A Remark on Positively Invariant Regions for Parabolic Systems with an Application Arising in Superconductivity*

0. Introduction. With regard to the maximum principle, elliptic systems have been treated as a particular case of parabolic systems (see, for example, [W]). Therefore, the smallest set (that this approach can offer) which localizes pointwise the steady states is necessarily positively invariant with respect to the flow that the associated parabolic system induces. It turns out that one can do better for the elliptic system and this for a very simple reason: by multiplying through by the inverse of the coefficient matrix corresponding to the highest derivatives, the system is reduced to one with equal diffusion coefficients. Thus, whenever this can be performed, we improve the rectangle to an inscribed ellipsoid. This observation, apparently not noticed before, has an interesting implication in the case that the parabolic system is gradient: the set that localizes the steady states as well as the family of its congruent sets that circumscribe it will not, in general, be positively invariant. Yet any solution corresponding to the initial condition lying in any of these sets has eventually to return to the set and stay in it. In a sense, in this case, the elliptic sets are eventually positively invariant for the parabolic system. We choose to illustrate these observations in terms of an example. We have not made any effort to identify the most general circumstances where our approach is applicable. The system we have in mind is 011 -r= DAu + [l (Au, «)] on £2, a (!) 8" -0, dn 3G