A remark on well-posedness for hyperbolic equations with singular coefficients. (Q1411746)

The authors are interested in well-posedness results (Gevrey or \(C^\infty\)) for the Cauchy problem \[ u_{tt}- \sum^n_{k,l=1} a_{kl}(t) u_{x_kx_l}= 0,\;u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x), \] under the following weak assumptions: \(\bullet\) the Cauchy problem is weakly hyperbolic, that is, \[ a(t,\xi):= \sum^n_{k,l= 1} a_{kl}(t)\xi_k \xi_l\geq \lambda_0|\xi|^2\quad \text{with} \quad \lambda_0\geq 0; \] \(\bullet\) \(a(\cdot,\xi)/|\xi|^2\in \bigcap_{\varepsilon> 0} W^{1,2}((0, t_0- \varepsilon)\cup (t_0+ \varepsilon, T))\) for all \(\xi\in \mathbb{R}^n\setminus\{0\}\); \(\bullet\) \(| t_0- \cdot|^p a'(\cdot,\xi)/|\xi|^2\in L^q(0,T)\) for all \(\xi\in\mathbb{R}^n\setminus \{0\}\). On the one hand the coefficients have a minimal regularity away from \(t_0\), on the other hand the behaviour of the derivative near \(t_0\) is described by the last assumption. Depending on \(p.q\) and \(\lambda_0\) (\(\lambda_0= 0\) or \(\lambda> 0\)), the critical Gevrey order is determined (in some cases one has even \(C^\infty\), but with a finite loss of derivatives). The results are changed if one can additionally control the behaviour of \(| t_0-\cdot|^r a(\cdot,\xi)/|\xi|^2\in L^s(0, T)\) for all \(\xi\in\mathbb{R}^n\setminus\{0\}\). The proofs are basing on approximate energies, on the finite speed of the propagation property and on a Paley-Wiener theorem.