Criteria for invariance of convex sets 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 Petrovskiǐ in the layer ℝⁿ × [0,T]. For such systems we obtain necessary and, separately, sufficient conditions for invariance of the closure of an arbitrary convex proper subdomain of ℝ^m. 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 set and the coefficients of the system. The criterion is concretized for certain classes of invariant convex sets: polyhedral angles, cylindrical and conical bodies.