Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle
HTML articles powered by AMS MathViewer
- by Romain Joly and Geneviève Raugel PDF
- Trans. Amer. Math. Soc. 362 (2010), 5189-5211 Request permission
Abstract:
In this paper, we show that, for scalar reaction-diffusion equations on the circle $S^1$, the property of hyperbolicity of all equilibria and periodic orbits is generic with respect to the non-linearity. In other words, we prove that in an appropriate functional space of non-linear terms in the equation, the set of functions, for which all equilibria and periodic orbits are hyperbolic, is a countable intersection of open dense sets. The main tools in the proof are the property of the lap number and the Sard-Smale Theorem.References
- 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
- Sigurd Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79–96. MR 953678, DOI 10.1515/crll.1988.390.79
- S. B. Angenent and B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc. 307 (1988), no. 2, 545–568. MR 940217, DOI 10.1090/S0002-9947-1988-0940217-X
- A. V. Babin and M. I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992. Translated and revised from the 1989 Russian original by Babin. MR 1156492
- Robert A. Bonic, Linear functional analysis, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. MR 0257686
- Pavol Brunovský and Shui-Nee Chow, Generic properties of stationary state solutions of reaction-diffusion equations, J. Differential Equations 53 (1984), no. 1, 1–23. MR 747403, DOI 10.1016/0022-0396(84)90022-6
- P. Brunovský and P. Poláčik, The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension, J. Differential Equations 135 (1997), no. 1, 129–181. MR 1434918, DOI 10.1006/jdeq.1996.3234
- Pavol Brunovsky and Geneviève Raugel, Genericity of the Morse-Smale property for damped wave equations, J. Dynam. Differential Equations 15 (2003), no. 2-3, 571–658. Special issue dedicated to Victor A. Pliss on the occasion of his 70th birthday. MR 2046732, DOI 10.1023/B:JODY.0000009749.10737.9d
- R. Czaja and C. Rocha, Transversality in scalar reaction-diffusion equations on a circle, Journal of Differential Equations \no245 (2008), pp. 692-721.
- Bernold Fiedler and John Mallet-Paret, A Poincaré-Bendixson theorem for scalar reaction diffusion equations, Arch. Rational Mech. Anal. 107 (1989), no. 4, 325–345. MR 1004714, DOI 10.1007/BF00251553
- Bernold Fiedler and Carlos Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Differential Equations 125 (1996), no. 1, 239–281. MR 1376067, DOI 10.1006/jdeq.1996.0031
- Bernold Fiedler and Carlos Rocha, Orbit equivalence of global attractors of semilinear parabolic differential equations, Trans. Amer. Math. Soc. 352 (2000), no. 1, 257–284. MR 1475682, DOI 10.1090/S0002-9947-99-02209-6
- Bernold Fiedler, Carlos Rocha, and Matthias Wolfrum, Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, J. Differential Equations 201 (2004), no. 1, 99–138. MR 2057540, DOI 10.1016/j.jde.2003.10.027
- M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518
- 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
- Dan Henry, Perturbation of the boundary in boundary-value problems of partial differential equations, London Mathematical Society Lecture Note Series, vol. 318, Cambridge University Press, Cambridge, 2005. With editorial assistance from Jack Hale and Antônio Luiz Pereira. MR 2160744, DOI 10.1017/CBO9780511546730
- Romain Joly, Generic transversality property for a class of wave equations with variable damping, J. Math. Pures Appl. (9) 84 (2005), no. 8, 1015–1066 (English, with English and French summaries). MR 2155898, DOI 10.1016/j.matpur.2005.01.002
- Romain Joly, Adaptation of the generic PDE’s results to the notion of prevalence, J. Dynam. Differential Equations 19 (2007), no. 4, 967–983. MR 2357534, DOI 10.1007/s10884-007-9097-7
- R. Joly and G. Raugel, Generic Morse-Smale property for the parabolic equation on the circle, submitted.
- John Mallet-Paret, Generic periodic solutions of functional differential equations, J. Differential Equations 25 (1977), no. 2, 163–183. MR 442995, DOI 10.1016/0022-0396(77)90198-x
- 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 and Ken-Ichi Nakamura, The global attractor of semilinear parabolic equations on $S^1$, Discrete Contin. Dynam. Systems 3 (1997), no. 1, 1–24. MR 1422536, DOI 10.3934/dcds.1999.5.1
- Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- M. M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967), 214–227. MR 209602, DOI 10.1016/0022-0396(67)90026-5
- Frank Quinn, Transversal approximation on Banach manifolds, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 213–222. MR 0264713
- J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations 4 (1979), no. 3, 293–319. MR 522714, DOI 10.1080/03605307908820096
- Björn Sandstede and Bernold Fiedler, Dynamics of periodically forced parabolic equations on the circle, Ergodic Theory Dynam. Systems 12 (1992), no. 3, 559–571. MR 1182662, DOI 10.1017/S0143385700006933
- J. Smoller and A. Wasserman, Generic bifurcation of steady-state solutions, J. Differential Equations 52 (1984), no. 3, 432–438. MR 744306, DOI 10.1016/0022-0396(84)90172-4
- C. Sturm, Sur une classe d’équations à différences partielles, J. Math. Pures Appl. \no1 (1836), pp. 373-444.
- T. I. Zelenjak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differencial′nye Uravnenija 4 (1968), 34–45 (Russian). MR 0223758
Additional Information
- Romain Joly
- Affiliation: Institut Fourier, UMR CNRS 5582, Université de Grenoble I, B.P. 74, 38402 Saint-Martin-d’Hères, France
- Email: Romain.Joly@ujf-grenoble.fr
- Geneviève Raugel
- Affiliation: CNRS, Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, F-91405 Orsay Cedex, France
- Email: Genevieve.Raugel@math.u-psud.fr
- Received by editor(s): May 20, 2008
- Published electronically: May 20, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 5189-5211
- MSC (2010): Primary 35B10, 35B30, 35K57, 37D05, 37D15, 37L45; Secondary 35B40
- DOI: https://doi.org/10.1090/S0002-9947-2010-04890-1
- MathSciNet review: 2657677