We deal with the Cauchy problem for a strictly hyperbolic second-order operator with nonregular coefficients in the time variable. It is well-known that the problem is well-posed in L 2 in case of Lipschitz continuous coefficients and that the log-Lipschitz continuity is the natural threshold for the well-posedness in Sobolev spaces which, in this case, holds with a loss of derivatives. Here, we prove that any intermediate modulus of continuity between the Lipschitz and the log-Lipschitz one leads to an energy estimate with arbitrary small loss of derivatives. We also provide counterexamples to show that the following classification: modulus of continuity → loss of derivatives * Corresponding author.

We consider the Cauchy problem for strictly hyperbolic m-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is L 2 well-posed in the case of Lipschitz continuous coefficients in time, H s well-posed in the case of Log-Lipschitz continuous coefficients in time (with an, in general, finite loss of derivatives) and Gevrey well-posed in the case of Hölder continuous coefficients in time (with an, in general, infinite loss of derivatives). Here, we use moduli of continuity to describe the regularity of the coefficients with respect to time, weight sequences for the characterization of their regularity with respect to space and weight functions to define the solution spaces. We establish sufficient conditions for the well-posedness of the Cauchy problem, that link the modulus of continuity and the weight sequence of the coefficients to the weight function of the solution space. The well-known results for Lipschitz, Log-Lipschitz and Hölder coefficients are recovered.Mathematics Subject Classification (2010). 35S05, 35L30, 47G30.Keywords. higher order strictly hyperbolic Cauchy problem, modulus of continuity, loss of derivatives, pseudodifferential operators.As to be expected from the know results of the above-mentioned authors, the modulus of continuity µ is linked to the weight function η. In this paper, we describe how µ and η are related to each other and give sufficient conditions for the well-posedness of problem (1.1) which link µ to η and the sequence K p .

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The goal of the paper is to study the loss of regularity for special p-evolution type models with bounded coefficients in the principal part. The obtained loss of regularity is related in an optimal way to some unboundedness conditions for the derivatives of coefficients up to the second-order with respect to t.

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