Homogeneous weakly hyperbolic equations with time dependent analytic coefficients (Q550020)

The authors consider the Cauchy problem for the following homogeneous hyperbolic equation of higher order \(N\): \[ \begin{aligned} L(t,\partial_t, \partial_x)u= 0\quad &\text{for }(t,x)\in (\delta, T+\delta)\times \mathbb{R}^n,\\ \partial^j_t u(t_0, x)= u_j(x)\quad &\text{for }x\in\mathbb{R}^n,\;j= 0,\dots, N-1,\;t_0\in (\delta, T+\delta),\end{aligned} \] where \[ L(t,\partial_t, \partial_x):= \partial^N_t- \sum_{1\leq j\leq N,|\alpha|= j} a_{\alpha,j}(t) \partial^{N-j}_t \partial^\alpha_x. \] Although we have no terms of lower order and only time-dependent analytic coefficients the problem of \(C^\infty\) well-posedness is quite delicate. A lot of authors describe necessary or sufficient conditions for the \(C^\infty\) well-posedness by the behavior of the characteristic roots of the symbol. The interesting goal of the authors is to propose only conditions with the aid of the coefficients \(a_{\alpha,j}\) of the operator to guarantee \(C^\infty\) well-posedness of the Cauchy problem. They apply the following procedure: {\parindent=5mm \begin{itemize}\item[1.]After partial Fourier transformation and introduction of a suitable energy they get the system \[ d_tV(t,\xi)= i|\xi|A(t,\xi|A(t,\xi) V(t,\xi),\;V(0,\xi)= V_0(\xi), \] where \(A\) is \(0\)-homogeneous in \(\xi\). \item[2.]The authors use the knowledge of a real symmetrizer \(Q\), whose entries \(Q_{jk}\) are fixed polynomials in the entries of the Sylvester matrix \(A(t,\xi)\). \item[3.] The authors consider the set \(\{Q_j,\, j= 1,\dots, N\}\) of principal \(j\times j\) minors by removing the first \(N-j\) rows and the first \(N-j\) columns of \(Q\). The determinant of \(Q_j\) is denoted by \(\Delta_f\). \item[4.]It is possible to describe the type of hyperbolicity by this set. The operator \(L\) is strictly hyperbolic if and only if \(\Delta_j> 0\); \(j= 1,\dots,N\). The operator \(L\) is weakly hyperbolic with exact \(r\) distinct roots if \(\Delta_j> 0\), \(j= 1,\dots, r\), \(\Delta_j= 0\) for \(j>r\). \item[5.]If \(\Delta_r(t,\xi)\) is not identically zero, the authors introduce \(\widetilde\Delta_r(t,\xi):= \Delta(t,\xi)+ {\Delta^{\prime 2}_r(t,\xi)\over \Delta_r(t,\xi)}\). \item[6.]Finally, the authors use the check function \(\psi_j\) of \(Q_j\) which is defined in the following way \[ \psi_j:={1\over 2} \text{trace}(Q_j'(Q^{\text{co}}_j)'). \] \end{itemize}} Then this procedure allows to formulate the following condition (besides some standard assumptions): For any \(\xi\) with \(|\xi|=1\) let \(r= r(\xi)\) be the largest integer with \(\Delta_r(t,\xi)\) is not identically zero in \((\delta, T+\delta)\). Then the authors assume \(|\psi_r(t,\xi)|\leq C\widetilde\Delta_r(t,\xi)\) for all \(t\in[a, b]\). If \(r< N\), then \(\psi_{r+1}(t,\xi)\equiv 0\). This condition implies \(C^\infty\) well-posedness of the Cauchy problem for \(t\in[a, b]\). In the one-dimensional case this condition is also necessary.