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For uniformly stable bounded analytic C 0-semigroups { T( t)} t≥0 of linear operators in a Banach space B, we study the behavior of their orbits T ( t) x, x ∈ B, at infinity. We also analyze the relationship between the order of approaching the orbit T ( t) x to zero as t → ∞ and the degree of smoothness of the vector x with respect to the operator A −1 inverse to the generator A of the semigroup { T( t)} t≥0. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than $$e^{ - at^\alpha } $$ , where a > 0, 0 < α < π/(2(π-θ)), θ is the angle of analyticity of { T( t)} t≥0, and the collection of these orbits is dense in the set of all orbits. [ABSTRACT FROM AUTHOR]

It is proved that if the coefficients of a $$\overrightarrow {2b} $$ -parabolic system admit analytic extension to a complex region in the space variables, then the fundamental matrix of solutions of the Cauchy problem and regular solutions of the system also possess the same property. [ABSTRACT FROM AUTHOR]