This book develops a framework of energy assessments for high-order evolutionary operators. This framework provides a unified presentation of mixed problems for strictly hyperbolic and parabolic differential equations with variable coefficients, as defined by Petrovsky. Additionally, the method allows for the exploration of a new, unconventional class of q-hyperbolic equations alongside these classical equations. In a supplementary section written by A. R. Shirikyan and the first author, hyperbolic equations are examined across the entire time axis. The study delves into the solvability within spaces of bounded, periodic, and almost periodic functions in time. It also investigates the properties of asymptotic stability and exponential dichotomy. This book is aimed at specialists in partial differential equations and mathematical physics, making it accessible to mathematicians, graduate students, and advanced undergraduates.