A generalization of Lions' theory to first-order evolution differential equations with smooth operator coefficients. I (Q2371640)

Part II, cf. Differ. Equ. 42, No. 6, 874--881 (2006); translation from Differ. Uravn. 42, No. 6, 820--826 (2006; Zbl 1130.35130). In the present paper, we generalize the well-known Theorem 1.1 (p. 129) in [\textit{J. L. Lions}, Équations différentielles opérationnelles et problèmes aux limites. Berlin etc.: Springer-Verlag (1961; Zbl 0098.31101)] on the existence and uniqueness of the solution of the Cauchy problem for a first-order operator-differential equation with variable domains of symmetric leading parts to the case of nonsymmetric leading parts of smooth operator coefficients. We construct a new class of even-order partial differential operators with symmetric leading parts and their variable domains which satisfy the assumption of the generalized theorem but do not necessarily satisfy Lions' assumptions. In the second part, we construct the corresponding odd-order differential operators with nonsymmetric leading parts and their variable domains.