## A Poincaré-Bendixson theorem for scalar reaction diffusion equations.(English)Zbl 0704.35070

Authors’ summary: “For scalar equations $$u_ t=u_{xx}+f(x,u,u_ x)$$ with $$x\in S^ 1$$ and $$f\in C^ 2$$ we show that the classical theorem of Poincaré and Bendixson holds: the $$\omega$$-limit set of any bounded solution satisfies exactly one of the following alternatives:
- it consists in precisely one periodic solution, or
- it consists of solutions tending to equilibrium as $$t\to \pm \infty.$$
This is surprising, because the system is genuinely infinite- dimensional.”
Reviewer: B.D.Sleeman

### MSC:

 35K57 Reaction-diffusion equations 35B99 Qualitative properties of solutions to partial differential equations
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### References:

 [1] S. Angenent 1, The Morse-Smale property for a semi-linear parabolic equation, J. Diff. Eq. 62 (1986), 427–442. · Zbl 0581.58026 [2] S. Angenent 2, The zeroset of a solution of a parabolic equation, J. reine angew. Math. 390 (1988), 79–96. · Zbl 0644.35050 [3] S. Angenent & B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. AMS 307 (1988), 545–568. · Zbl 0696.35086 [4] I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math. 24 (1901), 1–88. · JFM 31.0328.03 [5] J. E. Billoti & J. P. LaSalle, Periodic dissipative processes, Bull. AMS 6 (1971), 1082–1089. · Zbl 0274.34061 [6] P. Brunovský & B. Fiedler 1, Zero numbers on invariant manifolds in scalar reaction diffusion equations, Nonlin. Analysis TMA 10 (1986), 179–194. · Zbl 0594.35056 [7] P. Brunovský & B. Fiedler 2, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported 1 (1988), 57–89. · Zbl 0679.35047 [8] P. Brunovský & B. Fiedler 3, Connecting orbits in scalar reaction diffusion equations II: the complete solution, to appear in J. Diff. Eq. · Zbl 0699.35144 [9] X.-Y. Chen & H. Matano, Convergence of solutions of semilinear heat equations on S 1, in preparation. [10] E. A. Coddington & N. Levinson, Theory of Ordinary Differential Equations, McGraw Hill, New York 1955. · Zbl 0064.33002 [11] C. M. Dafermos, Asymptotic behavior of solutions of evolution equations, in ”Nonlinear Evolution Equations”, M. G. Crandall (ed.), Academic Press, New York 1978, 103–123. · Zbl 0499.35015 [12] B. Fiedler, Discrete Ljapunov functionals and {$$\omega$$}-limit sets, Mathematical Modelling and Numerical Analysis 23 (1989), 59–75. · Zbl 0688.58041 [13] B. Fiedler & J. Mallet-Paret, Connections between Morse sets for delay-differential equations, preprint 1987. [14] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall 1964. · Zbl 0144.34903 [15] G. Fusco & W. M. Oliva, Jacobi matrices and transversality, to appear in Proc. Royal Soc. Edinburgh. · Zbl 0639.34044 [16] G. Fusco & C. Rocha, in preparation. [17] J. K. Hale, Ordinary Differential Equations, John Wiley & Sons, New York 1969. · Zbl 0186.40901 [18] J. K. Hale, L. T. Magalhães & W. M. Oliva, An Introduction to Infinite Dimensional Dynamical Systems–Geometric Theory, Appl. Math. Sc. 47, Springer-Verlag, New York 1984. · Zbl 0533.58001 [19] J. K. Hale & J. Vegas, A nonlinear parabolic equation with varying domain, Arch. Rational Mech. Analysis 86 (1984), 99–123. · Zbl 0569.35048 [20] P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston 1982. [21] D. Henry 1, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math. 840, Springer-Verlag, New York 1981. · Zbl 0456.35001 [22] D. Henry 2, Some infinite dimensional Morse-Smale systems defined by parabolic differential equations, J. Diff. Eq. 59 (1985), 165–205. · Zbl 0572.58012 [23] M. W. Hirsch 1, Differential equations and convergence almost everywhere in strongly monotone semiflows, in ”Nonlinear Partial Differential Equations”, J. Smoller (ed.), AMS, Providence 1983, 267–285. · Zbl 0523.58034 [24] M. W. Hirsch 2, Systems of differential equations that are competitive or cooperative II. Convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), 423–439. · Zbl 0658.34023 [25] M. W. Hirsch 3, Stability and convergence in strongly monotone dynamical systems, to appear in J. reine angew. Math. [26] J. L. Kaplan & J. A. Yorke, On the stability of a periodic solution of a differential delay equation, SIAM J. Math. Anal. 6 (1975), 268–282. · Zbl 0293.34107 [27] S. Lefschetz, Differential Equations: Geometric Theory, Wiley & Sons, New York 1963. · Zbl 0107.07101 [28] J. Mallet-Paret 1, Morse decompositions and global continuation of periodic solutions for singularly perturbed delay equations, in ”Systems of Nonlinear Partial Differential Equations”, J. M. Ball (ed.), D. Reidel, Dordrecht 1983, 351–366 [29] J. Mallet-Paret 2, Morse decompositions for delay-differential equations, J. Diff. Eq. 72 (1988), 270–315. · Zbl 0648.34082 [30] J. Mallet-Paret & G. R. Sell, in preparation. [31] J. Mallet-Paret & H. Smith, The Poincaré-Bendixson theorem for monotone cyclic feedback systems, preprint 1987. · Zbl 0712.34060 [32] P. Massatt, The convergence of scalar parabolic equations with convection to periodic solutions, preprint 1986. [33] H. Matano 1, Convergence of solutions of one-dimensional parabolic equations, J. Math. Kyoto Univ. 18 (1978), 221–227. · Zbl 0387.35008 [34] H. Matano 2, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sc. Kyoto Univ. 15 (1979), 401–454. · Zbl 0445.35063 [35] H. Matano 3, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sec. IA 29 (1982), 401–441. · Zbl 0496.35011 [36] H. Matano 4, Existence of nontrivial unstable sets for equilibriums of strongly orderpreserving systems, J. Fac. Sc. Univ. Tokyo Sec. IA 30 (1984), 645–673. · Zbl 0545.35042 [37] H. Matano 5, Asymptotic behavior of solutions of semilinear heat equations on S1, in ”Nonlinear Diffusion Equations and Their Equilibrium States II”, W.-M. Ni & B. Peletier & J. Serrin (eds.), Springer-Verlag, New York 1988, 139–162. [38] H. Matano 6, Asymptotic behavior of nonlinear diffusion equations, Res. Notes Math., Pitman, to appear. [39] H. Matano 7, Strongly order-preserving local semi-dynamical systems–theory and applications, in ”Semigroups, Theory and Applications”, H. Brezis & M. G. Crandall & F. Kappel (eds.), John Wiley & Sons, New York 1986. · Zbl 0634.34031 [40] K. Mischaikow, Conley’s connection matrix, in ”Dynamics of Infinite Dimensional Systems”, S.-N. Chow & J. K. Hale (eds.), Springer-Verlag, Berlin 1987, 179–186. [41] K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleichungen, J. reine angew. Math. 211 (1962), 78–94. · Zbl 0127.31801 [42] H. Poincaré, OEuvres I, Gauthier-Villars, Paris 1928. [43] G. Pólya, Qualitatives über Wärmeausgleich, Z. Angew. Math. Mech. 13 (1933), 125–128. · Zbl 0006.29804 [44] G. Sansone & R. Conti, Non-Linear Differential Equations, Pergamon Press, Oxford 1964. · Zbl 0128.08403 [45] H. Smith, Monotone semiflows generated by functional differential equations, J. Diff. Eq. 66 (1987), 420–442. · Zbl 0612.34067 [46] R. A. Smith 1, The Poincaré-Bendixson theorem for certain differential equations of higher order, Proc. Roy. Soc. Edinburgh A 83 (1979), 63–79. [47] R. A. Smith 2, Existence of periodic orbits of autonomous ordinary differential equations, Proc. Roy. Soc. Edinburgh A 85 (1980), 153–172. · Zbl 0429.34040 [48] R. A. Smith 3, Existence of periodic orbits of autonomous retarded functional differential equations, Math. Proc. Camb. Phil. Soc. 88 (1980), 89–109. · Zbl 0435.34062 [49] J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, New York 1983. · Zbl 0508.35002 [50] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Appl. Math. Sc. 41, Springer-Verlag, New York 1982. · Zbl 0504.58001 [51] C. Sturm, Sur une classe d’équations à différences partielles, J. Math. Pure Appl. 1 (1836), 373–444. [52] T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Differential Equations 4, 1 (1968), 17–22. · Zbl 0232.35053
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