Shadowing for infinite dimensional Morse–Smale dynamical systems

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 $$\mathbb {R}^n$$ . In fact, we show that if $$\{{\mathcal {T}}(t):t\ge 0\}$$ is a Morse-Smale semigroup defined in a Hilbert space, with global attractor $$\mathcal {A}$$ and non-wandering set given by its equilibria, then $${\mathcal {T}}(1)|_{\mathcal {A}}:\mathcal {A}\rightarrow \mathcal {A} $$ admits the Lipschitz Shadowing property. Moreover, for any positively invariant bounded neighborhood $$\mathcal {U}\supset \mathcal {A}$$ of the global attractor, the map $${\mathcal {T}}(1)|_{\mathcal {U}}:\mathcal {U}\rightarrow \mathcal {U}$$ has the Hölder–Shadowing property. As applications, we obtain results related to the structural stability of Morse-Smale semigroups, that were only known on finite dimension, and continuity of global attractors.