On the strongly hyperbolic systems (Q1111773)

Let be \[ Lu:=\partial_ tu-\sum^{n}_{k=1}A_ k(x,t)\partial_{x_ k}u, \] where u is an m-vector and \(A_ k\) are \(m\times m\)-matrices in \(C^{\infty}({\mathbb{R}}^ n_ x\times (-T,T))\). The author proves the following theorem: Assume that there exist \((x_ 0,t_ 0;\xi^ 0)\in {\mathbb{R}}^ n_ x\times (-T,T)\times {\mathbb{R}}^ n_{\xi}\setminus \{0\}\) and \(\lambda^ 0\in {\mathbb{R}}\) such that \(rank(\lambda^ 0I-\sum^{n}_{k=1}A_ k(x_ 0,t_ 0)\xi^ 0_ k)=m-1.\) Then in order that L is a strongly hyperbolic system, it is necessary that the multiplicity of the characteristic root \(\lambda^ 0\) must be less than 3. This result shows that an assumption in a theorem of \textit{T. Nishitani} [Proc. Japan Acad., Ser. A 61, 193-196 (1985; Zbl 0579.35046)] is natural.