Criteria for validity of the maximum modulus principle for solutions of linear parabolic systems

We consider systems of partial differential equations of the first order in t and of order 2 s in the x variables, which are uniformly parabolic in the sense of Petrovskii. We show that the classical maximum modulus principle is not valid in R ⁿ×(0, T] for s≥2. For second order systems we obtain necessary and, separately, sufficient conditions for the classical maximum modulus principle, to hold in the layer R ⁿ×(0, T] and in the cylinder μ×(0, T], where μ is a bounded subdomain of R ⁿ. If the coefficients of the system do not depend on t, these conditions coincide. The necessary and sufficient condition in this case is that the principal part of the system is scalar and that the coefficients of the system satisfy a certain algebraic inequality. We show by an example that the scalar character of the principal part of the system everywhere in the domain is not necessary for validity of the classical maximum modulus principle when the coefficients depend both on x and t.