On a weak (algebraic) extremum principle for a second-order parabolic system

The notion of a weak “algebraic” extremum principle (WAEP) is introduced for second-order parabolic systems. It is based on the representation of the (coefficient) matrix of the system as a sum of matrices that are similar to diagonal matrices and nilpotent matrices, and on the reduction of the system to a single equation. The validity of the WAEP is proved for a rather broad class of second-order parabolic systems with “full mixing”. The WAEP is applied to prove the uniqueness of the solution of the first boundary-value problem for the parabolic systems in question.