Bi-spaces global attractors in abstract parabolic equations
Volume 60 / 2003
Banach Center Publications 60 (2003), 13-26
MSC: Primary 35B40, 35B41, 35K15, 35K45
DOI: 10.4064/bc60-0-1
Abstract
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 $\cal A$ 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 $R^n$ and to the partly dissipative system.
