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Lomovtsev, F. E.

doi:10.1023/a:1019278028676

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“…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.…”

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

Lomovtsev

2005

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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,

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“…Let the assumptions of Lemma 5 with m > 1 and Lemma 6 be satisfied, let the operators A s (t) satisfy inequalities (5), and let…”

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confidence: 99%

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

Lomovtsev

2005

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Quasuhyperbolic factorized differential-operator equations with variable domains of smooth operator coefficients were considered in [1]. In the case of discontinuous operator coefficients, only hyperbolic second-order differential-operator equations were investigated [2,3]. In the present paper, we prove the strong well-posedness and the smoothness of strong solutions of quasihyperbolic factorized differential-operator equations with variable domains of discontinuous operator coefficients; i.e., one of the main results for hyperbolic second-order differential-operator equations with discontinuous operators in [4] is generalized to the case of quasihyperbolic even-order differentialoperator equations.

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confidence: 99%