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Structured first order conservation models for pedestrian dynamics
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 |
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. |
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. |
[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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