bifurcation of diffused transition layers in degenerate inhomogeneous media, Part II, in Preprint, (2013). Google Scholar

[10]

J. E. Hutchinson and Y. Tonegawa, Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory, Calc. Var. Partial Differential Equations, 10 (2000), 49-84. doi: 10.1007/PL00013453.  Google Scholar

[11]

H. Ikeda, Singular perturbation approach to stability properties of traveling wave solutions of reaction-diffusion systems, Hiroshima Math. J., 19 (1989), 587-630.  Google Scholar

[12]

H. Ikeda, M. Mimura and Y. Nishirura, Global bifurcation phenomena of traveling wave solutions for some bistable reaction-diffusion systems, Nonlinear Analysis, Theory, Method and Application, 13 (1989), 507-526. doi: 10.1016/0362-546X(89)90061-8.  Google Scholar

[13]

H. Ikeda, Y. Nishiura and H. Suzuki, Stability of traveling waves and a relation between the Evans function and the SLEP equation, J. Reine Angew. Math., 475 (1996), 1-37.  Google Scholar

[14]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Royal Soc. Edinburgh Sect. A, 111 (1989), 69-84. doi: 10.1017/S0308210500025026.  Google Scholar

[15]

M. Kowalczyk, On the existence and morse index of solutions to Allen-Cahn equation in two dimensions, Annali di Mathematics Pura ed Applicata, 184 (2005), 17-52. doi: 10.1007/s10231-003-0088-y.  Google Scholar

[16]

H. Kukubu, Y. Nishirura and H. Oka, Heteroclinic and homoclinic bifurcation in bistable reaction-diffusion systems, Journal of Differential Equations, 86 (1990), 260-341. doi: 10.1016/0022-0396(90)90033-L.  Google Scholar

[17]

F. Li and K. Nakashima, Transition layers for a spatially inhomogeneous Allen-Cahn in a multidimensional domains, Disc. Cont. Dyn. Sys., 32 (2012), 1391-1420. doi: 10.3934/dcds.2012.32.1391.  Google Scholar

[18]

L. Modica and S. Mortola, Il limite nella $\Gamma$-convergenze di una famiglia di funzionali ellittici, Boll. Un. Mat. Ital. A, 14 (1977), 426-529.  Google Scholar

[19]

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.  Google Scholar

[20]

A. S. do Nascimento, Stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in $N$-dimensional domains, J. Diff. Eqns., 190 (2003), 16-38. doi: 10.1016/S0022-0396(02)00147-X.  Google Scholar

[21]

N. N. Nefedov and K. Sakamoto, Multi-dimensional stationary internal layers for spatially inhomogeneous reaction-diffusion equation with balanced nonlinearity, Hiroshima Math. J., 33 (2003), 391-432.  Google Scholar

[22]

Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593. doi: 10.1137/0513037.  Google Scholar

[23]

Y. Nishiura, Singular limit approach to stability and bifurcation for bistable reaction diffusion systems, Rocky Mountain J. Math., 21 (1991), 727-767. doi: 10.1216/rmjm/1181072964.  Google Scholar

[24]

Y. Nishiura, H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion systems, SIAM J. Math. Anal., 18 (1987), 1726-1770. doi: 10.1137/0518124.  Google Scholar

[25]

Y. Nishiura, M. Mimura, H. Ikeda and H. Fujii, Singular limit analysis of stability of traveling wave solutions to bistable reaction-diffusion systems, SIAM J. Math. Anal., 21 (1990), 85-122. doi: 10.1137/0521006.  Google Scholar

[26]

F. Pacard and M. Ritore, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, Journal of Differential Geometry, 64 (2003), 359-423.  Google Scholar

[27]

H. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Communications on Pure and Applied Mathematics, 51 (1998), 551-579. doi: 10.1002/(SICI)1097-0312(199806)51:6<551::AID-CPA1>3.0.CO;2-6.  Google Scholar

[28]

M. del Pino, M. Kowalczyk and J. Wei, The Toda system and clustering interfaces in the Allen-Cahn equation, Arch. Rational Mech. Anal., 190 (2008), 141-187. doi: 10.1007/s00205-008-0143-3.  Google Scholar

[29]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. doi: 10.1007/BF00253122.  Google Scholar

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