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The dynamics of rotating waves in scalar reaction diffusion equations. (English) Zbl 0696.35086

Summary: The maximal compact attractor for the reaction diffusion equation (RDE) \(u_ t=u_{xx}+f(u,u_ x)\) with periodic boundary conditions is studied. It is shown that any \(\omega\)-limit set contains a rotating wave, i.e., a solution of the form U(x-ct). A number of heteroclinic orbits from one rotating wave to another are constructed. Our main tool is the Nickel-Matano-Henry zero number. The heteroclinic orbits are obtained via a shooting argument, which relies on a generalized Borsuk- Ulam theorem.

MSC:

35K57 Reaction-diffusion equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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[1] M. Abramowitz and I. A. Stegun (Editors), Handbook of mathematical functions, Dover, New York, 1965. · Zbl 0171.38503
[2] Herbert Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math. 360 (1985), 47 – 83. · Zbl 0564.35060
[3] S. B. Angenent, The Morse-Smale property for a semilinear parabolic equation, J. Differential Equations 62 (1986), no. 3, 427 – 442. · Zbl 0581.58026
[4] Bernd Aulbach, Continuous and discrete dynamics near manifolds of equilibria, Lecture Notes in Mathematics, vol. 1058, Springer-Verlag, Berlin, 1984. · Zbl 0535.34002
[5] E. Brieskorn and H. Knörrer, Ebene algebraische Kurven, Birkhäuser, Basel, 1981. · Zbl 0508.14018
[6] P. Brunovský and B. Fiedler, Simplicity of zeros in scalar parabolic equations, J. Differential Equations 62 (1986), no. 2, 237 – 241. · Zbl 0549.35062
[7] Pavol Brunovský and Bernold Fiedler, Numbers of zeros on invariant manifolds in reaction-diffusion equations, Nonlinear Anal. 10 (1986), no. 2, 179 – 193. · Zbl 0594.35056
[8] -, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported (to appear). · Zbl 0679.35047
[9] Kuo Shung Chêng, Decay rate of periodic solutions for a conservation law, J. Differential Equations 42 (1981), no. 3, 390 – 399. · Zbl 0441.35011
[10] Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 251, Springer-Verlag, New York-Berlin, 1982. · Zbl 0487.47039
[11] Charles Conley and Joel Smoller, Topological techniques in reaction-diffusion equations, Biological growth and spread (Proc. Conf., Heidelberg, 1979) Lecture Notes in Biomath., vol. 38, Springer, Berlin-New York, 1980, pp. 473 – 483. · Zbl 0444.35005
[12] I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. · Zbl 0493.28007
[13] Constantine M. Dafermos, Applications of the invariance principle for compact processes. II. Asymptotic behavior of solutions of a hyperbolic conservation law, J. Differential Equations 11 (1972), 416 – 424. · Zbl 0252.35045
[14] A. Dold, Lectures on algebraic topology, Springer-Verlag, New York-Berlin, 1972 (German). Die Grundlehren der mathematischen Wissenschaften, Band 200. · Zbl 0234.55001
[15] James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. · Zbl 0144.21501
[16] J. J. Duistermaat, Stable manifolds, preprint no. 40, Math. Inst., Utrecht, 1976.
[17] John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. · Zbl 0515.34001
[18] Victor Guillemin and Alan Pollack, Differential topology, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. · Zbl 0361.57001
[19] Jack K. Hale, Infinite-dimensional dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 379 – 400.
[20] Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. · Zbl 0456.35001
[21] Daniel B. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations 59 (1985), no. 2, 165 – 205. · Zbl 0572.58012
[22] Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. · Zbl 0356.57001
[23] Morris W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, Nonlinear partial differential equations (Durham, N.H., 1982) Contemp. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1983, pp. 267 – 285. · Zbl 0523.58034
[24] Morris W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), no. 3, 423 – 439. · Zbl 0658.34023
[25] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. · Zbl 0355.58009
[26] H. Von Hopf and M. Rueff, Über faserungstreue Abbildungen der Sphären, Comment. Math. Helv. 11 (1938), no. 1, 49 – 61 (German). · Zbl 0019.37204
[27] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1980. · Zbl 0435.47001
[28] J. Kevorkian and Julian D. Cole, Perturbation methods in applied mathematics, Applied Mathematical Sciences, vol. 34, Springer-Verlag, New York-Berlin, 1981. · Zbl 0456.34001
[29] Gen Komatsu, Analyticity up to the boundary of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math. 32 (1979), no. 5, 669 – 720. · Zbl 0399.35061
[30] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537 – 566. · Zbl 0081.08803
[31] John Mallet-Paret, Morse decompositions and global continuation of periodic solutions for singularly perturbed delay equations, Systems of nonlinear partial differential equations (Oxford, 1982) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 351 – 365. · Zbl 0562.34060
[32] -, Morse decompositions for delay-differential equations, preprint. · Zbl 0648.34082
[33] J. Mallet-Paret and B. Fiedler, Connections of Morse sets for delay-differential equations, in preparation. · Zbl 0704.35070
[34] William S. Massey, Singular homology theory, Graduate Texts in Mathematics, vol. 70, Springer-Verlag, New York-Berlin, 1980. · Zbl 0442.55001
[35] Hiroshi Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. 18 (1978), no. 2, 221 – 227. · Zbl 0387.35008
[36] Hiroshi Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 401 – 441. · Zbl 0496.35011
[37] Hiroshi Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 645 – 673. · Zbl 0545.35042
[38] -, personal communication, 1986.
[39] E. F. Mishchenko and N. Kh. Rozov, Differential equations with small parameters and relaxation oscillations, Mathematical Concepts and Methods in Science and Engineering, vol. 13, Plenum Press, New York, 1980. Translated from the Russian by F. M. C. Goodspeed. · Zbl 0482.34004
[40] Karl Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleichungen, J. Reine Angew. Math. 211 (1962), 78 – 94. (1 insert) (German). · Zbl 0127.31801
[41] O. A. Oleinik and S. N. Kruzhkov, Quasilinear second order parabolic equations with many independent variables, Russian Math. Surveys 16 (1961), 105-146. · Zbl 0112.32604
[42] Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1983. · Zbl 0507.28010
[43] Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. · Zbl 0508.35002
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