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We consider the Cauchy problem for strictly hyperbolic $m$-th order partialdifferential equations with coefficients low-regular in time and smooth inspace. It is well-known that the problem is $L^2$ well-posed in the case ofLipschitz continuous coefficients in time, $H^s$ well-posed in the case ofLog-Lipschitz continuous coefficients in time (with an, in general, finite lossof derivatives) and Gevrey well-posed in the case of H\"older continuouscoefficients in time (with an, in general, infinite loss of derivatives). Here,we use moduli of continuity to describe the regularity of the coefficients withrespect to time, weight sequences for the characterization of their regularitywith respect to space and weight functions to define the solution spaces. Weestablish sufficient conditions for the well-posedness of the Cauchy problem,that link the modulus of continuity and the weight sequence of the coefficientsto the weight function of the solution space. The well-known results forLipschitz, Log-Lipschitz and H\"older coefficients are recovered.

Strictly Hyperbolic Equations with Coefficients Low-Regular in Time and Smooth in Space