Gi. Kresin et Vg. Mazya, CRITERIA FOR VALIDITY OF THE MAXIMUM MODULUS PRINCIPLE FOR SOLUTIONS OF LINEAR PARABOLIC-SYSTEMS, Arkiv for matematik, 32(1), 1994, pp. 121-155
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.