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 $X^α$. This semigroup possesses an $(X^α - Z)$-global attractor 𝓐 that is closed, bounded, invariant in $X^α$, and attracts bounded subsets of $X^α$ 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.
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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 $Ḣ¹_{lu}(ℝ ⁿ) × L̇²_{lu}(ℝ ⁿ)$. Asymptotic compactness of this semigroup and the existence of a global attractor are then shown.
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