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

Abstract

The present paper is a continuation of [1]. We continue the numbering of sections, assertions, remarks, and formulas in [1]. THE EXISTENCE OF STRONG SOLUTIONSThe solvability of the boundary value problems (1), (2) in the strong sense for all f ∈F −(m−1) is justified in the following assertion. Theorem 2. Let the assumptions of Theorem1 in [1] and Conditions II and IV be satisfied, and let d j A −1Proof. Since the a priori estimates (28) imply that the ranges R L m (λ m ) of the operators L m (λ m ) are closed inF −(m−1) , it follows that, to justify the solvability of the boundary value problems (1), (2) in the strong sense for all f ∈F −(m−1) , it suffices to show that the ranges R (L m (λ m )) of the operators L m (λ m ) are dense inF −(m−1) . Since the spacesF −(m−1) are reflexive, it suffices to show that v = 0 wheneverfor some function v ∈Ê m−1 . Identities (37) admit (m − 1)-fold integration by parts:By passing to the limit, we generalize the last identities to all u ∈ H such that d m+1 u/dt m+1 , A s (t)d [(s+1)/2] u/dt [(s+1)/2] ∈ H , s= 0, . . . , 2m − 1, d k u/dt k ∈ H m−k , k = 1, . . . , m,

Boundary value problems for complete partial differential equations of variable order

“…. , m. The property lim ε→0 B −1 ε (t)w − w = 0, w ∈ L 2 (R n ), and the commutativity of the operators A(t) and B(t) imply the existence of the limits ( 7) and (8). By virtue of the estimates (4), we have inequalities (12 ) from [11].…”

“…2 m−k,t → 0 by virtue of the estimates(8) and the inequality [which is a consequence of the estimate (9)]|B −1 (t)(dB(t)/dt)B −1 ε (t)v| 2 m−k,t ≤ c 2 m (0, p) B −1 ε (t)A (m−k+δ)/(2m) (t)v 2 < c 2 m (0, p)|v| 2 m−k+1,t for all v ∈ W m−k+1 (t), k = 1, .. . , m − 1, provided that δ = 2m /p(t) < 1 for 0 < ≤ n/(4m) and for i = 1.…”

“…Boundary value problems for parabolic equations in variable-order partial derivatives (in space variables) were earlier considered for higher derivatives of the first order [1][2][3] and odd order [4] in time, for hyperbolic equations with higher derivatives of the second order [3] and even order [5] in time, for partially pseudodifferential parabolic equations with higher derivatives of the first order [6] and higher order [7] in time, and for partially pseudodifferential hyperbolic equations with even-order derivatives in time [8]. In the present paper, we prove the strong well-posedness of new boundary value problems for partially pseudodifferential complete nonclassical equations of variable order with odd-order higher derivatives in time.…”

“…In the present paper, we prove the strong well-posedness of new boundary value problems for partially pseudodifferential complete nonclassical equations of variable order with odd-order higher derivatives in time. The variable differentiation order depended on the time in [1][2][3][4][5] and on space points at which the differentiation is performed in [6,7].…”

Euler-Poisson-Darboux differential-operator equation with variable domains of smooth operators

“…The first nonsingular hyperbolic differentialoperator equations with variable domains were studied in [2,3]. The paper [4] dealt with the existence, uniqueness, continuous dependence, and traces of strong solutions of two mixed problems for the equation of string vibrations with time singularities of a second-order differential operator in another variable. In the present paper, we use the method of energy inequalities to prove the existence and uniqueness of strong solutions of the Cauchy problem for a singular hyperbolic differential-operator equation of the Euler-Poisson-Darboux type with variable domains of variable unbounded operators and the continuous dependence of these solutions on the right-hand side of the equation.…”