The Cauchy problem for quasilinear hyperbolic equations with non-absolutely continuous coefficients in the time variable (Q2504068)

The author deals with a quasilinear hyperbolic Cauchy problem \[ P(t, x, D^{m-1} u, d_t, d_x) u(t, x)= f(t, x, D^{m-1}u),\quad (t,x)\in [0,T]\times\mathbb{R}^d, \] \[ J^j_t u(0,x)= u_j(x),\quad j= 0,\dots, m-1, \] where \[ P= D^m_t+ \sum^{m-1}_{j=0} \sum_{|\alpha|= m-j} a^{(j)}_\alpha(t, x,D^{m- 1}u)\,D^\alpha_x D^j_t \] with \(D^{m-1} u= (J^\alpha_{t,x} u)|_{|\alpha|\leq m-1}\) is a vector in \(\mathbb{R}^\ell\), \(\ell:= \#\{\alpha\mid|\alpha|\leq m-1\}\) and \(a^{(j)}_\alpha\), \(f\) are supposed to have a Gevrey behaviour with respect to the variables \(x\) and \(y= D^{m-1} u\). The author proves the existence and uniqueness of a local in time solution in Gevrey classes of index \(1<\sigma<\sigma_0= {qr\over qr-1}\) with a class of derivatives.