On energy inequalities and regularity of solutions to weakly hyperbolic Cauchy problems (Q789607)

The authors consider well-posedness of the Cauchy problem: \[ u_{tt}=\sum^{n}_{i,j=1}a_{ij}(t)u_{x_ ix_ j},\quad u(x,0)=\phi(x),\quad u_ t(x,0)=\psi(x) \] in \({\mathbb{R}}^ n\times n[0,T]\) under the weak hyperbolicity condition: \(\sum a_{ij}(t)\xi_ i\xi_ j\geq 0\) for \(\xi \in {\mathbb{R}}^ n\). They obtained two theorems. The first theorem asserts the problem is well posed in a Gevrey class \({\mathcal G}^ s\) with \(1\leq s<1+(k+\alpha)/2\) when each \(a_{ij}\) is a function \(\in C^{k,\alpha}([0,T])\), that is, k-times differentiable in [0,T] with the k-th derivative being \(\alpha\)-Hölder continuous for \(k\geq 0\) and \(0<\alpha \leq 1\). The second theorem affirms that the first theorem is the best result of well-posedness in Gevrey class, claiming: there exist \(a_{ij}\in C^{k,\alpha}([0,T])\) and \(\phi\),\(\psi\in {\mathcal G}^ s\) such that for any \(\tau>0\) the problem has no distribution value solution in \([0,\tau]\) for any k,\(\alpha\) and s satisfying \(s>1+(k+\alpha)/2\). As to the proofs, the plan is given for the first theorem in the one space dimensional case, but not to the second theorem.