On the solvability of a system of partial differential equations (Q1380256)

Let \(A(\xi)\), \(\xi\in \mathbb{R}^n\), be an \(m\times m\) polynomial matrix such that the roots \(\lambda_1(\xi), \dots, \lambda_m(\xi)\) of the characteristic equation \(\text{det} (\lambda E_m- A(\xi))=0\) satisfy the condition \(\forall\xi \in\mathbb{R}^n\) \(\lambda_1 (\xi), \dots, \lambda_r(\xi) \in\Pi^*_\alpha\), \(\lambda_{r+1} (\xi), \dots, \lambda_m (\xi)\in \Pi\setminus \Pi^*_\alpha\). Here \(E_m\) is the unit \(m\times m\) matrix, \(0<\alpha <\pi\), \(\Pi^*_\alpha =\{\lambda \mid\pi/2 <\arg \lambda<\frac 32 \pi- \alpha\}\), \(\Pi\) is the complex plane. The author investigates the existence of a solution of the system \[ \partial u(x,t)/ \partial t=A(D_x) u(x,t)+ f(x,t),\;(x,t)\in \mathbb{R}^n \times\Pi_\alpha, \] where \(u\) and \(f\) belong to certain function spaces.