Cauchy problems for quasi-hyperbolic factorized even-order differential equations with smooth operator coefficients having variable domains

Abstract

Cauchy problems for quasi-hyperbolic factorized operator-differential equations of higher even orders with constant domains were considered in [1]. The Cauchy problem for hyperbolic operatordifferential equations with variable domains was investigated in [2, 3] for second-order equations. The present paper deals with the proof of the well-posed solvability in the strong sense of Cauchy problems for some quasi-hyperbolic factorized operator-differential equations of higher even orders with unbounded operator coefficients whose domains vary; mixed problems for some hyperbolic partial differential equations with coefficients in the boundary conditions smoothly depending on time can be reduced to such problems. For the proof, we use modifications and generalizations of the functional method of energy inequalities in [1]. Unlike [1], in the present paper, the derivation of a priori estimates with the use of abstract smoothing operators is generalized to the case of variable domains of variable unbounded operator coefficients; the proof of the solvability by induction, decomposition of operators into operator factors, and the use of the Lemmas 5 and 6 below is a new technique; and a formula for their strong solutions is derived for the first time [see formula (25) below]. This formula generalizes a similar formula for smooth (classical) solutions and shows that, by analogy with smooth solutions, strong solutions of these Cauchy problems can be found in a recursive way on the basis of operator factors. In addition, unlike [1], in the present paper, we do not repeat any steps of the proofs in [4]. In conclusion, we consider an example of new well-posed mixed problems for hyperbolic factorized partial differential equations of even order with coefficients in the boundary conditions depending on t and smooth with respect to t.