Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The dynamics of rotating waves in scalar reaction diffusion equations
HTML articles powered by AMS MathViewer

by S. B. Angenent and B. Fiedler PDF
Trans. Amer. Math. Soc. 307 (1988), 545-568 Request permission

Abstract:

The maximal compact attractor for the 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.
References
    M. Abramowitz and I. A. Stegun (Editors), Handbook of mathematical functions, Dover, New York, 1965.
  • Herbert Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math. 360 (1985), 47–83. MR 799657, DOI 10.1515/crll.1985.360.47
  • S. B. Angenent, The Morse-Smale property for a semilinear parabolic equation, J. Differential Equations 62 (1986), no. 3, 427–442. MR 837763, DOI 10.1016/0022-0396(86)90093-8
  • Bernd Aulbach, Continuous and discrete dynamics near manifolds of equilibria, Lecture Notes in Mathematics, vol. 1058, Springer-Verlag, Berlin, 1984. MR 744191, DOI 10.1007/BFb0071569
    • E. Brieskorn and H. Knörrer, Ebene algebraische Kurven, Birkhäuser, Basel, 1981.
  • P. Brunovský and B. Fiedler, Simplicity of zeros in scalar parabolic equations, J. Differential Equations 62 (1986), no. 2, 237–241. MR 833419, DOI 10.1016/0022-0396(86)90099-9
  • Pavol Brunovský and Bernold Fiedler, Numbers of zeros on invariant manifolds in reaction-diffusion equations, Nonlinear Anal. 10 (1986), no. 2, 179–193. MR 825216, DOI 10.1016/0362-546X(86)90045-3
    • —, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported (U. Kirchgraber, ed.) (to appear).
  • Kuo Shung Chêng, Decay rate of periodic solutions for a conservation law, J. Differential Equations 42 (1981), no. 3, 390–399. MR 639229, DOI 10.1016/0022-0396(81)90112-1
  • Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633
  • 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. MR 609381
  • 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. MR 832433, DOI 10.1007/978-1-4615-6927-5
  • 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. MR 296476, DOI 10.1016/0022-0396(72)90055-1
  • A. Dold, Lectures on algebraic topology, Die Grundlehren der mathematischen Wissenschaften, Band 200, Springer-Verlag, New York-Berlin, 1972 (German). MR 0415602
  • James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
    • J. J. Duistermaat, Stable manifolds, preprint no. 40, Math. Inst., Utrecht, 1976.
  • John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
  • Victor Guillemin and Alan Pollack, Differential topology, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0348781
  • 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. MR 730278, DOI 10.1007/BFb0061425
  • Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
  • Daniel B. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations 59 (1985), no. 2, 165–205. MR 804887, DOI 10.1016/0022-0396(85)90153-6
  • Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
  • 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. MR 706104
  • 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. MR 783970, DOI 10.1137/0516030
  • M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
  • H. Von Hopf and M. Rueff, Über faserungstreue Abbildungen der Sphären, Comment. Math. Helv. 11 (1938), no. 1, 49–61 (German). MR 1509591, DOI 10.1007/BF01199691
    • T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1980.
  • J. Kevorkian and Julian D. Cole, Perturbation methods in applied mathematics, Applied Mathematical Sciences, vol. 34, Springer-Verlag, New York-Berlin, 1981. MR 608029
  • Gen Komatsu, Analyticity up to the boundary of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math. 32 (1979), no. 5, 669–720. MR 533297, DOI 10.1002/cpa.3160320504
  • P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
  • 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. MR 725532
    • —, Morse decompositions for delay-differential equations, preprint. J. Mallet-Paret and B. Fiedler, Connections of Morse sets for delay-differential equations, in preparation.
  • William S. Massey, Singular homology theory, Graduate Texts in Mathematics, vol. 70, Springer-Verlag, New York-Berlin, 1980. MR 569059
  • Hiroshi Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. 18 (1978), no. 2, 221–227. MR 501842, DOI 10.1215/kjm/1250522572
  • 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. MR 672070
  • 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. MR 731522
    • —, personal communication, 1986.
  • 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. MR 750298, DOI 10.1007/978-1-4615-9047-7
  • Karl Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleichungen, J. Reine Angew. Math. 211 (1962), 78–94. (1 insert) (German). MR 146534, DOI 10.1515/crll.1962.211.78
    • O. A. Oleinik and S. N. Kruzhkov, Quasilinear second order parabolic equations with many independent variables, Russian Math. Surveys 16 (1961), 105-146.
  • Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1983. MR 833286, DOI 10.1017/CBO9780511608728
  • Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K57, 58F12
  • Retrieve articles in all journals with MSC: 35K57, 58F12
Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 307 (1988), 545-568
  • MSC: Primary 35K57; Secondary 58F12
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0940217-X
  • MathSciNet review: 940217