On second order weakly hyperbolic equations with oscillating coefficients

Abstract

We study well-posedness issues in Gevrey classes for the Cauchy problem for wave equations of the form $\partial_t^2 u -a(t) \partial_x^2 u=0$. In the strictly hyperbolic case $a (t)\geq c(> 0)$, Colombini, De Giorgi and Spagnolo have shown that it is sufficient to assume H$\ddot { \rm o}$lder regularity of the coefficients in order to prove Gevrey well-posedness. Recently, assumptions bearing on the oscillations of the coefficient have been imposed in the literature in order to guarantee well-posedness. In the weakly hyperbolic case $a(t) \geq 0$, Colombini, Jannelli and Spagnolo proved well-posedness in Gevrey classes of order $1\leq s<s_0=1+(k+\alpha)/2$. In this paper, we put forward condition $ |a'(t)| \leq Ca(t)^p/t^q $ that bears both on the oscillations and the degree of degeneracy of the coefficient. we show that under such a condition, Gevrey well-posedness holds for $1\leq s<qs_0/\{ (k+\alpha) (1-p) +q-1\}$ if $q \geq 3-2p$. In particular, this improves on the result (corresponding to the case $p=0$) of Colombini, Del Santo and Kinoshita.