In this work we develop the shape Conley index theory for local semiflows on complete metric spaces by using a weaker notion of shape index pairs. This allows us to calculate the shape index of a compact isolated invariant set K by restricting the system on any closed subset that contains a local unstable manifold of K, and hence significantly increases the flexibility of the calculation of shape indices and Morse equations. In particular, it allows to calculate shape indices and Morse equations for an infinite dimensional system by using only the unstable manifolds of the invariant sets, without requiring the system to be two-sided on the unstable manifolds.

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In this paper, we mainly study the long-time dynamical behaviors of 2D nonlocal stochastic Swift–Hohenberg equations with multiplicative noise from two perspectives. First, by adopting the analytic semigroup theory, we prove the upper semi-continuity of random attractors in the Sobolev space H02(U), as the coefficient of the multiplicative noise approaches zero. Then, we extend the classical “stochastic Gronwall’s lemma,” making it more convenient in applications. Based on this improvement, we are allowed to use the analytic semigroup theory to establish the existence of ergodic invariant measures.

This paper is concerned with the dynamics of the following abstract retarded evolution equation: (/) () + () = ((− 1),. .. , (−)) + () in a Hilbert space , where : () ⊂ → is a self-adjoint positive-definite operator with compact resolvent and : () → (∈ [0, 1/2]) is a locally Lipschitz continuous mapping. The dissipativity and pullback attractors are investigated, and the existence of locally almost periodic solutions is established.