νν-loss of derivatives for an evolution type model

This paper is devoted to the study of loss of derivatives for an evolution type model, which allows for non-integer powers of (−Δ)(−Δ). On the one hand, we describe the influence from the orders of principle symbols. On the other hand, we explore the significant impact of νν-related oscillation from the pseudo-differential operators. In the final analysis, to demonstrate the sharpness of our estimates, counter-examples are constructed through the application of Floquet theory and instability arguments.