December  2015, 10(4): 897-948. doi: 10.3934/nhm.2015.10.897

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

1. 

School of Mathematics, Hunan University, Changsha 410082, China

2. 

Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, United States

Received  September 2014 Revised  August 2015 Published  October 2015

In this paper, we study the connection between the bifurcation of diffuse transition layers and that of the underlying limit interfacial problem in a degenerate spatially inhomogeneous medium. In dimension one, we prove the existence of bifurcation of diffuse interfaces in a pitchfork spatial inhomogeneity for a partial differential equation with bistable type nonlinearity. Bifurcation point is characterized quantitatively as well. The main conclusion is that the bifurcation diagram of the diffuse transition layers inherits mostly from that of the zeros of the spatial inhomogeneity. However, explicit examples are given for which the bifurcation of these two are different in terms of (im)perfection. This is a continuation of [8] which makes use of bilinear nonlinearity allowing the use of explicit solution formula. In the current work, we extend the results to a general smooth nonlinear function. We perform detail analysis of the principal eigenvalue and eigenfunction of some singularly perturbed eigenvalue problems and their interaction with the background inhomogeneity. This is the first result that takes into account simultaneously the interaction between singular perturbation, spatial inhomogeneity and bifurcation.
Citation: Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks and Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897
References:
[1]

N. Alikakos, G. Fusco and V. Stefanopoulos, Critical spectrum and stability of interfaces for a class of reaction-diffusion equations, Journal of Differential Equations, 126 (1996), 106-167. doi: 10.1006/jdeq.1996.0046.

[2]

S. B Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem, Journal of 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), 103-130. 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. doi: 10.1007/978-1-4612-5034-0.

[7]

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

[8]

C. Q. Huang and N. K. Yip, Singular perturbation and bifurcation of diffused transition layers in inhomogeneous media, part I, Networks and Heterogeneous Media, 8 (2013), 1009-1034. doi: 10.3934/nhm.2013.8.1009.

[9]

T. Iibun and K. Sakamoto, Internal layer intersecting the boundary of domains in the allen-cahn equation, Japan J. Indust. Appl. Math., 18 (2001), 697-738. doi: 10.1007/BF03167411.

[10]

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

[11]

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.

[12]

M. Ito, A remark on singular perturbation methods, Hiroshima Math. J., 14 (1985), 619-629.

[13]

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

[14]

B. Matkowsky and E. Reiss, Singular perturbations of bifurcations, SIAM J. Appl. Math., 33 (1977), 230-255. doi: 10.1137/0133014.

[15]

K. Nakamura, H. Matano, D. Hilhorst and R. Schätzle, Singular limit of a reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 95 (1999), 1165-1185. doi: 10.1023/A:1004518904533.

[16]

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

[17]

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.

[18]

M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems, Advances in Differential Equations, 8 (2003), 29-54.

[19]

M. Taniguchi, A remark on singular perturbation methods via the Lyapunov-Schmidt reduction, Publ. RIMS. Kyoto Univ., 31 (1995), 1001-1010. doi: 10.2977/prims/1195163593.

show all references

References:
[1]

N. Alikakos, G. Fusco and V. Stefanopoulos, Critical spectrum and stability of interfaces for a class of reaction-diffusion equations, Journal of Differential Equations, 126 (1996), 106-167. doi: 10.1006/jdeq.1996.0046.

[2]

S. B Angenent, J. Mallet-Paret and L. A. Peletier, Stable transition layers in a semilinear boundary value problem, Journal of 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), 103-130. 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. doi: 10.1007/978-1-4612-5034-0.

[7]

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

[8]

C. Q. Huang and N. K. Yip, Singular perturbation and bifurcation of diffused transition layers in inhomogeneous media, part I, Networks and Heterogeneous Media, 8 (2013), 1009-1034. doi: 10.3934/nhm.2013.8.1009.

[9]

T. Iibun and K. Sakamoto, Internal layer intersecting the boundary of domains in the allen-cahn equation, Japan J. Indust. Appl. Math., 18 (2001), 697-738. doi: 10.1007/BF03167411.

[10]

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

[11]

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.

[12]

M. Ito, A remark on singular perturbation methods, Hiroshima Math. J., 14 (1985), 619-629.

[13]

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

[14]

B. Matkowsky and E. Reiss, Singular perturbations of bifurcations, SIAM J. Appl. Math., 33 (1977), 230-255. doi: 10.1137/0133014.

[15]

K. Nakamura, H. Matano, D. Hilhorst and R. Schätzle, Singular limit of a reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 95 (1999), 1165-1185. doi: 10.1023/A:1004518904533.

[16]

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

[17]

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.

[18]

M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems, Advances in Differential Equations, 8 (2003), 29-54.

[19]

M. Taniguchi, A remark on singular perturbation methods via the Lyapunov-Schmidt reduction, Publ. RIMS. Kyoto Univ., 31 (1995), 1001-1010. doi: 10.2977/prims/1195163593.

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