Abstract
Recognition stems from realization of the separation boundary between two different physical or chemicalstates. In other words we can observe natural phenomena through the emergence and evolution of theinterface between these states as in solidification, combustion, chemical reaction, and biological patterns. The interface studied here results from the balance between two opposing tendencies: adiffusive effect and a (physical or chemical)separation kinetics built in the system. The former attempts to smooth out the inhomogeneity as in the heat equation, and the latter drives the system to one or the other pure state such as solid or liquid (see, for instance, Fife [19] for details). Turing’s contribution [58] is one of the pioneering works related to the onset of spatial patterns through a cooperative work of diffusion and separation kinetics. Besides the existence of these two tendencies, another key ingredient to produce interesting interfacial patterns is the differences of the strength of the above mixing and unmixing effects among species involved in the system. In fact, reaction diffusion systems for two componentsu and v, which are the main concern in this paper, can be classified formally as
-
(i)
There is a difference in the diffusion rates of u and v;
-
(ii)
There is a difference in the reaction rates of u and v;
-
(ii)
There are differences in the diffusion and reaction rates of u and v, i.e., a combination of (i) and (ii).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Alexander, R. Gardner and C. Jones, A topological invariant arising in the stability analysis of travelling waves, J. reine angew. Math.,410 (1990), 167–212.
N. D. Alikakos, P. W. Bates, and G. Fusco, Slow motion for the Cahn- Hilliard equation in one space dimension, J. Differential Equations,90 (1991), 81–135.
L. Bronsard and R. V. Kohn, On the slowness of the phase boundary motion in one space dimension, Comm. Pure Appl. Math.,43 (1990), 983–997.
L.Bronsard and R.V.Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Diff., Eqns.,90 (1991) 211–237.
P. Brunovský and B. Fiedler, Connection orbits in scalar reaction diffusion equations, Dynamics Reported vol1 (1988), 57–90, Wiley.
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal.,92 (1986), 205–245.
G. Caginalp and Y. Nishiura, The existence of travelling waves for phase field equations and convergence to sharp interface models in the singular limit, Quarterly of Appl. Math.,49 (1) (1991), 147–162.
J. Carr, M. E. Gurtin, and M. Slemrod, Structured phase transitions on a finite interval, Arch. Rat. Mech. Anal.,86 (1984), 317–351.
J. Carr and R. L. Pego, Metastable patterns in solutions ofut = ε2uxx —f(u), Comm. Pure Appl. Math.,42 (1989), 523–576.
R. G. Casten and C. J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Diff. Eqns., 27 (1978), 266–273.
X. -Y. Chen, Dynamics of interfaces in reaction diffusion systems, Hiroshima Math. J.,21 (1991), 47–83.
Y. -G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geometry33 (1991), 749–786.
S. -N. Chow, B. Deng, and D. Terman, The bifurcation of homoclinic and periodic orbits from two heteroclinic orbits, SIAM J. Math. Anal.,21 (1990), 179–204.
B. Deng, The bifurcations of countable connections from a twisted heteroclinic loop, SIAM J. Math. Anal.22 (1991), 653–679.
L. C. Evans and J. Spruck, Motion of level sets by mean curvature, J. Differential Geometry33 (1991), 601 - 633.
P. C. Fife, Boundary and interior transition layer phenomena for pairs of second order differential equations, J. Math. Anal.,54 (1976), 497–521.
P. C. Fife, Propagator-controller systems and chemical patterns, inNonequilibrium Dynamics in Chemical Systems, A. Pacault and C. Vidal, eds., Springer-Verlag (1984), 76–88.
P. C. Fife, Understanding the patterns in the BZ reagent, J. Statist. Phys.,39 (1985), 687–703.
P. C. Fife,Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics #53, SIAM, Philadelphia (1988).
P. C. Fife, Pattern dynamics for parabolic PDE’s, to appear in the Proc. of IMA Conference “Introduction to dynamical systems”, Sept. 1989.
P. C. Fife and W. M. Greenlee, Interior transition layers for elliptic boundary value problems with a small parameter, Russian Math. Surveys,29: 4 (1974), 103–131.
P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal.,65 (1977), 335–361.
H. Fujii and Y. Hosono, Neumann layer phenomena in nonlinear diffusion systems,Recent Topics in Nonlinear PDE, M. Mimura and T. Nishida, Math. Studies, 98, North-Holland, Amsterdam (1983), 21–38.
H. Fujii, M. Mimura and Y. Nishiura, A picture of the global bifurcation diagram in ecological interacting and diffusing systems, Phisica5D (1982), 1–42.
H. Fujii, Y. Nishiura and Y. Hosono, On the structure of multuple existence of stable stationary solutions in systems of reaction- diffusion equations, Patterns and Waves - Qualitative Analysis of Nonlinear Differential Equations, T. Nishida, M. Mimura and H. Fujii, Stud. Math. Appl.,18 (1986), 157–219.
G. Fusco and J. K. Hale, Slow-motion manifolds, dormant Instability, and singular perturbations, J. Dynamics and Diff. Eqns., 1 (1989), 75–94.
R. Gardner and C. K. R. T Jones, A stability index for steady state solutions of boundary value problems for parabolic systems, J. Diff. Eqns.,91 (1991), 181–203.
J. K. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math.,5 (1988), 367–405.
D. Henry,Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, New York, 1981.
D. Hilhorst, Y. Nishiura, and M. Mimura, A free boundary problem arising in some reacting-diffusing system, Proc. Roy. Soc. Edinburgh,118A (1991), 355–378.
T. Ikeda and Y. Nishiura, Pattern selection for two breathers, SLAM Appl. Math., in press.
M. Ito, A remark on singular perturbation methods, Hiroshima Math. J.,14 (1985), 619–629.
J. P. Keener and J. J. Tyson, Spiral waves in the Belousov- Zhabotinsky reaction, Physica,21D (1986), 307–324.
H. Kokubu, Homoclinic and heteroclinic bifurcation of vector fields, Japan J. Appl. Math.,5 (1987), 455–501.
H. Kokubu, Y. Nishiura, and H. Oka, Heteroclinic and homoclinic bifurcation in bistable reaction diffusion systems, J. Diff. Eqns.,86 (2) (1990), 260–341.
J. S. Langer, Instabilities and pattern formation in crystal growth, Review of Modern Physics,52 (1) (1980), 1–28.
H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. RIMS, Kyoto Univ.,15 (1979), 401–454.
H. Meinhardt,Models of biological pattern formation. Academic Press, 1982.
M. Mimura, M. Tabata, and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal.,11 (1980), 613–631.
J. D. Murray,Mathematical Biology. Springer-Verlag, 1989.
Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal.,13 (1982), 555–593.
Y. Nishiura, Singular limit approach to stability and bifurcation for bistable reaction diffusion systems. In “Proceeding of the Workshop on Nonlinear PDE’s, Provo, Utah, March 1987” (P. Bates and P. Fife,), Rocky Mountain J. Math.,21 (2) (1991), 727–767.
Y. Nishiura, Singular limit eigenvalue problem in higher dimensional space, Proceeding of International Symposium on Functional Differential Equations and Related Topics, Kyoto, Japan, 30 August - 2 September, T. Yoshizawa and J. Kato, World Scientific, 1991.
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction diffusion equations, SIAM J. Math. Anal.,18 (1987), 1726–1770.
Y. Nishiura and H. Fujii, SLEP method to the stability of singularly perturbed solutions with multiple internal transition layers in reaction-diffusion systems, Proc. NATO Workshop “Dynamics of Infinite Dimensional Systems”, J. Hale and S.N. Chow, NATO ASI SeriesF37 (1986), 211–230.
Y. Nishiura and M. Mimura, Layer oscillations in reaction- diffusion systems, SIAM J. Appl. Math.,49 (1989), 481–514.
Y. Nishiura, M. Mimura, H. Ikeda, and H. Fujii, Singilar limit analysis of stability of travelling wave solutions in bistable reaction-diffusion systems, SIAM J. Math. Anal.,21 (1990), 85–122.
Y. Nishiura and H. Suzuki, Coalescence,repulsion, and nonuniqueness for singular limit reaction diffusion systems, to appear in the Proceedings of the International Taniguchi Symposium on Nonlinear Partial Differential Equations and Applications, Katata, 23 August - 29 August, 1989 ( Eds. T.Nishida and M.Mimura).
Y. Nishiura and H. Suzuki, Stability of generic interface of reaction diffusion systems in higher space dimensions, in preparation.
Y. Nishiura and T. Tsujikawa, Instability of singularly perturbed Neumann layer solutions in reaction-diffusion systems, Hiroshima Math. J.,20 (1990), 297–329.
T. Ohta and M. Mimura, Pattern dynamics in excitable reaction- diffusion media,Formation, Dynamics and Stabilities of Patterns, K. Kawasaki, M. Suzuki, and A. Onuki, 1, World Scientific, 1990.
T. Ohta, M. Mimura, and R. Kobayashi, Higher-dimensional localized pattern in excitable media, Physics 34D (1989), 115–144.
P. Pelce (editor),Dynamics of curved fronts, Perspective in Physics, Academic Press, 1988.
J. Rinzel and D. Terman, Propagation phenomena in a bistable reaction-diffusion systems, SIAM J. Appl. Math., 42 (1982) 1111–1137.
K. Sakamoto, Construction and stability analysis of transition layer solutions in reaction-diffusion systems, Tohoku Math. J.,42 (1990), 17–44.
H. Suzuki, Y. Nishiura, and H. Ikeda, Stability of traveling waves and a relation between the Evans function and the SLEP equation, submitted for publication.
M. Taniguchi and Y. Nishiura, Instability of planar interfaces in reaction diffusion systems, SIAM Math. Anal., in press.
A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Lond.,B237 (1952), 37–72.
R. S. Varga,Matrix iterative analysis, Prentice-Hall, 1962.
J. H. Wilkinson,The algebraic eigenvalue problem, Oxford, Clarendon Press, 1965.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Nishiura, Y. (1994). Coexistence of Infinitely Many Stable Solutions to Reaction Diffusion Systems in the Singular Limit. In: Jones, C.K.R.T., Kirchgraber, U., Walther, HO. (eds) Dynamics Reported. Dynamics Reported, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-78234-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-78234-3_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-78236-7
Online ISBN: 978-3-642-78234-3
eBook Packages: Springer Book Archive