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. STATEMENT OF THE PROBLEMSIn a Hilbert space H with inner product (· , ·) and norm |·|, we consider boundary value problemson a bounded interval ]0, T [ , where u and f are functions of the variable t ranging in H and λ m ≥ 1 is a numerical parameter. The linear unbounded closed operators A s (t) in H with t-dependent domains D (A s (t)), s = 0, . . . , 2m − 1, are subjected to the following conditions. I. For all t ∈ [0, T ], the operators A 0 (t) are self-adjoint in H and satisfy the inequality (A 0 (t)u, u) ≥ c 0 (t)|u| 2 for all u ∈ D (A 0 (t)), c 0 (t) > 0, and their inverses are A −1 0 (t) ∈ B ([0, T ], L (H)) (where B ([0, T ], L (H)) is the set of linear operators in L (H), which are bounded with respect to t ∈ [0, T ] and in the norm) and have the strong t-derivative [6, p. 22] dA −1To state constraints for A s (t), s > 0, we introduce appropriate spaces. By [6], for A 0 (t) with each t ∈ [0, T ], we introduce the fractional powers A γ 0 (t), |γ| ≤ 1, with domains D (A γ 0 (t)). If we equip D A α/(2m) 0 (t) with the Hermitian norms |υ| α,t = |A α/(2m) 0 (t)υ|, then we obtain Hilbert spaces W α (t), |α| ≤ 2m, W 0 (t) = H.II. For each t ∈ [0, T ], the operators dA −1 0 (t)/dt have the strong derivatives d j A −1 0 (t)/dt j ∈ B ([0, T ], L (H)), 2 ≤ j ≤ m + 1,

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

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The present paper is a continuation of [1]. We continue the numbering of sections, assertions, remarks, and formulas in [1].

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“…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 whenever …”

Boundary value problems for complete partial differential equations of variable order

“…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].…”