Maximum principles for parabolic systems coupled in both first-order and zero-order terms (Q1338441)

\textit{G. N. Hile} and \textit{M. H. Protter} [J. Differ. Equations 24, 136-151 (1977; Zbl 0351.35007)] proved that the Euclidean length of the solution vector \(u\in C^2(D)\cap C(\overline D)\) of a second-order elliptic system can be bounded by a constant times the maximum of its boundary values under a ``small'' condition which requires that either the domain \(D\) or the coefficients \(b_{sij}\) and \(c_{sj}\) are sufficiently small. In this paper, we establish the same kind of maximum principle for the second-order parabolic system \[ \sum^n_{i,k=1} a_{ik}(x,t){\partial^2 u_s\over\partial x_i\partial x_k}- {\partial u_s\over\partial t}+\sum^n_{i=1}\sum^m_{j=1} b_{sij}(x,t){\partial u_j\over\partial x_i}+\sum^m_{j=1} c_{sj}(x,t)u_j=0,\quad 1\leq s\leq m. \] Moreover, our parabolic version of the maximum principle holds without any ``small'' condition.