Criteria for Invariance of Convex Bodies for Linear Parabolic Systems

We consider systems of linear partial differential equations, which contain only second and first derivatives in the $x$ variables and which are uniformly parabolic in the sense of Petrovski\v{\i} in the layer ${\mathbb R}^n\times [0,T]$. For such systems we obtain necessary and, separately, sufficient conditions for invariance of a convex body. These necessary and sufficient conditions coincide if the coefficients of the system do not depend on $t$. The above mentioned criterion is formulated as an algebraic condition describing a relation between the geometry of the invariant convex body and coefficients of the system. The criterion is concretized for certain classes of invariant convex sets: polyhedral angles, cylindrical and conical bodies.