We consider a boundary value problem in the half-space for a linear parabolicequation of fourth order with a degeneration on the boundary of the half-space.The equation under consideration is substantially a linearized thin filmequation. We prove that, if the right hand side of the equation and theboundary condition are polynomials in the tangential variables and time, thesame property has any solution of a power growth. It is shown also that thespecified property does not apply to normal variable. As an application, wepresent a theorem of uniqueness for the problem in the class of functions ofpower growth.
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