Abstract
We have already seen that the structure of the attractor of an autonomous gradient semigroup can be completely described: it is given by the union of the unstable sets of the equilibria (Theorem 2.43). However, key to the definition of a gradient semigroup (Definition 2.38) is the existence of a Lyapunov function, and this is a very delicate matter.
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References
Caraballo T, Carvalho AN, Langa JA, Rivero F (2010a) Existence of pullback attractors for pullback asymptotically compact processes. Nonlinear Anal 72:1967–1976
Caraballo T, Langa JA, Liu Z (2012) Gradient infinite-dimensional random dynamical systems. SIAM J Appl Dyn Syst, to appear
Carvalho AN, Langa JA (2007) The existence and continuity of stable and unstable manifolds for semilinear problems under non-autonomous perturbation in Banach spaces. J Differential Equations 233:622–653
Carvalho AN, Langa JA (2009) An extension of the concept of gradient semigroups which is stable under perturbation. J Differential Equations 246:2646–2668
Conley C (1978) Isolated invariant sets and the Morse index. CBMS regional conference series in mathematics, vol 38. American Mathematical Society, Providence, RI
Gordon P (1974) Paths connecting elementary critical points of dynamical systems. SIAM J Appl Math 26:35–102
Hale JK, Raugel G (1992a) A damped hyperbolic equation on thin domains. Trans Amer Math Soc 329:185–219
Hurley M (1995) Chain recurrence, semiflows and gradients. J Dynam Differential Equations 7:437–456
Kloeden PE, Rasmussen M (2011) Nonautonomous dynamical systems. Mathematical surveys and monographs. American Mathematical Society, Providence, RI
Langa JA, Lukaszewicz G, Real J (2007a) Finite fractal dimension of pullback attractors for non-autonomous 2D Navier–Stokes equations in some unbounded domains. Nonlinear Anal 66:735–749
Langa JA, Robinson JC, Rodríguez-Bernal A, Suárez A (2009) Permanence and asymptotically stable complete trajectories for non-autonomous Lotka–Volterra models with diffusion. SIAM J Math Anal 40:2179–2216
Liu Z (2007) The random case of Conley’s theorem. III: Random semiflow case and Morse decomposition. Nonlinearity 20:2773–2791
Liu Z, Ji S, Su M (2008) Attractor-repeller pair, Morse decomposition and Lyapunov function for random dynamical systems. Stoch Dyn 8:625–641
Mallet-Paret J (1988) Morse decompositions for delay-differential equations. J Differential Equations 72:270–315
Mischaikow K, Smith H, Thieme HR (1995) Asymptotically autonomous semiflows: chain recurrent and Lyapunov functions. Trans Amer Math Soc 347:1669–1685
Norton DE (1995) The fundamental theorem of dynamical systems. Comment Math Univ Caro 36:585–597
Rasmussen M (2007b) Morse decompositions of nonautonomous dynamical systems. Trans Amer Math Soc 359:5091–5115
Rybakowski KP (1987) The homotopy index and partial differential equations. Universitext, Springer Berlin Heidelberg New York
Sell GR, You Y (2002) Dynamics of evolutionary equations. Applied mathematical sciences, vol 143. Springer, Berlin Heidelberg New York
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Carvalho, A.N., Langa, J.A., Robinson, J.C. (2013). Gradient semigroups and their dynamical properties. In: Attractors for infinite-dimensional non-autonomous dynamical systems. Applied Mathematical Sciences, vol 182. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4581-4_5
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DOI: https://doi.org/10.1007/978-1-4614-4581-4_5
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