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The article is devoted to semilinear Schrödinger equations in bounded domains. A unified semigroup approach is applied following a concept of Trotter-Kato approximations.Critical exponents are exhibited and global solutions are constructed for nonlinearities satisfying even a certain critical growth condition in $ H^1_0(Ω)$.

$u_t_t + 2\nA^(1/2)u_t + A\u = \f(u)$

in $W^(1,p)_0 (\Omega) \times L^p(\Omega), p \in (1,\infty)$, where $\Omega \subset \mathbb {R}^N$ is a bounded smooth domain, $\n$ > 0 and -$\A$ is the Dirichlet Laplacian in $L^p(\Omega)$. We prove local well posedness result for nonlinearities $\f : \mathbb {R} \rightarrow \mathbb {R}$ satisfying $|f(s) - f(t)| \<= C|s - t|(1 + |s|^(p - 1) + |t|^(p - 1))$ with $\p < (N+p)/(N - p) (N > p)$, and show that the solutions are classical. If $f$ is dissipative and $p < (N+2)/(N - 2) (N \>= 3)$, we show that the associated semigroup has a global attractor $\cc{A}_(n,p)$ in $W^(1,p)_0(\Omega)\times L^p(\Omega)$, $p \in [2,\infty)$, which coincides with the attractor $\bb{A}_(n,2) =: \bb{A}_n$. We also obtain that $\bb{A}_n$ is compact in $C^(2+\mu)(bar(\Omega)) \times C^(1+\mu)(bar(\Omega))$ and attracts bounded subsets of $H^1_0(\Omega) \times L^2(\Omega)$ in $C^(2+\mu)(bar(\Omega)) \times C^(1+\mu)(bar(\Omega))$ for each $\mu \in (0, 1)$.

*Discrete and Continuous Dynamical Systems - Series B*on the asymptotic dynamics of non-autonomous systems.

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If the global attractor of an autonomous dynamical system is given as the union of a finite number of unstable manifolds of equilibria, a detailed understanding of the continuity of the local dynamics under perturbation leads to important results on the lower-semicontinuity and topological structural stability for the pullback attractors of evolution processes that arise from small non-autonomous perturbations, with respect to the limit regime. Finally, continuity with respect to global dynamics under non-autonomous perturbation is also studied, for which appropriate concepts for Morse decomposition of attractors and non-autonomous Morse--Smale systems are introduced. All of these results will also be considered for uniform attractors. As a consequence, this paper also makes connections between different approaches to the qualitative theory of non-autonomous differential equations, which are often treated independently.

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