Boundary Value Problems for Complete Quasi-Hyperbolic Differential Equations with Variable Domains of Smooth Operator Coefficients: I

Abstract

Complete quasi-hyperbolic operator-differential equations of even order with constant domains were considered in [1, 2]. Quasi-hyperbolic operator-differential equations of even order with variable domains in the case of a two-term leading part were analyzed in [3]. Complete hyperbolic operator-differential equations of the second order with variable domains were investigated in [4, 5]. In the present paper, we generalize and improve the results of all above-mentioned papers and consider complete quasi-hyperbolic operator-differential equations of even order with variable domains. In applications, such equations include hyperbolic equations such that the coefficients in the equations and in the boundary conditions [3] smoothly depend on time, singular hyperbolic equations [4], and “hyperbolic” equations of higher-order in the space variables, represented in the second part of the present paper.