The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term

In this paper, the initial-boundary value problems for the two-dimensional nonlinear Schrödinger equation with a special gradient term with purely imaginary coefficients in the nonlinear part, when the coefficients of the equation are measurable bounded functions, are considered. The existence and u...

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Дата:	2018
Автори:	Yagub, G., Ibrahimov, N.S., Zengin, M.
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Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2018
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Цитувати:	The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term / G. Yagub, N.S. Ibrahimov, M. Zengin // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 214-232. — Бібліогр.: 19 назв. — англ.

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spelling	irk-123456789-1458692019-02-02T01:23:17Z The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term Yagub, G. Ibrahimov, N.S. Zengin, M. In this paper, the initial-boundary value problems for the two-dimensional nonlinear Schrödinger equation with a special gradient term with purely imaginary coefficients in the nonlinear part, when the coefficients of the equation are measurable bounded functions, are considered. The existence and uniqueness of solutions of the first and second initial-boundary value problems is proved almost everywhere. У статтi розглядаються початково-крайовi задачi для двовимiрного нелiнiйного рiвняння Шредiнгера iз спецiальним градiєнтним членом з чисто уявними коефiцiєнтами в нелiнiйнiй частинi, коли коефiцiєнти рiвняння є вимiрними обмеженими функцiями. Доведено iснування i єднiсть розв язкiв першо i друго початково-крайово задачi майже скрiзь. 2018 Article The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term / G. Yagub, N.S. Ibrahimov, M. Zengin // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 214-232. — Бібліогр.: 19 назв. — англ. 1812-9471 DOI: https://doi.org/10.15407/mag14.02.214 Mathematics Subject Classification 2010: 35D, 35M, 35Q http://dspace.nbuv.gov.ua/handle/123456789/145869 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this paper, the initial-boundary value problems for the two-dimensional nonlinear Schrödinger equation with a special gradient term with purely imaginary coefficients in the nonlinear part, when the coefficients of the equation are measurable bounded functions, are considered. The existence and uniqueness of solutions of the first and second initial-boundary value problems is proved almost everywhere.
format	Article
author	Yagub, G. Ibrahimov, N.S. Zengin, M.
spellingShingle	Yagub, G. Ibrahimov, N.S. Zengin, M. The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term Журнал математической физики, анализа, геометрии
author_facet	Yagub, G. Ibrahimov, N.S. Zengin, M.
author_sort	Yagub, G.
title	The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term
title_short	The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term
title_full	The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term
title_fullStr	The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term
title_full_unstemmed	The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term
title_sort	solvability of the initial-boundary value problems for a nonlinear schrödinger equation with a special gradient term
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2018
url	http://dspace.nbuv.gov.ua/handle/123456789/145869
citation_txt	The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term / G. Yagub, N.S. Ibrahimov, M. Zengin // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 214-232. — Бібліогр.: 19 назв. — англ.
series	Журнал математической физики, анализа, геометрии
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last_indexed	2025-07-10T22:46:09Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2018, Vol. 14, No. 2, pp. 214–232 doi: https://doi.org/10.15407/mag14.02.214 The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term G. Yagub, N.S. Ibrahimov, and M. Zengin In this paper, the initial-boundary value problems for the two-dimen- sional nonlinear Schrödinger equation with a special gradient term with purely imaginary coefficients in the nonlinear part, when the coefficients of the equation are measurable bounded functions, are considered. The exi- stence and uniqueness of solutions of the first and second initial-boundary value problems is proved almost everywhere. Key words: Schrödinger equation, special gradient term, existence and uniqueness, first and second initial-boundary value problems. Mathematical Subject Classification 2010: 35D, 35M, 35Q. 1. Introduction In this paper we study the correct formulation of the initial-boundary value problems for the nonlinear Schrödinger equation with a special gradient term. As it is well known, the Schrödinger equation with a special gradient term and the initial-boundary value problems for this equation appear in quantum mechanics, nuclear physics, nonlinear optics and other fields of modern physics and engi- neering [3, 15, 19]. Especially in quantum mechanics and nonlinear optics in the study of the motion of charged particles in a nonhomogeneous environment, the Schrödinger equation has a special gradient term. Therefore, the study of bound- ary value problems of this type of Schrödinger equations is of interest both for theoretical and practical problems. It should be noted that the initial-boundary value problems for the linear and nonlinear Schrödinger equations in various for- mulations were previously studied in detail in [1, 2, 4–8, 16, 18]. However, even for the linear Schrödinger equation with a special gradient term, initial-boundary value problems are poorly investigated [10,17]. In this paper, we study the ques- tions of the existence and uniqueness of solutions of boundary value problems for the linear one-dimensional and two-dimensional Schrödinger equations with a special gradient term, where the coefficients are square integrable functions. Note that the initial-boundary value problem for the nonlinear Schrödinger equation with a special gradient term has not been studied yet. Therefore, the study of the c© G. Yagub, N.S. Ibrahimov, and M. Zengin, 2018 https://doi.org/10.15407/mag14.02.214 . . . Initial-Boundary Value Problems for a Nonlinear Schrödinger. . . 215 existence and uniqueness of the initial-boundary value problems for the nonlinear Schrödinger equation with a special gradient term is of scientific and practical interest. 2. The existence and uniqueness of a solution of the first initial-boundary value problem In this paper, we will first examine the first initial-boundary value problem for the nonlinear Schrödinger equation with a special gradient term with purely imaginary coefficients in the nonlinear part in the case when the coefficients of the equation are bounded measurable functions. Let D be a bounded convex domain in R2, with the boundary Γ, that is assumed to be smooth enough; x = (x1, x2) be an arbitrary point of the domain D; T > 0 be a given number; 0 ≤ t ≤ T ; Ωt = D×(0, t); Ω = ΩT ; S = Γ×(0, T ) be the lateral surface of Ω; Ck ([0, T ] , B) be a Banach space of the functions, k-times differentiable in the interval [0, T ] with values in the Banach space B; Lp (D) be a Lebesque space of the functions summerable over the module with an order p ≥ 1 of functions for which the p-th power of the absolute value is summable; L2 (0, T ;B) be a Banach space of the functions defined and square-summable on the interval [0, T ] with values from the Banach space B; L∞ (0, T ;B) be a Banach space of measurable bounded functions on (0, T ) with values from the Banach space B. The Sobolev spaces W k p (0, l), W k,m p (Ω) , p ≥ 1, k ≥ 0, m ≥ 0 are defined as in [11,13]. Consider the initial-boundary problem of determination of a function ψ = ψ (x, t) in the domain Ω from the conditions i ∂ψ ∂t + a0∆ψ + ia1 (x)∇ψ − a (x)ψ + v (x)ψ + ia2 \|ψ\|2 ψ = f (x, t) , (x, t) ∈ Ω, (2.1) ψ (x, 0) = ϕ (x) , x ∈ D, ψ \|S = 0 , (2.2) where i = √ −1; a0 > 0, a2 > 0 are given numbers; ∆ = ∂2 ∂x21 + ∂2 ∂x22 is the Laplace operator; ∇ = ( ∂ ∂x1 , ∂ ∂x2 ) ; a (x) , v (x) are measurable bounded functions satisfying the conditions 0 ≤ a (x) ≤ µ0, x ∈ D, µ0 = const > 0 ; (2.3) \|v (x)\| ≤ b0, x ∈ D, b0 = const > 0; (2.4) a1 (x) = (a11 (x) , a12 (x)) is a given vector-function with the components satisfy- ing the conditions \|a1j (x)\| ≤ µ1, ∣∣∣∣∂a1j (x) ∂xk ∣∣∣∣ ≤ µ2, x ∈ D, j, k = 1, 2, µ1, µ2 = const > 0; (2.5) φ (x) , f (x, t) are complex valued functions satisfying the conditions ϕ ∈W 2 2 ◦ (D) , f ∈W 0,1 2 (Ω) . (2.6) 216 G. Yagub, N.S. Ibrahimov, and M. Zengin It is clear that the problem of determination of ψ = ψ (x, t) from conditions (2.1), (2.2) is an initial-boundary problem for the two-dimensional nonlinear Schrödinger equation of the form (2.1). Definition 2.1. The function ψ = ψ (x, t) from the space B0 ≡ C0 ( [0, T ] ,W 2 2 ◦ (D) )⋂ C1 ([0, T ] , L2 (D)) is called a generalized solution to (2.1), (2.2) if it satisfies equation (2.1) for almost all x ∈ D and for any t ∈ [0, T ], and initial and boundary conditions (2.2) for almost all x ∈ D and almost all (ξ, t) ∈ S, correspondingly. Theorem 2.2. Let the functions a (x), v (x), a1 (x), ϕ (x), f (x, t) satisfy con- ditions (2.3)–(2.6). Then the initial-boundary problem (2.1), (2.2) has a unique solution in the space B0, and for this solution the estimate ‖ψ (·, t)‖ W 2 2 ◦ (D) + ∥∥∥∥∂ψ (·, t) ∂t ∥∥∥∥ L2(D) (2.7) ≤ c0 ( ‖φ‖ W 2 2 ◦ (D) + ‖f‖ W 0,1 2 (Ω) + ‖φ‖3 W 2 2 ◦ (D) ) , t ∈ [0, T ] , (2.8) is valid, where the constant c0 > 0 does not depend on φ, f and t. Proof. We choose the fundamental in W 2 2 ◦ (D) and orthonormal in L2 (D) system of functions uk = uk (x) , k = 1, 2, . . ., for example, the system of eigen- functions of the following spectral problem: LX (x) = λX (z) , x ∈ D, X \|Γ = 0 (2.9) at λ = λk, k = 1, 2, . . ., where the operator L is defined by the formula L = −a0∆ + a (x) (2.10) with Dirichlet boundary conditions. Notice that (2.9) is a spectral problem for the two-dimensional equation of elliptic type studied in [15]. Therefore, by the help of the result obtained in this paper, we can state that the spectral problem (2.10) has nontrivial solutions X = uk (x), k = 1, 2, . . ., at λ = λk, k = 1, 2, . . ., forming a spectra of the problem, and these solutions form a basis in the spaces W 1 2 ◦ (D) ,W 2 2 ◦ (D) . The conditions of orthonormality in L2 (D) and orthogonality in W 1 2 ◦ (D) , W (D) are given in the form (uk, um)L2(D) = ∫ D uk (x)um (x) dx = δmk , (2.11) where δmk are Kronecker symbols δmk = { 1 k = m, 0, k 6= m , k,m = 1, 2, . . . (2.12) . . . Initial-Boundary Value Problems for a Nonlinear Schrödinger. . . 217 It is clear that the functions uk (x), k = 1, 2, . . . are orthogonal in the following sense: [uk, um] = (uk, um) W 1 2 ◦ (D) = (Luk, um)L2(D) = e ∫ D a0 2∑ j ∂uk ∂xj ∂um ∂xj + a (x)ukum ) dx = λkδ m k , k,m = 1, 2, . . . ; (2.13) {uk, um} = (Luk, Lum)L2(D) = (uk, um) W 2 2 ◦ (D) = λkδ m k , k,m = 1, 2, . . . . (2.14) Due to the assumption a (x) ≥ 0, all eigenvalues λ = λk , k = 1, 2, . . . are real, positive and numbered in the increasing order 0 ≤ λ1 ≤ λ2 ≤ λ3 ≤ · · · ≤ λk ≤ · · · , λk →∞ as k →∞. (2.15) We additionally assume that ‖uk‖ W 2 2 ◦ (D) ≤ d̃k < +∞, k = 1, 2, . . . , (2.16) where d̃k, k = 1, 2, . . ., are positive constants. By Galerkin’s method, we seek the approximate solution in the form ψN (x, t) = N∑ k=1 cNk (t)uk (x) , (2.17) where cNk (t) = ( ψN (·, t) , uk ) L2(∂) , k = 1, 2, . . . , N , are defined by the conditions i d dt ( ψN (·, t) , uk ) L2(D) − ( LψN (·, t) , uk ) L2(D) + i ( a1 (·)∇ψN (·, t) , uk ) L2(D) + ( v (·)ψN (·, t) , uk ) L2(D) + i ( a2 ∣∣ψN ∣∣2 ψN , uk) = fk (t) , k = 1, 2, . . . , N, t ∈ [0, T ] , (2.18) cNk (0) = ( ψN (·, 0) , uk ) L2(D) = ϕk, k = 1, 2, . . . , N. (2.19) Here fk (t) = (f (·, t) , uk) L2(D) , φk = (φ, uk) L2(D) , k = 1, 2, . . . , N . System (2.18) consists of the system of N nonlinear ordinary differential equations. It fol- lows from assumptions (2.3)–(2.6) and the properties of uk (x), k = 1, 2, . . . that the second, third, fourth and fifth terms in the left- and the right-hand sides are continuous on each set { t ∈ [0, T ] , ∣∣cNk ∣∣ ≤ const } of the functions t, cNk , k = 1, 2, . . . , N . Therefore, for the existence of a solution to the Cauchy problem, it is sufficient to show that the solutions are bounded uniformly with respect to t ∈ [0, T ] for any T > 0 (see [9, 12–14]). To establish the boundedness, we have to prove the following. 218 G. Yagub, N.S. Ibrahimov, and M. Zengin Lemma 2.3. For the solutions of problem (2.18), (2.19), the estimate N∑ k=1 ∣∣cNk (t) ∣∣2 + N∑ k=1 ∣∣∣∣dcNk (t) dt ∣∣∣∣2 ≤ ∥∥ψN (·, t) ∥∥ W 2 2 ◦ (D) + ∥∥∥∥∂ψN (·, t) ∂t ∥∥∥∥2 L2(D) ≤ c1 ( ‖φ‖2 W 2 2 ◦ (D) + ‖f‖2 W 0,1 2 (Ω) + ‖φ‖6 W 1 2 ◦ (D) ) , t ∈ [0, T ] , N = 1, 2, . . . . (2.20) is valid. Proof. Multiplying each k-th equation from (2.18) by c̄Nk (t), summing the obtained equalities over k from 1 to N , integrating over t from zero to t ≤ T and then using the formula of integration by parts and the condition uk\|Γ = 0, k = 1, 2, . . ., we get∫ Ωt ( i ∂ψN ∂t ψ̄N − a0 (x) ∣∣∇ψN ∣∣2 + ia1 (x)∇ψN ψ̄N − a (x) ∣∣ψN ∣∣2 + v (x) ∣∣ψN ∣∣2 + ia2 ∣∣ψN ∣∣4) dxdτ = 2i ∫ Ωt Im ( fψ̄N ) dx dτ, t ∈ [0, T ] . Subtracting from this equality its complex conjugate, we get the validity of the equality i ∫ Ωt ( ∂ψN ∂t ψ̄N + ∂ψ̄N ∂t ψN ) dx dτ + i ∫ Ωt ( a1 (x)∇ψN ψ̄N + a1 (x)∇ψ̄N ψN ) dx dτ + 2ia2 ∫ Ωt ∣∣ψN ∣∣4 dxdτ = 2i ∫ Ωt Im ( fψ̄N ) dx dτ, t ∈ [0, T ] . Using differentiability of the functions a1j (x) , j = 1, 2, the last one may be written as∫ Ωt ∂ ∂t ∣∣ψN ∣∣2 dx dτ + ∫ Ωt 2∑ j=1 ∂ ∂xj ( a1j (x) ∣∣ψN ∣∣2) dx dτ + 2a2 ∫ Ωt ∣∣ψN ∣∣4 dx dτ = ∫ Ωt 2∑ j=1 ∂a1j (x) ∂xj ∣∣ψN ∣∣2 dx dτ + 2 ∫ Ωt Im ( fψ̄N ) dx dτ, t ∈ [0, T ] . (2.21) Considering that the functions uk = uk (x) , k = 1, 2, . . . satisfy the homogeneous boundary conditions uk\|Γ = 0, k = 1, 2, . . . , from decomposition (2.17) we have ψN (x, t) \|Γ = 0, t ∈ (0, T ) , N = 1, 2, . . . . (2.22) . . . Initial-Boundary Value Problems for a Nonlinear Schrödinger. . . 219 Taking into account the second terms and the conditions imposed on the coeffi- cients of the equation from equality (2.21), one can easily obtain the validity of the inequality∥∥ψN (·, t) ∥∥2 L2(D) + 2a2 ∫ Ωt ∣∣ψN ∣∣4 dx dτ ≤ ∥∥ψN (·, 0) ∥∥2 L2(D) + ‖f‖2L2(Ω) + (2µ2 + 1) ∫ t 0 ∥∥ψN (·, τ) ∥∥2 L2(D) dτ (2.23) for all t ∈ [0, T ] . Using formula (2.17), we can write the relation ∥∥ψN (·, 0) ∥∥2 L2(D) = N∑ k=1 ∣∣cNk (0) ∣∣2 ≤ ∞∑ k=1 \|ϕk\|2 = ‖ϕ‖2L2(D) . (2.24) With the help of this relation, from (2.23), we get ∥∥ψN (·, t) ∥∥2 L2(D) + 2a2 ∫ Ωt ∣∣ψN ∣∣4 dx dτ ≤ ‖φ‖2L2(D) + ‖f‖2L2(Ω) + (2µ2 + 1) ∫ t 0 ∥∥ψN (·, τ) ∥∥2 L2(D) dτ, t ∈ [0, T ] . Using this inequality and Gronwall’s lemma, it is not difficult to get the estimate∥∥ψN (·, t) ∥∥2 L2(D) + 2a2 ∫ Ωz ∣∣ψN ∣∣4 dx dτ ≤ c2 ( ‖φ‖2L2(D) + ‖f‖2L2(Ω) ) , t ∈ [0, T ] . (2.25) Now we estimate ∂ψN ∂t . For this purpose, we write system (2.18) in the form i d dt ( ψN (·, t) , uk ) L2(D) − ( a0∇ψN (·, t) ,∇uk ) L2(D) − ( a (·)ψN (·, t) , uk ) L2(D) + ( v (·)ψN (·, t) , uk ) L2(D) + i ( a1 (·)∇ψN (·, t) , uk ) L2(D) + ( ia2 ∣∣ψN (·, t) ∣∣2 ψN (·, t) , uk ) L2(D) = fk (t) , k = 1, 2, . . . , N. (2.26) We differentiate both sides of this system with respect to t and multiply the k-th equation of the obtained system by dc̄Nk (t) dt , and then sum the obtained equations over k from 1 to N . Then, integrating the obtained equality on the interval (0, t), we have ∫ Ωt ( i ∂2ψN ∂t2 ∂ψ̄N ∂t − a0 2∑ j=1 ∣∣∣∣∂2ψN ∂t∂xj ∣∣∣∣2 + i 2∑ j=1 a1j (x) ∂2ψN ∂xj∂t ∂ψ̄N ∂t − a (x) ∣∣∣∣∂ψN∂t ∣∣∣∣2 + v (x) ∣∣∣∣∂ψN∂t ∣∣∣∣2 ) dx dτ 220 G. Yagub, N.S. Ibrahimov, and M. Zengin + ia2 ∫ Ωt ∂ ∂t (∣∣ψN ∣∣2 ψN) ∂ψ̄N ∂t dx dτ = ∫ Ωt ∂f (x, τ) ∂t ∂ψ̄N (x, τ) ∂t dx dτ, t ∈ [0, T ] . Subtracting the complex conjugate from this equality, we get∫ Ωt ∂ ∂t ∣∣∣∣∂ψN∂t ∣∣∣∣2 dx dτ + ∫ Ωt 2∑ j=1 ∂ ∂xj ( a1j (x) ∣∣∣∣∂ψN∂t ∣∣∣∣2 ) dx dτ + ∫ Ωz a2 ( ∂ ∂t (∣∣ψN ∣∣2 ψN) ∂ψ̄N ∂t + ∂ ∂t (∣∣ψN ∣∣2 ψ̄N) ∂ψN ∂t ) dx dτ = − ∫ Ωt 2∑ j=1 a1j (x) ∣∣∣∣∂ψN∂t ∣∣∣∣2 dx dτ + 2 ∫ Ωt Im ( ∂f ∂t ∂ψ̄N ∂t ) dx dτ, t ∈ [0, T ] . (2.27) It is clear that the equality ∂ ∂t (∣∣ψN ∣∣2 ψN) ∂ψ̄N ∂t + ∂ ∂t (∣∣ψN ∣∣2 ψ̄N) ∂ψN ∂t = 4 ∣∣ψN ∣∣2 ∣∣∣∣∂ψN∂t ∣∣∣∣2 + 2 Re [( ψN )2(∂ψ̄N ∂t )] (2.28) holds true. From the other hand, using equality (2.17) and the condition uk\|Γ = 0, k = 1, 2, . . . , we can write ∂ψN ∂t ∣∣∣∣ S = 0, N = 1, 2, .... (2.29) Using (2.28), (2.29), the Cauchy–Bunyakovsky–Schwartz inequality and estimate (2.25), it can be obtained from (2.27) that the inequality∥∥∥∥∂ψN (·, t) ∂t ∥∥∥∥2 L2(D) + 2a2 ∫ Ωt ∣∣ψN ∣∣2 ∣∣∣∣∂ψN∂t ∣∣∣∣2 dx dτ ≤ ∥∥∥∥∂ψN (·, 0) ∂t ∥∥∥∥2 L2(D) + c3 ( ‖φ‖2L2(D) + ‖f‖2L2(Ω) ) + (2µ2 + 1) ∫ t 0 ∥∥∥∥∂ψN (·, t) ∂t ∥∥∥∥2 L2(D) dx dτ, t ∈ [0, T ] (2.30) holds true. To estimate the first term of the right-hand side of this inequality, we use system (2.18) and establish the inequality∥∥∥∥∂ψN (·, 0) ∂t ∥∥∥∥2 L2(D) ≤ 5 ∥∥LψN (·, 0) ∥∥2 L2(D) + 5a2 2 ∥∥ψN (·, 0) ∥∥6 L6(D) . . . Initial-Boundary Value Problems for a Nonlinear Schrödinger. . . 221 + 5 ‖f (·, 0)‖2L2(D) + 20µ2 1 ∥∥∇ψN (·, 0) ∥∥2 + 5b20 ∥∥ψN (·, 0) ∥∥2 L2(D) . (2.31) Using the Gagliardo–Nirenberg inequality (see [16, p.79]), for n = 2 we have∥∥ψN (·, t) ∥∥ L6(D) ≤ β ∥∥∇ψN (·, t) ∥∥ 2 3 L2(D) ∥∥ψN (·, t) ∥∥ 1 3 L2(D) , (2.32) where β > 0 is some constant. With the help of formula (2.17), from this we get∥∥ϕN∥∥ L6(D) ≤ β ∥∥∇ϕN∥∥ 2 3 L2(D) ∥∥ϕN∥∥ 1 3 L2(D) . (2.33) Since f ∈W 0,1 2 (Ω), it is easy to set ‖f (·, t)‖2L2(D) ≤ c4 ‖f‖W 0,1 2 (Ω) , t ∈ [0, T ] . (2.34) It is clear that LψN (x, 0) = N∑ k=1 cNk (0)uk (x) = N∑ k=1 (Lϕ, uk)L2(D) uk (x) . (2.35) Then we get ∥∥LψN (·, 0) ∥∥2 L2(D) = N∑ k=1 ∣∣∣(Lϕ, uk)L2(D) uk (x) ∣∣∣2 ≤ ‖Lϕ‖2l2(D) . It follows from the last inequality, the condition ϕ ∈ W 2 2 ◦ (D) and the conditions set on the coefficients of equation (2.1) that we can get the estimate∥∥LψN (·, 0) ∥∥2 L2(D) ≤ c5 ‖φ‖2 W 2 2 ◦ (D) . (2.36) In a similar way, we obtain∥∥∇ψN (·, 0) ∥∥2 L2(D) ≤ c6 ‖φ‖2 W 1 2 ◦ (D) . (2.37) Considering inequalities (2.32)–(2.37) and (2.31), we have∥∥∥∥∂ψN (·, 0) ∂t ∥∥∥∥2 L2(D) ≤ c7 ( ‖φ‖2 W 2 2 ◦ (D) + ‖f‖2 W 0,1 2 (Ω) + ‖φ‖6 W 1 2 ◦ (D) ) . (2.38) Consideration of (2.30) gives∥∥∥∥∂ψN (·, t) ∂t ∥∥∥∥2 L2(D) + 2a2 ∫ Ωt ∣∣ψN ∣∣2 ∣∣∣∣∂ψN∂t ∣∣∣∣2 dx dτ ≤ c8 ( ‖φ‖2 W 2 2 ◦ (D) + ‖f‖2 W 0,1 2 (Ω) + ‖φ‖6 W 1 2 ◦ (D) ) + (2µ2 + 1) ∫ t 0 ∥∥∥∥∂ψN (·, t) ∂t ∥∥∥∥2 L2(D) dx dτ, t ∈ [0, T ] . (2.39) 222 G. Yagub, N.S. Ibrahimov, and M. Zengin Using Gronwall’s lemma, one can derive from (2.39):∥∥∥∥∂ψN (·, t) ∂t ∥∥∥∥2 L2(D) + 2a2 ∫ Ωt ∣∣ψN ∣∣2 ∣∣∣∣∂ψN∂t ∣∣∣∣2 dx dτ ≤ c9 ( ‖φ‖2 W 2 2 ◦ (D) + ‖f‖2 W 0,1 2 (Ω) + ‖φ‖6 W 1 2 ◦ (D) ) , t ∈ [0, T ] . (2.40) To estimate ∇ψN (x, t) in the L2 (D) norm for any t ∈ [0, T ], we multiply each k-th equation of system (2.18) by dc̄Nk (t) dt and take a sum of all obtained equalities over k = 1 up to k = N . Then, integrating the obtained equation on the interval (0, t), we get∫ Ωt ( i ∣∣∣∣∂ψN∂t ∣∣∣∣2 − a0∇ψN ∂ ∂t ( ∇ψ̄N ) + ia1(x)∇ψN ∂ψ̄ N ∂t +v(x)ψN ∂ψ̄N ∂t + ia2 ∣∣ψN ∣∣2 ψN ∂ψ̄N ∂t ) dx dτ ∫ Ωt f ∂ψ̄N ∂t dxdτ, t ∈ [0, T ]. Summing this equality with its complex conjugate and applying to the obtained Cauchy–Bunyakovsky–Schwartz inequality and using then the conditions on the coefficients and estimates (2.25), (2.40), one can easily get the inequality∥∥∇ψN (·, t) ∥∥2 L2(D) ≤ ∥∥∇ψN (·, 0) ∥∥2 L2(D) + c10 ( ‖φ‖2 W 2 2 ◦ (D) + ‖f‖2 W 0,1 2 (Ω) + ‖φ‖6 W 1 2 ◦ (D) ) , t ∈ [0, T ]. Here the constant c10 > 0 does not depend on N . The last inequality and (2.37) give the estimate∥∥∇ψN (·, t) ∥∥2 L2(D) ≤ c11 ( ‖φ‖2 W 2 2 ◦ (D) + ‖f‖2 W 0,1 2 (Ω) + ‖φ‖6 W 1 2 ◦ (D) ) , t ∈ [0, T ]. (2.41) Here the constant c11 > 0 does not depend on N . Now we estimate ψN (x, t) in the norm of W 2 2 ◦ (D). For this purpose, we multiply each k-th equation of system (2.18) by λk c̄ N k (t) and sum all obtained equalities over k = 1 up to k = N . Then we get∫ D ∣∣LψN (x, t) ∣∣2 dx = ∫ D [ i ∂ψN (x, t) ∂t + ia1∇ψN (x, t) + v (x)ψN (x, t) +ia2 ∣∣ψN (x, t) ∣∣2 ψN (x, t)− f (x, t) ] Lψ̄N (x, t) dx, t ∈ [0, T ]. (2.42) From this equation, using the Cauchy–Bunyakovsky–Schwartz inequality, we ob- tain the inequality ∥∥LψN (·, t) ∥∥2 L2(D) ≤ 5 ∥∥∥∥∂ψN (·, t) ∂t ∥∥∥∥2 L2(D) + 5µ2 2 ∥∥∇ψN (·, t) ∥∥2 L2(D) . . . Initial-Boundary Value Problems for a Nonlinear Schrödinger. . . 223 + 5a2 2 ∥∥ψN (·, t) ∥∥6 L6(D) + 5b20 ∥∥ψN (·, t) ∥∥2 L2(D) + 5 ‖f(·, t)‖2L2(D) , t ∈ [0, T ]. (2.43) With the help of inequalities (2.32), (2.34) and estimates (2.25), (2.40), (2.41), from (2.43) we get∥∥LψN (·, t) ∥∥2 L2(D) ≤ c12 ( ‖φ‖2 W 2 2 ◦ (D) + ‖f‖2 W 0,1 2 (Ω) + ‖φ‖6 W 1 2 ◦ (D) ) , t ∈ [0, T ]. (2.44) Here the constant c12 > 0 does not depend on N . By the definition of the operator L, we have∥∥LψN (·, t) ∥∥ L2(D) = ∥∥−a0∆ψN (·, t) + a(·)ψN (·, t) ∥∥ L2(D) ≥ a0 ∥∥∆ψN (·, t) ∥∥ L2(D) − µ0 ∥∥ψN (·, t) ∥∥ L2(D) . This implies∥∥∆ψN (·, t) ∥∥ L2(D) ≤ 1 a0 ∥∥LψN (·, t) ∥∥ L2(D) + µ0 a0 ∥∥ψN (·, t) ∥∥ L2(D) . Substituting (2.25) and (2.44) in this inequality, we get the validity of∥∥∆ψN (·, t) ∥∥2 L2(D) ≤ c13 ( ‖φ‖2 W 2 2 ◦ (D) + ‖f‖2 W 0,1 2 (Ω) + ‖φ‖6 W 2 2 ◦ (D) ) . (2.45) Here the constant c13 > 0 does not depend on N . Using the well-known inequality (see [11, p. 124]), for the convex domain D we obtain∥∥ψN (·, t) ∥∥2 W 2 2 ◦ (D) ≤ c14 ∥∥∆ψN (·, t) ∥∥ , t ∈ [0, T ]. (2.46) Consideration of (2.45) and (2.46) gives∥∥ψN (·, t) ∥∥2 W 2 2 ◦ (D) ≤ c15 ( ‖φ‖2 W 2 2 ◦ (D) + ‖f‖2 W 0,1 2 (Ω) + ‖φ‖6 W 1 2 ◦ (D) ) , t ∈ [0, T ]. (2.47) Here the constant c15 > 0 does not depend on N . Thus, taking into account estimates (2.40) and (2.47), we finally get∥∥ψN (·, t) ∥∥2 W 2 2 ◦ (D) + ∥∥∥∥∂ψN (·, t) ∂t ∥∥∥∥2 L2(D) ≤ c16 ( ‖φ‖2 W 2 2 ◦ (D) + ‖f‖2 W 0,1 2 (Ω) + ‖φ‖6 W 1 2 ◦ (D) ) , t ∈ [0, T ], (2.48) where the constant c16 > 0 does not depend on N . Using this estimate and the inequality N∑ k=1 ∣∣cNk (t) ∣∣2 + N∑ k=1 ∣∣∣∣dcNk (t) dt ∣∣∣∣2 ≤ ∥∥ψN (·, t) ∥∥2 W 2 2 ◦ (D) + ∥∥∥∥∂ψN (·, t) ∂t ∥∥∥∥2 L2(D) , t ∈ [0, T ] , denoting c1 = c16, we come to the statement of the lemma. Lemma 2.3 is proved. 224 G. Yagub, N.S. Ibrahimov, and M. Zengin Now we continue the proof of the theorem. Let us consider the functions lN,k (t) = ( ψN (·, t) , uk ) L2(D) , N, k = 1, 2, . . .. It follows from (2.20) and orthog- onality of the functions uk = uk (x), k = 1, 2, . . ., that the families of functions lN,k (t), N, k = 1, 2, . . ., and their derivatives dlN,k(t) dt , N, k = 1, 2, . . ., are uni- formly bounded on the interval [0, T ], \|lN,k (t)\| ≤ c17, ∣∣∣∣dlN,k (t) dt ∣∣∣∣ ≤ c18, N, k = 1, 2, . . . , t ∈ [0, T ] . (2.49) Let us show that for the fixed k and any N ≥ k, the functions lN,k (t) , N, k = 1, 2, . . ., are equicontinuous on the interval [0, T ]. Indeed, integrating the k-th equation from (2.18) on the interval [t, t+ ∆t], we get \|lN,k (t+ ∆t)− lN,k (t)\| ≤ ∫ t+∆t t ∣∣∣∣∫ D a0∆ψN (x, τ)uk (x) dx ∣∣∣∣ dτ + ∫ t+∆t t ∣∣∣∣∫ D ia1 (x)∇ψN (x, τ)uk (x) dx ∣∣∣∣ dτ + ∫ t+∆t t ∣∣∣∣∫ D a (x)ψN (x, τ)uk (x) dx ∣∣∣∣ dτ + ∫ t+∆t t ∣∣∣∣∫ D v (x)ψN (x, τ)uk (x) dx ∣∣∣∣ dτ + ∫ t+∆t t ∣∣∣∣∫ D ia2 ∣∣ψN (x, τ) ∣∣2 ψN (x, τ)uk (x) dx ∣∣∣∣ dτ + ∫ t+∆t t ∣∣∣∣∫ D f (x, τ)uk (x) dx ∣∣∣∣ dτ. Together with the Cauchy–Bunyakovsky–Schwartz inequality this gives the in- equality \|lN,k (t+ ∆t)− lN,k (t)\| ≤ a0 ∫ t+∆t t ∥∥∆ψN (·, τ) ∥∥ L2(D) ‖uk‖L2(D) dτ + √ 2µ1 ∫ t+∆t t ∥∥∇ψN (·, τ) ∥∥ L2(D) ‖uk‖L2(D) dτ + (µ0 + b0) ∫ t+∆t t ∥∥ψN (· , τ) ∥∥ L2(D) ‖uk‖L2(D) dτ + a2 ∫ t+∆t t ∥∥ψN (·, τ) ∥∥3 L6(D) ‖uk‖L2(D) dτ + ∫ t+∆t t ‖f (·, τ)‖L2(D) ‖uk‖L2(D) dτ. Therefore, taking into account (2.32), estimates (2.20), (2.45) and assumption (2.16), we get the relation \|lN,k (t+ ∆t)− lN,k (t)\| ≤ c19dk∆t, N, k = 1, 2, . . . , (2.50) . . . Initial-Boundary Value Problems for a Nonlinear Schrödinger. . . 225 where the constant c19 > 0 does not depend on N, k, t. Performing the integration by parts in the second and third terms of the left-hand side of equations (2.18) and differentiating the obtained relations with respect to t, and then integrating on the interval [t, t+ ∆t], one can get∣∣∣∣dlN,k (t+ ∆t) dt − dlN,k (t) dt ∣∣∣∣ ≤ ∫ t+∆t t ∣∣∣∣∫ D a0 ∂ψN (x, τ) ∂τ ∆uk (x) dx ∣∣∣∣ dτ + ∫ t+∆t t ∣∣∣∣∫ D i ∂ψN (x, τ) ∂τ ∇ (a1 (x)uk (x)) dx ∣∣∣∣ dτ + ∫ t+∆t t ∣∣∣∣∫ D a (x) ∂ψN (x, τ) ∂τ uk (x) dx ∣∣∣∣ dτ + ∫ t+∆t t ∣∣∣∣∫ D v (x) ∂ψN (x, τ) ∂τ uk (x) dx ∣∣∣∣ dτ + ∫ t+∆t t ∣∣∣∣∫ D ia2 ∂ ∂τ (∣∣ψN (x, τ) ∣∣2 ψN (x, τ) ) uk (x) dx ∣∣∣∣ dτ + ∫ t+∆t t ∣∣∣∣∫ D ∂f (x, τ) ∂τ uk (x) dx ∣∣∣∣ dτ. From this, by virtue of Cauchy–Bunyakovsky–Schwartz inequality, we get∣∣∣∣dlN,k (t+ ∆t) dt − dlN,k (t) dt ∣∣∣∣ ≤ a0 ∫ t+∆t t ∥∥∥∥∂ψ (·, τ) ∂τ ∥∥∥∥ L2(D) dτ ‖∆uk‖L2(D) + √ 2µ1 ∫ t+∆t t ∥∥∥∥∂ψ (·, τ) ∂τ ∥∥∥∥ L2(D) dτ ‖∇uk‖L2(D) + ( µ0 + b0 + √ 2µ2 )∫ t+∆t t ∥∥∥∥∂ψ (·, τ) ∂τ ∥∥∥∥ L2(D) dτ ‖uk‖L2(D) + 3a2 ∫ t+∆t t ∫ D ∣∣∣∣∂ψN (x, τ) ∂τ ∣∣∣∣ \|ψ (x, τ)\|2 \|uk (x)\| dx dτ + ∫ t+∆t t ∥∥∥∥∂f (·, τ) ∂τ ∥∥∥∥ L2(D) dτ ‖uk‖L2(D) . (2.51) Now, let us estimate the fourth term of the right-hand side of this inequality. Then, by virtue of the Cauchy–Bunyakovsky–Schwartz inequality, we get 3a2 ∫ t+∆t t ∫ D ∣∣∣∣∂ψN (x, τ) ∂τ ∣∣∣∣ ∣∣ψN (x, τ) ∣∣2 \|uk (x)\| dx dτ ≤ 3a2 ∫ t+∆t t (∫ D ∣∣∣∣∂ψN (x, τ) ∂τ ∣∣∣∣2 ∣∣ψN (x, τ) ∣∣2 dx) 1 2 × (∫ D ∣∣ψN (x, τ) ∣∣2 \|uk (x)\|2 dx ) 1 2 dτ. (2.52) If we apply the Cauchy–Bunyakovsky–Schwartz inequality to the second multi- plier in the integrant in the right-hand side of this inequality, then we get 226 G. Yagub, N.S. Ibrahimov, and M. Zengin (∫ D ∣∣ψN (x, τ) ∣∣2 \|uk (x)\|2 dx ) 1 2 ≤ (∫ D ∣∣ψN (x, τ) ∣∣4 dx) 1 4 (∫ D \|uk (x)\|4 dx ) 1 4 . (2.53) By virtue of the inequalities from [13, pp. 84 and 88], we have ‖uk‖L4(D) ≤ c20 ‖∇uk‖L2(D) , (2.54)∥∥ψN (·, τ) ∥∥ L4(D) ≤ c21 ∥∥ψN (·, τ) ∥∥ W 1 2 ◦ (D) . (2.55) Then, taking into account (2.53)–(2.55), from (2.52) we get 3a2 ∫ t+∆t t ∫ D ∣∣∣∣∂ψN (x, τ) ∂τ ∣∣∣∣ ∣∣ψN (x, τ) ∣∣2 \|uk (x)\| dx dτ ≤ 3c20c21a2 ∫ t+∆t t (∫ D ∣∣∣∣∂ψN (x, τ) ∂τ ∣∣∣∣2 ∣∣ψN (x, τ) ∣∣2 dx) 1 2 × ∥∥ψN (·, τ) ∥∥ W 1 2 ◦ (D) dτ ‖∇uk‖L2(D) . (2.56) Substituting (2.56) into (2.51), with the help of (2.20), (2.40) and assumption (2.16) it is easy to establish the inequality∣∣∣∣dlN,k (t+ ∆t) dt − dlN,k (t) dt ∣∣∣∣ ≤ c22dk (∆t) 1 2 , N, k = 1, 2, . . . , (2.57) where the constant c22 > 0 does not depend on N, k, t. It follows from (2.50) and (2.57) that the families of functions {lN,k (t)},{ dlN,k(t) dt } , N, k = 1, 2, . . ., are equicontinuous on the interval [0, T ] for a fixed k and arbitrary N ≥ k. Then, by a standard diagonal procedure, we can choose a subsequence Nm, m = 1, 2, . . ., such that the corresponding functions lNm,k (t), m = 1, 2, . . .. and their derivatives dlNm,k(t) dt , m = 1, 2, . . ., converge uniformly on the interval [0, T ] to the continuous functions lk (t), dlk(t) dt for each k = 1, 2, . . .. The functions lk (t), k = 1, 2, . . ., and their derivatives define the functions ψ (x, t) = ∞∑ k=1 lk (t)uk (x) , ∂ψ (x, t) ∂t = ∞∑ k=1 dlk (t) dt uk (x) . (2.58) Then, as in [4, 8], we can state the subsequences { ψNm (x, t) } , { ∂ψNm (x,t) ∂t } , de- fined by formulas (2.58), converge weakly in W 2 2 ◦ (D) , L2 (D) to the functions ψ (x, t) , ∂ψ(x,t) ∂t , respectively, uniformly with respect to t ∈ [0, T ]. The limit function ψ (x, t) belongs to the space B0. Now we show that the limit function ψ (x, t) is a solution of problem (2.1), (2.2) in the sense of Definition 2.1. For this purpose, we first prove that this . . . Initial-Boundary Value Problems for a Nonlinear Schrödinger. . . 227 function satisfies equation (2.1) for almost all x ∈ D and arbitrary t ∈ [0, T ]. We set N = Nm and multiply the k-th equation from (2.18) by a continuous function η̄k (t) and sum up the obtained equations with respect to k from k = 1 to N ′ ≤ Nm. Then we get∫ D ( i ∂ψNm (x, t) ∂t − a0∆ψNm (x, t) + ia1 (x)∇ψNm (x, t)− a (x)ψNm (x, t) + v (x)ψNm (x, t) + ia2 ∣∣ψNm (x, t) ∣∣2 ψNm (x, t) −f (x, t) ) η̄N ′ (x, t) dx = 0, t ∈ [0, T ] , (2.59) for any function η̄N ′ k (x, t) = ∑N ′ k=1 η̄k (t)uk (x), N ′ ≤ Nm. The sequence { ψNm (x, t) } converges uniformly to the function ψ = ψ (x, t) as m→∞, and the space W 2 2 ◦ (D) is compact embedded into L2 (D) (see [15, 16, 19]). Therefore, there exists a subsequence of { ψNm (x, t) } which converges strongly in L2 (D) to the function ψ = ψ (x, t) as m→∞, i.e.,∥∥ψNm (·, t)− ψ (·, t) ∥∥ L2(D) → 0 (2.60) uniformly with respect to t ∈ [0, T ] as m → ∞. Consequently, there exists a subsequence of { ψNm (x, t) } which converges to the function ψ = ψ (x, t) almost everywhere in D. For the sake of simplicity, this subsequence is denoted again by { ψNm (x, t) } . Then we can write ψNm (x, t)→ ψ (x, t) almost everywhere in D (2.61) uniformly with respect to t ∈ [0, T ] as m→∞. Besides, due to uniform estimate (2.20) and inequality (2.32), for N = Nm, the inequality∥∥∥∣∣ψNm (·, t) ∣∣2 ψ (·, t) ∥∥∥ L2(D) ≤ ∥∥ψNm (·, t) ∥∥3 L2(D) holds true. From the known lemma (see [13, pp. 530–531]), we obtain that {∣∣ψNm (x, t) ∣∣2 ψNm (x, t) } converges weakly in L2 (D) to the function \|ψ (x, t)\|2 ψ (x, t) uniformly with respect to t ∈ [0, T ] as m→∞, i.e.,∫ D ∣∣ψNm (x, t) ∣∣2 ψNm (x, t) η̄N ′ (x, t) dx → ∫ D \|ψ (x, t)\|2 ψ (x, t) η̄N ′ (x, t) dx as m→∞ t ∈ [0, T ] (2.62) for any continuous on the interval [0, T ] in the L2 (D) norm function η̄N ′ k (x, t) =∑N ′ k=1 η̄k (t)uk (x), N ′ ≤ Nm. Using this limit relation and the convergence of the subsequence { ψNm (x, t) } to the function ψ (x, t), passing to limit as m→∞, in (2.59), we get∫ D ( i ∂ψ (x, t) ∂t − a0∆ψ (x, t) + ia1 (x)∇ψ (x, t)− a (x)ψ (x, t) + v (x)ψ (x, t) 228 G. Yagub, N.S. Ibrahimov, and M. Zengin + ia2 \|ψ (x, t)\|2 ψ (x, t)− f (x, t) ) η̄N ′ (x, t) dx = 0, t ∈ [0, T ] (2.63) for any function η̄N ′ k (x, t) = ∑N ′ k=1 η̄k (t)uk (x), N ′ ≤ Nm. Since all functions of the form η̄N ′ k (x, t) = ∑N ′ k=1 η̄k (t)uk (x) are dense in C0 ([0, T ] , L2 (D)), we obtain immediately from identity (2.63) that the limit function ψ (x, t) satisfies equation (2.1) for any t ∈ [0, T ] and for almost all x ∈ D. The fulfillment of the initial and boundary conditions (2.2) for the limit function ψ (x, t) follows from the limit relation (2.60) for t = 0 and the fact that the space B0 is compactly embedded into L2 (S). Thus, we have proved that the limit function ψ (x, t) is a solution of the initial-boundary problem (2.1), (2.2), and this solution belongs to the space B0 and satisfies (2.7), which follows immediately from (2.20) after passing to the lower limit over the weakly convergent subsequence { ψNm (x, t) } from B0 to the function ψ (x, t). Now, continuing the proof of the theorem, we prove the uniqueness of the solution of the initial-boundary value problem (2.1), (2.2). Let ψ (x, t) and Φ (x, t) be two arbitrary solutions for problem (2.1), (2.2). Let w (x, t) = ψ (x, t)−Φ (x, t). Then it is clear from condition (2.1), (2.2) that w = w (x, t) can be a solution of the following initial-boundary problem: i ∂w ∂t + a0∆w + ia1 (x)∇w − a (x)w + v (x)w + ia2 ( \|ψ\|2 + \|Φ\|2 ) w + ia1ψΦw̄ = 0, (x, t) ∈ Ω, (2.64) w (x, 0) = 0, x ∈ D, w\|S = 0. (2.65) To establish the estimate for the solution of this problem, we multiply (2.64) by the function w̄ (x, t) and integrate on the domain Ωt. Using the boundary condition from (2.65) and integrating by parts, we get∫ Ωt ( i ∂w ∂t w̄ − a0 \|∇w\|2 + ia1 (x)∇ww̄ − a (x) \|w\|2 + v (x) \|w\|2 + ia2 ( \|ψ\|2 + \|Φ\|2 ) \|w\|2 + ia1ψΦ (w̄)2 ) dx dτ = 0, t ∈ [0, T ] . Subtracting from this equality its complex conjugate and using boundary condi- tion (2.65), we obtain∫ Ωt i ( ∂w ∂τ w̄ + ∂w̄ ∂τ w ) dx dτ + i2a2 ∫ Ωt ( \|ψ\|2 + \|Φ\|2 ) \|w\|2 dx dτ = −i2a2 ∫ Ωt Im [ ψΦ (w̄)2 ] dx dτ − i ∫ Ωt 2∑ j=1 ∂a1j (x) ∂xj \|w\|2 dx dτ for any t ∈ [0, T ]. Together with (2.65) this gives ‖w (·, t)‖2L2(D) + 2a2 ∫ Ωt ( \|ψ\|2 + \|Φ\|2 ) \|w\|2 dx dτ . . . Initial-Boundary Value Problems for a Nonlinear Schrödinger. . . 229 + 2a2 ∫ Ωt \|ψ\| \|Φ\| \|w\|2 dxdτ + 2µ2 ∫ Ωt \|w\|2 dx dτ for any t ∈ [0, T ]. Application of the inequality 2 \|ψ\| \|Φ\| ≤ \|ψ\|2 + \|Φ\|2 implies ‖w (·, t)‖2L2(D) + a2 ∫ Ωt ( \|ψ\|2 + \|Φ\|2 ) \|w\|2 dxdτ ≤ 2µ2 ∫ t 0 ‖w (·, τ)‖2L2(D) dτ for any t ∈ [0, T ]. With the help of Gronwall’s lemma, we get the relation ‖w (·, t)‖2L2(D) = 0, t ∈ [0, T ] , which proves the validity of w (x, t) = 0, x ∈ D, t ∈ [0, T ] . The uniqueness of the solution of the initial-boundary problem (2.1), (2.2) follows immediately. Theorem 2.2 is proved. Remark 2.4. A similar result can be established when the set D belongs to R3. 3. The existence and uniqueness of a solution of the second initial-boundary value problem Consider the initial-boundary problem on determining the function ψ = ψ (x, t) in the domain Ω subject to i ∂ψ ∂t + a0∆ψ + ia1 (x)∇ψ − a (x)ψ + v (x)ψ + ia2 \|ψ\|2 ψ = f, (x, t) ∈ Ω, (3.1) ψ (x, 0) = φ (x) , x ∈ D, ∂ψ ∂ν \|S = 0 , (3.2) where i = √ −1; a0 > 0, a2 > 0 are given numbers; ν is an outward normal to the boundary Γ; ∆ = ∂2 ∂x21 + ∂2 ∂x22 is the Laplace operator; ∇ = ( ∂ ∂x1 , ∂ ∂x2 ) , a (x) , v (x) are measurable bounded functions satisfying the conditions µ0 ≤ a (x) ≤ µ1, x ∈ D, µ0, µ1 = const > 0 , (3.3) \|v (x)\| ≤ b0, x ∈ D, b0 = const > 0; (3.4) a1 (x) = (a11 (x) , a12 (x)) is a given vector-function whose components satisfy the conditions \|a1j (x)\| ≤ µ2, ∣∣∣∣∂a1j (x) ∂xk ∣∣∣∣ ≤ µ3, x ∈ D, j, k = 1, 2, µ2, µ3 = const > 0; (3.5) φ (x) , f (x, t) are complex valued functions satisfying the conditions φ ∈W 2 2 (D) , ∂φ ∂ν ∣∣∣∣ Γ = 0, f ∈W 0,1 2 (Ω) . (3.6) 230 G. Yagub, N.S. Ibrahimov, and M. Zengin Definition 3.1. The function ψ = ψ (x, t) from the space B1 ≡ C0 ( [0, T ] ,W 2 2 (D) )⋂ C1 ([0, T ] , L2 (D)) is called a generalized solution of (3.1), (3.2) if it satisfies equation (3.1) for almost all x ∈ D and any t ∈ [0, T ], and initial and boundary conditions (3.2) for almost all x ∈ D and for almost all (ξ, t) ∈ S, respectively. Theorem 3.2. Let the functions a (x), v (x),a1 (x), ϕ (x), f (x, t) satisfy con- ditions (3.3)–(3.6). Then the initial-boundary problem (3.1), (3.2) has the only solution from the space B1, and for this solution the estimate ‖ψ (·, t)‖W 2 2 (D) + ∥∥∥∥∂ψ (·, t) ∂t ∥∥∥∥ L2(D) ≤ c23 (‖φ‖W 2 2 (D) + ‖f‖ W 0,1 2 (Ω) + ‖φ‖3W 1 2 (D) ) , t ∈ [0, T ] , (3.7) where the constant c23 > 0 does not depend on φ, f , and t, is valid. This theorem can be proved by using Galerkin’s approximations in the same way as Theorem 1. In this case, as a fundamental in W 2 2 (D) system of functions we take an orthonormal in L2 (D) and orthogonal in W 2 2 (D) system uk = uk (x), k = 1, 2, . . ., of eigenfunction of the spectral problem LX (x) = λX (z) , x ∈ D, ∂X ∂ν \|Γ = 0 (3.8) at λ = λk, k = 1, 2, . . ., where the operator L is defined as L = −a0∆ + a (x) (3.9) with the Neumann boundary conditions. Remark 3.3. A similar result is valid when the set D lies in R3. References [1] G.D. Akbaba, The Optimal Control Problem with the Lions Functional for the Schrödinger Equation Including Virtual Coefficient Gradient, Master’s thesis, Kars (Turkey), 2011 (Turkish). [2] L. Baudouin, O. Kavian, and J.P. Puel, Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control, J. Differential Equa- tions 216 (2005), 188–222. [3] A.G. Butkovskiy and Y.I. Samojlenko, Control of Quantum-Mechanical Processes and Systems. Mathematics and its Applications (Soviet Series), 56, Kluwer Aca- demic Publishers Group, Dordrecht, 1990. [4] N.S. Ibragimov, The solvability of the initial-boundary value problems for the non- linear stationary equation of quasi-optics with purely imaginary coefficient in the nonlinear part, News of Baku State University, Ser. Physics and Math. Sciences (2010), No. 3, 72–84. . . . Initial-Boundary Value Problems for a Nonlinear Schrödinger. . . 231 [5] A.D. Iskenderov and G.Y. Yagubov, A variational method for solving an inverse problem of determining the quantum mechanical potential, Dokl. Akad. Nauk SSSR 303 (1988), 1044–1048 (Russian); Engl. transl.: Soviet Math. Dokl. 38 (1989), 637– 641. [6] A.D. Iskenderov and G.Y. Yagubov, Optimal control of nonlinear quantum- mechanical systems, Avtomat. i Telemekh. (1989), No. 12, 27–38 (Russian); Engl. transl.: Automat. Remote Control 50 (1989), No. 12, Part 1, 1631–1641 (1990). [7] A. Iskenderov and G. Yagubov, Optimal control of the unbounded potential in the multidimensional nonlinear nonstationary Schrödinger equation, Bulletin of Lankaran State University, Ser. Natural Sciences (2007), 3–56. [8] A.D. Iskenderov, G.Y. Yagubov, and M.A. Musayeva, Identification of the Quantum Potentials, Chashyoglu, Baku, 2012 (Azerbaijani). [9] M. Jahanshahi, S. Ashrafi, and N. Aliev, Boundary layer problem for the system of the first order ordinary differential equations with constant coefficients by general nonlocal boundary conditions, Adv. Math. Models Appl. 2 (2017), 107–116. [10] O.A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics, Nauka, Moscow, 1973 (Russian). [11] O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Ural’tseva, Linear and Quasi- Linear Equations of Parabolic Type, Nauka, Moscow, 1967 (Russian); Engl. transl.: Translations of Mathematical Monographs, 23, Amer. Math. Soc., Providence, R.I., 1968. [12] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Ap- plications, I, Die Grundlehren der mathematischen Wissenschaften, 181, Springer- Verlag, New York–Heidelberg, 1972. [13] L.S. Pontryagin, Ordinary Differential Equations, Nauka, Moscow, 1982. [14] F.P. Vasilyev, Numerical Methods for Solving of the Extremal Problems, Nauka, Moscow, 1980. [15] M.A. Vorontsov and V.I. Schmalhausen, The Principles of Adaptive Optics, Nauka, Moscow, 1985 (Russian). [16] G.Y. Yagubov and M.A. Musayeva, On an identification problem for nonlinear Schrödinger equation, Differ. Uravn. 33 (1997), 1691–1698. [17] G. Yagubov, F. Toyğolu, and M. Subaşı, An optimal control problem for two- dimensional Schrödinger equation, Appl. Math. Comput. 218 (2012), 6177–6187. [18] K. Yajima and G. Zhang, Smoothing property for Schrödinger equations with po- tential superquadratic at infinity, Comm. Math. Phys. 221 (2001), 573–590. [19] V.M. Zhuravlev, Nonlinear Waves in Multicomponent Systems Dispersion and Dif- fusion, Ulyanovsk State University, Ulyanovsk, 2001 (Russian). Received February 13, 2017, revised January 10, 2018. G. Yagub, Kafkas University, Paşaçayırı Campus, Kars, 36040, Turkey, E-mail: gabilya@mail.ru mailto:gabilya@mail.ru 232 G. Yagub, N.S. Ibrahimov, and M. Zengin N.S. Ibrahimov, Baku State University, 23 Academic Zahid Khalilov St., Baku, AZ 1148, Azerbaijan; Lankaran State University, 50 Hazi Aslanov St., Lankaran, AZ 4200, Azerbaijan, E-mail: natiq ibrahimov@mail.ru M. Zengin, Kafkas University, Paşaçayırı Campus, Kars, 36040, Turkey, E-mail: merveezengin14@gmail.com Розв’язок початково-крайової задачi для нелiнiйного рiвняння Шредiнгера iз спецiальним градiєнтним членом G. Yagub, N.S. Ibrahimov, and M. Zengin У статтi розглядаються початково-крайовi задачi для двовимiрно- го нелiнiйного рiвняння Шредiнгера iз спецiальним градiєнтним членом з чисто уявними коефiцiєнтами в нелiнiйнiй частинi, коли коефiцiєнти рiвняння є вимiрними обмеженими функцiями. Доведено iснування i єд- нiсть розв’язкiв першої i другої початково-крайової задачi майже скрiзь. Ключовi слова: рiвняння Шредiнгера, спецiальний градiєнтний член, iснування та єднiсть, перша i друга початково-крайовi задачi. mailto:natiq_ibrahimov@mail.ru mailto:merveezengin14@gmail.com Introduction The existence and uniqueness of a solution of the first initial-boundary value problem The existence and uniqueness of a solution of the second initial-boundary value problem

The Solvability of the Initial-Boundary Value Problems for a Nonlinear Schrödinger Equation with a Special Gradient Term

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