Modulus of continuity of the coefficients and (non)quasianalytic solutions in the strictly hyperbolic Cauchy problem

In the strictly hyperbolic Cauchy problem, we investigate the relation between the modulus of continuity in the time variable of the coefficients and the well-posedness in Beurling-Roumieu classes of ultra-differentiable functions and functionals. We find well-posedness in nonquasi analytic classes assuming that the coefficients have modulus of continuity t omega(1/t) such that integral(1)(0) omega(1/t)dt < +infinity. This condition is sharp because, in thecase integral(1 omega)(0)(1/t)dt = +infinity, we provideexamples of Cauchy problems which are well-posed only in quasianalytic classes. (c) 2006 Elsevier Inc. All rights reserved.