Criteria for the Well-Posedness of the Cauchy Problem for Differential Operator Equations of Arbitrary Order

Abstract

In Banach spaces, we investigate the differential equation \(\mathop \sum \nolimits_{j = 0}^n \;A_j u^{(j)} (t) = 0\) with closed linear operators A _j (generally speaking, the operator coefficient A _n of the higher derivative is degenerate). We obtain well-posedness conditions that characterize the continuous dependence of solutions and their derivatives on initial data. Abstract results are applied to partial differential equations.