The Cauchy problem for complete second-order hyperbolic differential equations with variable domains of operator coefficients

Lomovtsev, F. E.

doi:10.1007/bf02754255

Cited by 5 publications

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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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confidence: 79%

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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confidence: 79%

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

Lomovtsev

2005

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“…The aim of the present paper is to develop a method for studying strong well-posedness of hyperbolic differential equations with variable domains of unbounded operator coefficients in the case of nonlocal initial conditions. Our method generalizes and develops the well-known energy inequality method [1][2][3]. In the present paper, we prove the existence, uniqueness, and continuous dependence of strong solutions of complete second-order hyperbolic operator-differential equations with variable domains of unbounded operator coefficients under nonlocal initial conditions.…”

Section: Introductionmentioning

confidence: 86%

“…An abstract Love equation with an unbounded constant operator coefficient under nonlocal initial conditions was studied in [1]. Second-order hyperbolic operator-differential equations with variable domains of unbounded operator coefficients were investigated in [2,3] only under local initial conditions.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal problem for complete second-order hyperbolic operator-differential equations with variable domains

Khatimtsov

Lomovtsev

2011

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We prove the well-posed solvability (in the strong sense) of complete second-order hyperbolic operator-differential equations with variable domains of unbounded operator coefficients under nonlocal initial conditions. We are the first to establish the well-posed solvability of the mixed problem for the complete string vibration equation with nonstationary boundary conditions and nonlocal initial conditions.

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“…The main objective of this research work is to develop one of the powerful methods of functional analysis, namely, the energy inequality method for a certain classes of partial differential equations with nonlocal constraints of convolution type in some functional spaces of Sobolev type. This method, based on the ideas of Petrovski [1], Leray [2], Garding [3], and presented on a method form by Dezin [4], was used to investigate and study different categories of mixed problems related to elliptic, parabolic, and hyperbolic equations [5][6][7][8][9][10][11][12], mixed equations [13][14][15], nonclassical equations [16,17], and operational equations [18,19], with classical conditions of types: Cauchy, Dirichlet, Neumann, and Robinson.…”

Section: Introductionmentioning

confidence: 99%

On an Initial Boundary Value Problem for a Class of Odd Higher Order Pseudohyperbolic Integrodifferential Equations

Mesloub

2014

Journal of Applied Mathematics

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This paper is devoted to the study of the well-posedness of an initial boundary value problem for an odd higher order nonlinear pseudohyperbolic integrodifferential partial differential equation. We associate to the equationnnonlocal conditions andn+1classical conditions. Upon some a priori estimates and density arguments, we first establish the existence and uniqueness of the strongly generalized solution in a class of a certain type of Sobolev spaces for the associated linear mixed problem. On the basis of the obtained results for the linear problem, we apply an iterative process in order to establish the well-posedness of the nonlinear problem.

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The Cauchy problem for complete second-order hyperbolic differential equations with variable domains of operator coefficients

Cited by 5 publications

References 0 publications

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

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

Nonlocal problem for complete second-order hyperbolic operator-differential equations with variable domains

On an Initial Boundary Value Problem for a Class of Odd Higher Order Pseudohyperbolic Integrodifferential Equations

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