On the maximum principle for a class of linear parabolic differential systems (Q1117100)

The author studies to which extent the maximum principle holds for a linear parabolic system of the form \[ (1)\quad D_ tu=\sum A_{jk}D_{x_ j}uD_{x_ k}u+\sum B_ jD_{x_ j}u\quad in\quad {\mathbb{R}}^ n\times [0,T] \] where the matrices \(A_{jk}\) are subjected to the condition \[ Re(\sum A_{jk}\sigma_ j\sigma_ k\zeta,\zeta)\geq \delta | \sigma |^ 2| \zeta |^ 2 \] for some \(\delta >0.\) The main result asserts that the inequality \[ \sup \{| u(x,t)|:\quad x\in {\mathbb{R}}^ n,\quad t\in [0,T]\}\leq \sup \{| u(x,0)|:\quad x\in {\mathbb{R}}^ n\} \] holds for every solution u of (1) if and only if \(A_{jk}=a_{jk}E\), \(B_ j=b_ jE\) for suitable real-valued functions \(a_{jk}\), \(b_ j\), where E denotes the unit matrix. This theorem extends a previous result of \textit{V. G. Maz'ya} and \textit{G. I. Kresin} [Mat. Sb., Nov. Ser. 125(167), No.4(12), 458-48 (1984; Zbl 0577.35003)].