2025/02/12 by Arrieta, José M., Carvalho, Alexandre N., Takaessu, Carlos R.
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.08315
In this paper we extend to an infinite dimensional setting some results on the shadowing property that are known on finite dimensional compact manifolds without border and in ℝn. In fact, we show that if \\T(t):t≥ 0\ is a Morse-Smale semigroup defined in a Hilbert space with a global attractor A and non-wandering set given only by its equilibria, then \T(1)|A:A→ A admits the Lipschitz Shadowing property. Moreover, for any positively invariant bounded neighborhood \U\supsetA of the global attractor, the map \T(1)|\U:\U→ \U has the Hölder-Shadowing property. We obtain results related to the structural stability of Morse-Smale semigroups, that were only known on finite dimension and continuity of global attractors.