Non-uniqueness in the Cauchy problem for partial differential operators with multiple characteristics. II (Q5902923)

[For part I see Commun. Partial Differ. Equations 9, 63-106 (1984; Zbl 0559.35001).] In this note, we consider non-uniqueness of \(C^{\infty}\)-solutions of the non-characteristic Cauchy problem for the following operators: \[ P=(\partial_ t-t^{\ell}C(t,x;D_ x))^ p+t^ kA(t,x;D_ x)-t^ mB(t,x;D_ x)+ \] \[ \sum^{p}_{j=1}\sum_{i\leq j}t^{m(j,i)} B_{j,i}(t,x;D_ x)\partial_ t^{p-j\quad}, \] where \(A,B,C,B_{j,i}\) are partial differential operators with \(C^{\infty}\)- coefficients, homogeneous order q, q-r, 1, i w.r.t. \(D_ x\) respectively and \(p\geq q>r\geq 1\). Our main result is as follows: Suppose \(k>(pr\ell +(p-q)m)/(p-q+r)\), \(m<(\ell +1)(q-r)-p\), \(m(j,i)>j+((m-p\ell)/(p-q+r))(j-i)\). We also assume that there exist \(\xi^ 0\in {\mathbb{R}}^ d\setminus (0)\) and a branch \(B(0,0;\xi^ 0)^{1/p}\) satisfying \[ Re C(0,0;\xi^ 0)+Re B(0,0;\xi^ 0)^{1/p}>0,\quad pRe C(0,0;\xi^ 0)+(q-r)Re B(0,0;\xi^ 0)^{1/p}<0. \] Then there exist \(C^{\infty}\)-functions u, f near the origin s.t. Pu-\(fu=0\), (0,0)\(\in \sup p u\subset \{t\geq 0\}\).