ON SECOND ORDER WEAKLY HYPERBOLIC EQUATIONS WITH OSCILLATING COEFFICIENTS

We study well-posedness issues in Gevrey classes for the Cauchy problem for wave equations of the form partial derivative(2)(t)u-a(t)partial derivative(2)(x)u = 0. lathe strictly hyperbolic case a(t) >= c(> 0), Colombini, De Giorgi and Spagnolo have shown that it is sufficient to assume Holder 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) >= 0, Colombini, Jannelli and Spagnolo proved well-posedness in Gevrey classes of order 1 <= s <= s(0) = 1 + (k + alpha)/2. In this paper, we put forward condition vertical bar a'(t)vertical bar <= 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-posalness holds for 1 <= s < qs(0)/{(k + alpha)(1 - p) + q - 1} if q >= 3 - 2p. In particular, this improves on the result (corresponding to the case p = 0) of Colombini, Del Santo and Kinoshita.