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Equilibrium-Like Solutions of Asymptotically Autonomous Differential Equations

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Abstract

We analyze the chain recurrent set of skew product semiflows obtained from nonautonomous differential equations—ordinary differential equations or semilinear parabolic differential equations. For many gradient-like dynamical systems, Morse–Smale dynamical systems e.g., the chain recurrent set contains only isolated equilibria. The structure in the asymptotically autonomous setting is richer but still close to the structure of a Morse–Smale dynamical system. The main tool used in this paper is a nonautonomous flavour of Conley index theory developed by the author. We will see that for a class of good equations, the Conley index can be understood in terms of equilibria (in a generalized meaning) and their connections. This allows us to find specific solutions of asymptotically autonomous equations and generalizes properties of Morse–Smale dynamical systems to the asymptotically autonomous setting.

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Notes

  1. A solution u is called full or entire if u is defined for all \(t\in \mathbb {R}\).

  2. \(N\subset C^k(\mathbb {R}\times E, E)\) is a neighbourhood of f iff there are finitely many sub-basis elements \(N_1,\ldots ,N_k\) which are neighborhoods of f and whose intersection is a subset of N.

  3. The set of all such h is residual i.e., a superset of a countable intersection of open and dense subsets.

  4. Conley [2] points out that a finest Morse-decomposition need not exist because the would-be finest Morse-decomposition might consist of infinitely many Morse sets. The union of all finest Morse-sets is then called chain recurrent set. In our case there are finest Morse-sets, so the distinction is meaningless.

  5. Strictly speaking, it is not required to assume that f is continuously differentiable in t.

  6. Every index pair in the sense of Definition 3.10 is assumed to be strongly admissible.

  7. \((\tilde{N}_1, \tilde{N}_2)\) is an index pair for \(K(f^\infty )\) in the sense of Franzosa and Mischaikow or an FM-index pair as defined by Rybakowski.

References

  1. Brunovsky, P., Polacik, P.: The Morse–Smale structure of a generic reaction–diffusion equation in higher space dimensions. J. Differ. Equ. 135, 129–181 (1997)

    Article  MathSciNet  Google Scholar 

  2. Conley, C.C.: Isolated Invariant Sets and the Morse Index, vol. 38. American Mathematical Society, New York (1978)

    Book  Google Scholar 

  3. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics IV, vol. 840, p. 348. Springer, Berlin (1981)

    Book  Google Scholar 

  4. Jänig, A.: The generic gradient-like structure of certain asymptotically autonomous semilinear parabolic equations. J. Differ. Equ. 264, 5713–5733 (2018)

    Article  MathSciNet  Google Scholar 

  5. Jänig, A.: Nonautonomous conley index theory: the connecting homomorphism. ArXiv e-prints. arXiv:1801.03427 [math.DS]

  6. Jänig, A.: Nonautonomous conley index theory: the homology index and attractor–repeller decompositions. In: Topological Methods in Nonlinear Analysis. ArXiv e-prints. arXiv:1711.04631 [math.DS]

  7. Jänig, A.: The Conley index along heteroclinic solutions of reaction–diffusion equations. J. Differ. Equ. 252, 4410–4454 (2012)

    Article  Google Scholar 

  8. Jänig, A.: A non-autonomous Conley index. J. Fixed Point Theory Appl. 19, 1825–1870 (2017)

    Article  MathSciNet  Google Scholar 

  9. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  Google Scholar 

  10. Rybakowski, K.P.: The Homotopy Index and Partial Differential Equations. Springer, Berlin (1987)

    Book  Google Scholar 

  11. Sacker, R.J., Sell, G.R.: Dichotomies for linear evolutionary equations in Banach spaces. J. Differ. Equ. 113, 17–67 (1994)

    Article  MathSciNet  Google Scholar 

  12. Sacker, R.J., Sell, G.R.: Dichotomies for linear evolutionary equations in Banach spaces. J. Differ. Equ. 113(1), 17–67 (1994)

    Article  MathSciNet  Google Scholar 

  13. Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Applied Mathematical Sciences, vol. 143, p. 670. Springer, New York (2002)

    Book  Google Scholar 

  14. Spanier, E.H.: Algebraic Topology. McGraw-Hill, New York (1966)

    MATH  Google Scholar 

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Correspondence to Axel Jänig.

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Jänig, A. Equilibrium-Like Solutions of Asymptotically Autonomous Differential Equations. J Dyn Diff Equat 32, 1249–1272 (2020). https://doi.org/10.1007/s10884-019-09785-8

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