On the Cauchy problem for weakly hyperbolic operators (Q2756116)

The author considers in \(\mathbb{R}^2\) the Cauchy problem for an operator \(L+M\) as follows: \((L+M)u=f\), \(\partial^j_t u(0,x)= u_j(x)\), \(j=0, \dots, m-1\), where \(L=\partial^m_t +\sum^{m-j}_{j=0} a_{m-j}\partial^j_x \partial_t^{m-j}\) and \(M=\sum_{j+k \leq m-1}b_{jk}(t) \partial^k_x\). It is called that \(L\) is strongly hyperbolic in the Gevrey class of index \(s\) if for any \(u_j\in \gamma^s\), any \(f\in C^0([0,T]; \gamma^s)\) and any lower order term \(M\) there is a unique solution \(u\in C^m([0,T]; \gamma^s)\) of the above equation. Let \(\tau_1(t) \leq\tau_2(t) \leq\cdots \leq\tau_m(t)\) be the roots of the equation \(L (t, \tau,1)= \tau_m+ \sum^{m-1}_{j=0} a_{m-j}\tau^j=0\) and \(1\leq k_1\leq k_2\leq \cdots\leq k_l\) be integers. Suppose that \(\tau_j(0)=0\) \((j=1,\dots,m)\) and \(L (t,\tau, 1)=L_1(t,\tau) L_2(t,\tau)\cdots L_{l-1} (t,\tau)L_l' (t,\tau)\), where the roots \(\tau_p(t)\), \(\tau_q(t)\) in the term \(L_j\) satisfy \(|\tau_p(t)-\tau_q (t)|\geq t^{k_j}\) for \(p\neq q\), and that \(r_j=\) order of \(L_j\), \(r_l' =\) order of \(L_l'\) satisfy \(r_1k_1+\cdots +r_{l-1}k_{l-1} <r_l'\). Then \(L\) is strongly hyperbolic in \(\gamma^s\), if \(1<s<(1- {k_l+1\over r_1 k_1+ \cdots +r_l'k_l})^{-1}\).