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

Abstract

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 ultradifferentiable functions and functionals. We find well-posedness in nonquasianalytic classes assuming that the coefficients have modulus of continuity t ω ( 1 / t ) such that ∫ 0 1 ω ( 1 / t ) d t < + ∞ . This condition is sharp because, in the case ∫ 0 1 ω ( 1 / t ) d t = + ∞ , we provide examples of Cauchy problems which are well-posed only in quasianalytic classes.