CRITERIA FOR VALIDITY OF THE MAXIMUM MODULUS PRINCIPLE FOR SOLUTIONS OF LINEAR PARABOLIC-SYSTEMS

We consider systems of partial differential equations of the first ord er in t and of order 2s in the x variables, which are uniformly parabo lic in the sense of Petrovskii. We show that the classical maximum mod ulus principle is not valid in R(n) x (0, T] for s > 2. For second ord er systems we obtain necessary and, separately, sufficient conditions for the classical maximum modulus principle to hold in the layer Rn x (0, T] and in the cylinder Q x (0, T], where OMEGA is a bounded subdom ain of R(n). If the coefficients of the system do not depend on t, the se 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 t he classical maximum modulus principle when the coefficients depend bo th on x and t.