On Solvability of Third-Order Operator Differential Equation with Parabolic Principal Part in Weighted Space

Aliev, Araz R.; Mirzoev, Sabir S.; Soylemezo, Mustafa A.

doi:https://doi.org/10.1155/2017/2932134

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Research Article | Open Access

Volume 2017 | Article ID 2932134 | https://doi.org/10.1155/2017/2932134

On Solvability of Third-Order Operator Differential Equation with Parabolic Principal Part in Weighted Space

Araz R. Aliev,^1,2Sabir S. Mirzoev ,^2,3and Mustafa A. Soylemezo⁴

Academic Editor: Hugo Leiva

Received30 Apr 2017

Accepted27 Sept 2017

Published31 Oct 2017

Abstract

Sufficient conditions are found for the correct and unique solvability of a class of third-order parabolic operator differential equations, whose principal parts have multiple characteristics, in a Sobolev-type space with exponential weight. The estimates for the norms of intermediate derivative operators are obtained and the relationship between these estimates and solvability conditions is established. Besides, the connection is found between the order of exponential weight and the lower bound for the spectrum of abstract operator appearing in the principal part of the equation.

1. Problem Statement

Let be a separable Hilbert space with the scalar product , , and let be a self-adjoint positive definite operator in (, , is a unit operator). Denote by () a scale of Hilbert spaces generated by the operator ; that is, , , . For , we assume that , , .

Denote by () a space of measurable (see [1]) functions with the values in equipped with the normand by a space of functions with the values in such that , , equipped with the normFor more details on the space see [2, Chapter 1].

Note that throughout this paper all the derivatives are understood in the sense of the theory of distributions, and the operator is defined by the spectral decomposition of the operator ; that is, where is a decomposition of the unit of the operator .

Now, let us recall one fact related to the space .

It is known that if , then the inequalitiesare valid, where , , are constants independent of function . This fact is referred to as the intermediate derivatives theorem (see [2, Chapter 1]). Also, these inequalities are usually referred to as Kolmogorov-type inequalities.

Let . For the functions defined on with the values in , we introduce the following spaces with the weight :Obviously, in case we get the spaces , .

In the sequel, by we will mean a set of linear bounded operators from the Hilbert space to another Hilbert space . If , we will write instead of . By we will denote the spectrum of the operator .

Consider the operator differential equationwhere , , , are linear and, in general, unbounded operators, , .

Definition 1. If for every there exists a vector function which satisfies (5) almost everywhere withthen this vector function is called a regular solution of (5), and (5) is said to be regularly solvable.

The principal part of (5) has multiple characteristics, so, according to the classification of [3], this equation belongs to the class of parabolic operator differential equations. The equations of form (5) characterize the problems of diffusion or heat conductivity in viscoelastic media [4]. Besides, such equations are also interesting in view of the fact that some classes of equations, which can be useful in modeling the problems of world population growth, can be reduced to them [5].

Note that the solvability issues for the operator differential equations in the spaces without weights have been studied quite widely (see Krein [6], Lions and Magenes [2], V. I. Gorbachuk and M. L. Gorbachuk [7], Goldstein [8], Yakubov [9, 10], S. Yakubov and Y. Yakubov [11], and the references therein). Fundamental contribution to this field was made by Gasymov in [12–14]. These issues remain as relevant today as they were long before. A lot of researches appeared in this field over the last years; see, for example, Aliev [15], Aliev and Mirzoev [16], Favini et al. [17], Favini and Yakubov [18, 19], Favini et al. [20], Mirzoev et al. [21, 22], Aliev and Yakubov [23, 24], and Mirzoev et al. [25].

Despite the number of works covering the solvability of operator differential equations in weighted spaces being relatively small (see, e.g., Dubinskiĭ [3], Shkalikov [26], Mirzoev [27], and Aliev [28]), recently a lot of works appeared dedicated to the issues of regular and normal solvability in weighted spaces for operator differential equations with multiple characteristics (see Aliev [29], Mirzoev and Humbataliev [30], Aliev and Elbably [31], and Aliev and Lachinova [32]). But the equations considered in the above-mentioned works, except for [32], belong to the class of quasi-elliptic operator differential equations [3]. In [32], the initial-boundary value problem was studied for parabolic equation on the half-axis .

In this work, we find the regular solvability conditions for (5) on the entire axis. We also obtain the estimates for the norms of intermediate derivative operators in a Sobolev-type space through the norm of the operator generated by the principal part of (5) (it should be noted here that the estimates for the norms of intermediate derivatives for scalar functions have been obtained in [33, 34] and the references therein). Moreover, we establish the relationship between these estimates and the regular solvability conditions for (5). The found regular solvability conditions are expressed in terms of operator coefficients of (5), which makes them easily verifiable and convenient for use both in theoretical problems and in applications.

2. Solvability of (5) in When ,

We first consider (5) in case .

Denote by the operator acting from to as follows:

The following theorem is true.

Theorem 2. Let be a self-adjoint positive definite operator with the lower bound for its spectrum and . Then the operator performs an isomorphism between the spaces and .

Proof. Let us make a change in the equation, . Then . Aswe haveTaking into account the fact that , we can rewrite (10) in the formin the space ; that is, , .
DenoteThen (11) can be rewritten as follows:where , . To solve the last equation, we make use of the Fourier transform:where , are Fourier transforms of the vector functions , , respectively. Let us show that for the operator pencilis invertible. In fact, let . Then the characteristic polynomial (15) has the formHence we havethat is, the spectral decomposition of the operator implies the invertibility of the operator pencil for . Consequently, from (14) we can find :Thus,It is clear that satisfies (11) almost everywhere.
Let us prove that . In fact, by virtue of well-known Plancherel theorem, it suffices to show that and . Obviously,Aslet us estimate the norm for . Spectral theory of self-adjoint operators impliesConsequently,Similarly we haveHence, for and we obtainConsequently,Thus, .
It is clear that the vector function belongs to and is a regular solution of (8).
It is also clear that the equation has only trivial solution in the space .
Now let us show that is a bounded operator from the space to the space . In fact, by virtue of Cauchy-Schwarz and Young inequalities, for , we haveThen, taking into account intermediate derivatives theorem [2, Chapter 1], we getthat is,So we obtain that the operator : is one-to-one and bounded. Then, by the Banach theorem on inverse operator, the operator : is bounded.
Thus, is an isomorphism between the spaces and . Theorem is proved.

Theorem 2 has the following.

Corollary 3. When , the norms and are equivalent in the space .

Remark 4. For , the operator is noninvertible.
As is known, the intermediate derivative operatorsare continuous [2]. By virtue of this fact and Corollary 3, the norms of the operators (30) can be estimated through .

Theorem 5. Let be a self-adjoint positive definite operator with the lower bound for its spectrum and . Then, for every the following inequalities are true:where

Proof. As the mapping is an isomorphism between the spaces and , to prove inequalities (31) it suffices to estimate the norms , , by . Then, making a change and applying Fourier transform, we obtainConsequently, for and , we need to estimate the normsConsider the case . Solving extremal problem, we havewhere if , and if .
For , we havewhere if , and if .
Thus, considering the obtained estimates in inequalities (33), we haveThe latter inequalities, in turn, are equivalent to the following ones:Theorem is proved.

3. Solvability of (5) in When ,

Denote by the operator acting from to as follows:

The following lemma is true.

Lemma 6. Let be a self-adjoint positive definite operator in and let , . Then the operator is bounded from to .

Proof. By the conditions of lemma, for , we haveUsing intermediate derivatives theorem [2, Chapter 1], we getLemma is proved.

The obtained results allow studying (5) in case , .

Denote by the operator which acts from to as follows:

The lemma below can be proved with the help of Theorem 2 and Lemma 6.

Lemma 7. Let be a self-adjoint positive definite operator in and let , . Then the operator acts boundedly from to .

Now let us state the main result of this work.

Theorem 8. Let , , , and , , and let the inequalitybe true, where the numbers and are defined as in Theorem 5. Then (5) is regularly solvable.

Proof. Let us write (5) in the form of operator equation:where , .
By Theorem 2, (8) is regularly solvable. Make a change and rewrite (44) as follows:By Theorem 5, for every , we haveAs the operator is invertible in the space , whenwe can define by the formulaand we haveTheorem is proved.

Theorem 8 has the following.

Corollary 9. Under the conditions of Theorem 8, the operator is an isomorphism between the spaces and .

In conclusion, let us illustrate our main result by the example of a problem for a partial differential equation. Consider the following problem on the strip :where , and , are the functions bounded on . The problem (50) can be easily reduced to (5), with , , , and the operator being defined in by the equality and the conditions . In this case, . Applying Theorem 8, we obtain that if andwherethen problem (50) has a unique solution in the space .

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Copyright © 2017 Araz R. Aliev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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