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

Abstract

We consider the Cauchy problem \[\left\{ \begin{array}{lll} P\bigl(t,x,D^{m-1}u,D_{t},D_{x}\bigr)u(t,x)=f(t,x,D^{m-1}u) \\ \partial^j_{t} u(0,x)=u_{j}(x),\ \ \ j=0,...,m-1, \end{array} \right.\] in $[0,T]\times{\mathbb{R}}^n$ for a quasilinear weakly hyperbolic operator \[ P\bigl(t,x,D^{m-1}u,D_{t},D_{x}\bigr)=D_{t}^{m}+\sum_{j=0}^{m-1} \sum_{|\alpha|={m-j}}a_{\alpha}^{(j)}(t,x,D^{m-1}u)D_x^{\alpha}D_t^j\] with coefficients $a_{\alpha}^{(j)}$ having the first time derivative with singular behavior of the type $t^{-q}$, $q>1$, as $t\to 0$. We show that for $t\leq T_0^\ast$, $T_0^\ast$ sufficiently small, given Cauchy data in a Gevrey class $G^\sigma$ there exists a unique solution $u\in C^{m-1}([0,T_0^\ast];G^\sigma)$ provided that $\sigma < \frac{qr}{qr-1}$ where $r$ denotes the largest multiplicity of the characteristic roots.