Abstract: In this paper we prove Phragmen-Lindelöf type alternative.for foe initial boundaryproblem of Stokes equation, i. e. we show that the energy expression for the solutionof the initial boundary problem must either grow exponentially or decay exponentiallywilh axial distance from the end of a semi-infinite strip. For the case of decay, we alsoestablish the pointwise estimate for the maximum module of the Stokes.flow andpresent a method for obtaining explicit bounds for the total energy.

Key words: Stokes equation, initial boundary problem, phragmen-Lindelöf alternative theorem, estimate of energy dissipation