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Generic Morse-Smale property for the parabolic equation on the circle. (English) Zbl 1213.35046
The following scalar reaction-diffusion equation on \(S^1\) is considered
\[ \begin{cases} u_t(x,t)= u_{xx}(x,t)+f(x,u(x,t),u_x(x,t)), &(x,t)\in S^1\times(0,\infty),\\ u(x,0)=u_0(x), &x\in S^1, \end{cases} \]
where \(f\) belongs to the space \(C^2(S^1\times\mathbb{R}\times\mathbb{R},\mathbb{R})\), and \(u_0\) is a given function in the Sobolev space \(H^s(S^1)\), \(s\in(3/2,2)\), so that \(H^s(S^1)\) is continuously embedded into \(C^{1+\alpha}\) for \(\alpha=s-3/2\).
On the basis of the lap number property, exponential dichotomies, Sard-Smale theorem and analysis of the asymptotic behaviour of solutions of the linearized equations along the connecting orbits the authors complete their previous results [R. Joly and G. Raugel, Trans. Am. Math. Soc. 362, No. 10, 5189–5211 (2010; Zbl 1205.35151)] and show that any connecting orbit between two hyperbolic equilibria with distinct Morse indices or between a hyperbolic equilibrium and hyperbolic periodic orbit is automatically transverse. Generically, with respect to \(f\), there does not exist any connection between equilibria with the same Morse index. The main result of the article is that generically with respect to \(f\) the nonwandering set consists of a finite number of hyperbolic equilibria and periodic orbits.

MSC:
35B10 Periodic solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K57 Reaction-diffusion equations
37D05 Dynamical systems with hyperbolic orbits and sets
37D15 Morse-Smale systems
37L45 Hyperbolicity, Lyapunov functions for infinite-dimensional dissipative dynamical systems
35B40 Asymptotic behavior of solutions to PDEs
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References:
[1] Agmon, S., Unicité et convexité dans LES problèmes différentiels, (1966), Presses de l’Université de Montreal · Zbl 0147.07702
[2] Andronov, A.; Pontrjagin, L.S., Systèmes grossiers, Dokl. akad. nauk, 14, 247-250, (1937) · JFM 63.0728.01
[3] Angenent, S., The Morse-Smale property for a semi-linear parabolic equation, J. differential equations, 62, 427-442, (1986) · Zbl 0581.58026
[4] Angenent, S., The zero set of a solution of a parabolic equation, Journal für die reine und angewandte Mathematik, 390, 79-96, (1988) · Zbl 0644.35050
[5] Angenent, S.; Fiedler, B., The dynamics of rotating waves in scalar reaction diffusion equations, Trans. amer. math. soc., 307, 545-568, (1988) · Zbl 0696.35086
[6] Bardos, C.; Tartar, L., Sur l’unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. rational mech. analysis, 50, 10-25, (1973) · Zbl 0258.35039
[7] P. Brunovský, R. Joly, G. Raugel, Genericity of the Kupka-Smale property for scalar parabolic equations, manuscript.
[8] Brunovský, P.; Poláčik, P., The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension, J. differential equations, 135, 129-181, (1997) · Zbl 0868.35062
[9] Brunovský, P.; Raugel, G., Genericity of the Morse-Smale property for damped wave equations, Journal of dynamics and differential equations, 15, 2, 571-658, (2003) · Zbl 1053.35099
[10] Chen, M.; Chen, X.-Y.; Hale, J.K., Structural stability for time periodic one dimensional parabolic equations, J. differential equations, 96, 355-418, (1992) · Zbl 0779.35061
[11] Chow, S.-N.; Hale, J.K.; Mallet-Paret, J., An example of bifurcation to homoclinic orbits, J. differential equations, 37, 351-373, (1980) · Zbl 0439.34035
[12] Coppel, W.A., Dichotomies in stability theory, Lecture notes in math., vol. 629, (1978), Springer-Verlag · Zbl 0376.34001
[13] Czaja, R.; Rocha, C., Transversality in scalar reaction-diffusion equations on a circle, J. differential equations, 245, 692-721, (2008) · Zbl 1157.35004
[14] Fiedler, B.; Mallet-Paret, J., A Poincaré-Bendixson theorem for scalar reaction-diffusion equations, Arch. rational mech. analysis, 107, 325-345, (1989) · Zbl 0704.35070
[15] Fiedler, B.; Rocha, C.; Wolfrum, M., Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, J. differential equations, 201, 99-138, (2004) · Zbl 1064.35076
[16] Fusco, G.; Oliva, W.M., Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems, Journal of dynamics and differential equations, 2, 1-17, (1990) · Zbl 0702.34038
[17] Golubitsky, M.; Guillemin, V., Stable mappings and their singularities, Graduate texts in mathematics, vol. 14, (1973), Springer-Verlag New York, Heidelberg · Zbl 0294.58004
[18] J.K. Hale, R. Joly, G. Raugel, book in preparation.
[19] Hale, J.K.; Lin, X.-B., Heteroclinic orbits for retarded functional differential equations, J. differential equations, 65, 175-202, (1986) · Zbl 0611.34074
[20] Hale, J.K.; Magalhães, L.T.; Oliva, W.M., Dynamics in infinite dimensions, Applied mathematical sciences, vol. 47, (2002), Springer-Verlag · Zbl 1002.37002
[21] J.K. Hale, G. Raugel, Behaviour near a non-degenerate periodic orbit, manuscript, 2010.
[22] Henry, D., Geometric theory of semilinear parabolic equations, Lecture notes in math., vol. 840, (1981), Springer-Verlag · Zbl 0456.35001
[23] Henry, D., Some infinite dimensional Morse-Smale systems defined by parabolic differential equations, J. differential equations, 59, 165-205, (1985) · Zbl 0572.58012
[24] Henry, D., Exponential dichotomies, the shadowing lemma and homoclinic orbits in Banach spaces, Resenhas IME-USP, 1, 381-401, (1994) · Zbl 0906.58034
[25] Henry, D., Perturbation of the boundary for boundary value problems of partial differential operators, London mathematical society lecture note series, vol. 318, (2005), Cambridge University Press Cambridge, UK, with editorial assistance from Jack Hale and Antonio Luiz Pereira
[26] Hirsch, M.W., Stability and convergence on strongly monotone dynamical systems, J. reine angew. math., 383, 1-53, (1988) · Zbl 0624.58017
[27] Hirsch, M.W.; Pugh, C.C.; Shub, M., Invariant manifolds, Bull. amer. math. soc., 76, 1015-1019, (1970) · Zbl 0226.58009
[28] Hirsch, M.W.; Pugh, C.C.; Shub, M., Invariant manifolds, Lecture notes in math., vol. 583, (1977), Springer-Verlag Berlin, New York · Zbl 0355.58009
[29] Joly, R., Generic transversality property for a class of wave equations with variable damping, Journal de mathématiques pures et appliquées, 84, 1015-1066, (2005) · Zbl 1082.35109
[30] Joly, R., Adaptation of the generic PDE’s results to the notion of prevalence, Journal of dynamics and differential equations, 19, 967-983, (2007) · Zbl 1130.35013
[31] Joly, R.; Raugel, G., Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle, Trans. amer. math. soc., 362, 5189-5211, (2010) · Zbl 1205.35151
[32] R. Joly, G. Raugel, A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations, preprint, submitted for publication.
[33] Kupka, I.; Kupka, I., Contribution à la théorie des champs génériques, Contributions to differential equations, Contributions to differential equations, 3, 411-420, (1964), Addendum and corrections · Zbl 0149.41003
[34] Lang, S., Introduction to differentiable manifolds, (1962), John Wiley and Sons USA · Zbl 0103.15101
[35] Lax, P., A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Communications on pure and applied mathematics, IX, 747-766, (1956) · Zbl 0072.33004
[36] Lin, X.B., Exponential dichotomies and homoclinic orbits in functional differential equations, J. differential equations, 63, 227-254, (1986) · Zbl 0589.34055
[37] Matano, H., Convergence of solutions of one-dimensional semilinear parabolic equations, Journal of mathematics of Kyoto university, 18, 221-227, (1978) · Zbl 0387.35008
[38] Matano, H., Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation, J. fac. sci. univ. Tokyo sec. IA, 29, 401-441, (1982) · Zbl 0496.35011
[39] Nickel, K., Gestaltaussagen über Lösungen parabolischer differentialgleichungen, Journal für die reine und angewandte Mathematik, 211, 78-94, (1962) · Zbl 0127.31801
[40] Oliva, W.M., Morse-Smale semiflows. openness and A-stability, (), 285-307 · Zbl 1220.37068
[41] Ott, W.; Yorke, J.A., Prevalence, Bulletin of the American mathematical society, 42, 263-290, (2005) · Zbl 1111.28014
[42] Palis, J., On Morse-Smale dynamical systems, Topology, 8, 385-405, (1969) · Zbl 0189.23902
[43] Palis, J.; de Melo, W., Geometric theory of dynamical systems, (1982), Springer-Verlag Berlin
[44] Palis, J.; Smale, S., Structural stability theorems, (), 223-231 · Zbl 0214.50702
[45] Palmer, K.J., Exponential dichotomies and transversal homoclinic points, J. differential equations, 55, 225-256, (1984) · Zbl 0508.58035
[46] Palmer, K.J., Exponential dichotomies, the shadowing lemma and transversal homoclinic points, (), 265-306
[47] Palmer, K.J., Shadowing in dynamical systems. theory and applications, (2000), Kluwer Dordrecht, Boston, London · Zbl 0997.37001
[48] Peixoto, M.M., Structural stability on two-dimensional manifolds, Topology, 1, 101-120, (1962) · Zbl 0107.07103
[49] Peixoto, M.M., On an approximation theorem of kupka and Smale, J. differential equations, 3, 214-227, (1967) · Zbl 0153.40901
[50] Poláčik, P., Parabolic equations: asymptotic behavior and dynamics on invariant manifolds, (), 835-883 · Zbl 1002.35001
[51] Ruelle, D., Elements of differentiable dynamics and bifurcation theory, (1989), Academic Press London · Zbl 0684.58001
[52] Quinn, F., Transversal approximation on Banach manifolds, (), 213-222 · Zbl 0206.25705
[53] Robbin, J.W., Algebraic kupka-Smale theory, (), 286-301 · Zbl 0487.58018
[54] Saut, J.-C.; Temam, R., Generic properties of nonlinear boundary value problems, Communications in PDE, 4, 293-319, (1979) · Zbl 0462.35016
[55] Sacker, R.J., Existence of dichotomies and invariant splitting for linear differential systems IV, J. differential equations, 27, 106-137, (1978) · Zbl 0359.34044
[56] Salamon, D., Morse theory, the Conley index and Floer homology, Bulletin of London mathematical society, 22, 113-140, (1990) · Zbl 0709.58011
[57] Sandstede, B.; Fiedler, B., Dynamics of periodically forced parabolic equations on the circle, Ergodic theory and dynamical systems, 12, 559-571, (1992) · Zbl 0754.35066
[58] Smale, S., Morse inequalities for a dynamical system, Bulletin of the AMS, 66, 43-49, (1960) · Zbl 0100.29701
[59] Smale, S., Stable manifolds for differential equations and diffeomorphisms, Annali Della scuola normale superiore di Pisa, 17, 97-116, (1963) · Zbl 0113.29702
[60] Smale, S., Diffeomorphisms with many periodic points, (), 63-80 · Zbl 0142.41103
[61] Sturm, C., Sur une classe d’équations à différences partielles, Journal de mathématiques pures et appliquées, 1, 373-444, (1826)
[62] Zelenyak, T.J., Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Diff. equations, 4, 17-22, (1968) · Zbl 0232.35053
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