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An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on $X^{\alpha }$. This semigroup possesses an $(X^{\alpha }-Z)$-global attractor that is closed, bounded, invariant in $X^{\alpha }$, and attracts bounded subsets of $X^{\alpha }$ in a ’weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in Rⁿ and to the partly dissipative system.

The paper is devoted to the Cauchy problem for a semilinear damped wave equation in the whole of ℝ ⁿ. Under suitable assumptions a bounded dissipative semigroup of global solutions is constructed in a locally uniform space $Ḣ{¹}_{lu}\left(ℝⁿ\right)\times L̇{²}_{lu}\left(ℝⁿ\right)$. Asymptotic compactness of this semigroup and the existence of a global attractor are then shown.