On well-posedness of the Cauchy problem for p-parabolic systems. II (Q908445)

[For part I see ibid. 27, 569-586 (1987; Zbl 0636.35033).] Let A(x,D) be a matrix pseudo-differential operator of order p in the form \[ (1)\quad A(x,D)=H(x,D)\Lambda^ p+B(x,D),\quad x\in {\mathbb{R}}^{\ell}, \] where H(x,\(\xi)\) is \(m\times m\) homogeneous matrix of degree 0 in \(\xi\) (\(| \xi | \geq 1)\) and smooth in x and \(\xi\). B(x,\(\xi)\) belongs to the class \(S^{P_ 0}_{1,0}\), \(0\leq p_ 0\leq p\), modulo smoothing operators and \(\Lambda\) is a pseudodifferential operator, where the symbol belongs to \(S^ 1_{1,0}\) and coincides with \(| \xi |\) for \(| \xi | \geq 1\), p is a positive number. The purpose of this paper is to show that the conditions \[ (2)\quad \sup_{x\in {\mathbb{R}}^{\ell}},_{\xi \in S_{\xi}^{\ell - 1}}\lambda;\quad (x,\xi)<0,\quad \ell \leq \forall i\leq m \] is necessary and sufficient in order that there exist positive constants a, b and \(\beta\) such that the estimate \[ (3)\quad \| (\lambda I-A(x,D))U\| \geq a(| \lambda | -\beta_ 0)\| U\| +b\| U\|_ p, \] for \(\forall U\in H^ p\), \(\forall \lambda\), Re \(\lambda\geq \beta_ 0\) holds. Here U(x) is an m-vector, \(\| \cdot \|\), \(\| \cdot \|_ p\) denote \(L^ 2\) and \(H^ p\)-norm respectively, \(\lambda_ i(x,\xi)\) \((i=1,...,m)\) are the roots of the characteristic equation \[ \det (\lambda I-H(x,\xi))=0. \] The sufficiency of condition (2) was proved in Part I. In this part the author shows the necessity of condition (2) for the correctness of inequality (3).