# American Institute of Mathematical Sciences

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

 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.)
Citation: Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I. Networks and 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. [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. [3] S. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. I, Arch. Rational Mech. Anal., 59 (1975), 159-188. [4] S. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. II, Arch. Rational Mech. Anal., 62 (1976), 209-235. [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. [6] M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Volume I, Applied Mathematical Sciences 51, Spring-Verlag, New York 1985. [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. [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. [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). [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. [11] H. Ikeda, Singular perturbation approach to stability properties of traveling wave solutions of reaction-diffusion systems, Hiroshima Math. J., 19 (1989), 587-630. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### References:
 [1] G. Alberti, Variational models for phase transitions, an approach via $\Gamma$-convergence, Calculus of Variations and Partial Differential Equations, (2000), 95-114. [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. [3] S. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. I, Arch. Rational Mech. Anal., 59 (1975), 159-188. [4] S. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. II, Arch. Rational Mech. Anal., 62 (1976), 209-235. [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. [6] M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Volume I, Applied Mathematical Sciences 51, Spring-Verlag, New York 1985. [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. [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. [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). [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. [11] H. Ikeda, Singular perturbation approach to stability properties of traveling wave solutions of reaction-diffusion systems, Hiroshima Math. J., 19 (1989), 587-630. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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