Some results on the well-posedness for systems with time dependent coefficients (Q735081)

The authors consider the Cauchy problem for weakly hyperbolic systems whose coefficients depend only on the time variable of dimension 2 and 3: \[ \begin{aligned} LU\equiv\partial_tU-\sum_{j=1}^n A_j(t)\partial_{x_j}U-B(t)U&=f(t,x),\quad (t,x)\in[0,T]\times{\mathbb R}^n,\\ U(0,x)&=U_0(x),\quad x\in{\mathbb R}^n,\end{aligned} \] where: \(A(x,\xi)=|\xi|^{-1}\sum_{j=1}^nA_j(t)\xi_j\) is real valued and (weakly) hyperbolic, that is its eigenvalues are real (not necessarily distinct) for any \(\xi\in{\mathbb R}^n\setminus\{0\}\). The coefficients of \(B\) may be complex valued. The authors obtain some sufficient conditions in order the Cauchy problem to be well-posed in \(C^\infty\) and in Gevrey spaces. In the proof of the main results they used the Fourier transforms, some estimates, and the fact from theory of hyperbolic systems.