On a sharp Levi condition in Gevrey classes for some infinitely degenerate hyperbolic equations and its necessity
The authors consider the Cauchy problem in Gevrey class for a second-order linear weakly hyperbolic operator \(L= \partial^2_t- \sum a_{jk}(t) \partial_j\partial_k- \sum a_j(t)\partial_j\) with time dependent coefficients in \(C^1\) class. Let \(\lambda(t)\in C^1([0,T])\) be a weight function satisfying \(\lambda(0)= \lambda'(0)= 0\), \(\lambda'(t)> 0\), \(t> 0\) and for \(t\in [0,T]\) \[ c_0{\lambda(t)\over \Lambda(t)}\leq {\lambda'(t)\over \lambda(t)}\leq c_1{\lambda'(t)\over \Lambda(t)}, \] where \(\Lambda(t)= \int^t_0 \lambda(\tau) d\tau\). They prove that, if the coefficients of \(L\) satisfy for \(t\in [0,T]\) \[ d_0\lambda(t)^2 |\xi|^2\leq \sum a_{jk}(t) \xi_j\xi_k\leq d_1\lambda(t)^2 |\xi|^2, \] \[ |\partial_t \sum a_{jk}(t) \xi_j\xi_k|\leq d_3\lambda(t)^3 \Lambda(t)^{-s/(s- 1)}|\xi|^2, \] \[ \max_{j=1,2,\dots,n} |\partial^k_t a_j(t)|\leq d_1\lambda(t)^{-(k+1)s/(s+ 1)}\quad (k= 0,1), \] then the Cauchy problem for \(L\) is well posed in Gevrey class of index \(s\).
second-order linear weakly hyperbolic operator
scientific article; zbMATH DE number 1833276