On stability of solutions to the Cauchy problem for hyperbolic systems in two independent variables (Q1288068)

The authors consider the Cauchy problem for a hyperbolic system with multiple characteristics, \[ \frac{\partial u}{\partial t}=A\frac{\partial u}{\partial x}+Bu, \quad u|_{t=t_0}=h(x). \] Here \(u\:\mathbb R^2\to \mathbb C^N\), \(A=\sum_{k=1}^na_kP_k\), \(P_k=\text{diag}\{0,\dots ,0,1,\dots ,1\), \(0,\dots ,0\}\), \(\sum P_k=I\), \(a_k:\mathbb R^2\to \mathbb R\), \(a_1<\dots <a_n\), \(A\) and \(B\) are smooth matrices, \(h(x)\) is a given matrix in the Hilbert space of functions \(\mathbb R\to\mathbb C^N\) with inner product \(\langle u,v\rangle =\int v^*u dx\). The main result of the article consists in some sufficient conditions for validity of the estimate \[ \| v(t)\|_H\leqslant\mu e^{-\nu(t-t_0)}\| v(t_0)\|_H \quad (\mu>0,\;\nu>0,\;t>t_0), \] where \(v=Tu\), \(T\) is an invertible bounded linear operator from \(H\) into \(H\). The method of the authors is based on reduction of a hyperbolic system to an ordinary differential equation and subsequent application of the Lyapunov function method.