Higher-Order Stationary Dispersive Equations on Bounded Intervals

Abstract

A boundary value problem for a stationary nonlinear dispersive equation of order on an interval was considered. The existence, uniqueness, and continuous dependence of a regular solution have been established.

1. Introduction

This work concerns the existence, uniqueness, and continuous dependence of regular solutions to a boundary value problem for one class of nonlinear stationary dispersive equations posed on bounded intervals, where is a positive constant. This class of stationary equations appears naturally while one wants to solve a corresponding evolution equation making use of an implicit semidiscretization scheme:where , [1]. Comparing (3) with (1), it is clear that and

For , we have the well-known generalized KdV equation and for the Kawahara equation. Initial value problems for the Kawahara equation, which had been derived in [2] as a perturbation of the Korteweg-de Vries (KdV) equation, have been considered in [3–12] and attracted attention due to various applications of those results in mechanics and physics such as dynamics of long small-amplitude waves in various media [13–15]. On the other hand, last years appeared publications on solvability of initial-boundary value problems for dispersive equations (which included the KdV and Kawahara equations) in bounded and unbounded domains [16–23]. In spite of the fact that there is not some clear physical interpretation for the problems on bounded intervals, their study is motivated by numerics [24]. The KdV and Kawahara equations have been developed for unbounded regions of wave propagations; however, if one is interested in implementing numerical schemes to calculate solutions in these regions, there arises the issue of cutting off a spatial domain approximating unbounded domains by bounded ones. In this case, some boundary conditions are needed to specify a solution. Therefore, precise mathematical analysis of mixed problems in bounded domains for dispersive equations is welcome and attracts attention of specialists in this area [16–19, 21, 25].

As a rule, simple boundary conditions at and such as for the Kawahara equation were imposed. Different kind of boundary conditions was considered in [19, 26]. Obviously, boundary conditions for (1) are the same as for (2). Because of that, study of boundary value problems for (1) helps to understand solvability of initial-boundary value problems for (2).

Last years, publications on dispersive equations of higher orders appeared [7, 9, 10, 21, 27]. Here, we propose (1) as a stationary analog of (2) because the last equation includes classical models such as the KdV and Kawahara equations.

The goal of our work is to formulate a correct boundary value problem for (1) and to prove the existence, uniqueness, and continuous dependence on perturbations of for regular solutions.

The paper has the following structure. Section 1 is Introduction. Section 2 contains formulation of the problem and main results of the article. In Section 3 we give some useful facts. Section 4 is devoted to the boundary value problem for a complete linear equation, necessary to prove in Section 5 the existence of regular solutions for the original problem. Finally, in Section 6 uniqueness is proved which provided certain restriction on as well as continuous dependence of solutions.

2. Formulation of the Problem and Main Results

For real , consider the following one-dimensional stationary higher-order equation:subject to boundary conditionswhere , , , are the derivatives of order , and is the given function.

Throughout this paper we adopt the usual notation , , and , , for the inner product and the norm in and the norm in , respectively [28]. Symbols , mean positive constants appearing during the text.

The main results of the article are the following theorem.

Theorem 1. Let . Then for fixed , problem (4)-(5) admits at least one regular solution such that with depending only on , , , and . Moreover, if and with , then the solution is uniquely defined and depends continuously on . For the uniqueness and continuous dependence are satisfied if is sufficiently small.

3. Preliminary Results

Lemma 2. For all , such that for some , one has

Proof. Let , such that . Then for any From this, the result follows immediately.

We will use the following version of the Gagliardo-Nirenberg’s inequality [29–31].

Theorem 3. For suppose belongs to and its derivatives of order belong to . Then for the derivatives the following inequalities hold: where for all (the constants , depending only on , , , , and ).

We will use the following fixed point theorem [32].

Theorem 4 (Schaefer’s fixed point theorem). Let be a real Banach Space. Suppose is a compact and continuous mapping. Assume further that the set is bounded. Then has a fixed point.

We start with the linearized version of (4), (5).

4. Linear Problem

Consider the linear equationsubject to boundary conditions (5).

Theorem 5. Let . Then problem (12)-(5) admits a unique regular solution such that with depending only on and .

Proof. Denote where is the identity matrix of order . Suppose and consider the following problem:as well as the associated homogeneous problemIt is known, [33, 34], that (15) has a unique classical solution if and only if (16)-(17) has only the trivial solution.
Let be a nontrivial solution of (16)-(17). Multiplying (16) by and integrating over , we have By integration by parts and the principle of finite induction, we calculate for all . Fixing and making use of (5), we find that Thus thereforewhich implies . Since , it follows that .
Therefore, (15) has a unique classical solution given by where is Green’s function associated with problem (16)-(17), [33, 34]. That is, with where are the coefficients of (12). The function is a unique solution to the following initial value problem: and the continuous real functions are determined by , , and (5).
We prove the following estimates.
Estimate I. Multiplying (12) by , we obtainBy Cauchy-Schwarz’s inequality, we getEstimate II. Multiplying (12) by and integrating over we haveIntegration by parts and the principle of finite induction give for all . Fixing and making use of (5), we get Therefore Applying Schwarz’s inequality on the right-hand side of (29), we concludewith depending only on and .
Estimate III. Rewriting (12) in the form we estimateFor we have and for denote and Hence we can write Then (35) becomesMaking use of (33), we getOn the other hand, for all . Hence, by Theorem 3, there are , , depending only on and , such that Making use of Young’s inequality with , , and arbitrary , we get where . Summing over and making use of (28), we findSubstituting (39), (42) into (38), we obtain Taking , we concludewhere depends only on , , and .
Again by Theorem 3, for all , there are , depending only on and such that Making use of (28), (44), we obtainTaking into account (33), (44), and (46), we conclude that andwith depending only on , , and . Uniqueness of follows from (28). In fact, such calculations must be performed for smooth solutions and the general case can be obtained via density arguments. Therefore, the proof of the Theorem 5 is complete.

5. Nonlinear Case

Given , set . Clearly, and by Lemma 2, By the Young inequality with , , and , we obtainLet be a unique solution of the linear equationsubject to boundary conditions (5). By (47), we know additionally that Let us henceforth write whenever is derived from via (50), (5). We assert that is compact and continuous.

Indeed, if is a bounded sequence in , then, in view of estimate (51), we have that sequence is bounded in . Since is compactly embedded in , then there exists a convergent in subsequence ; therefore is compact.

Similarly, let in , then there are a subsequence and a function such that in . Write (50) in the form for all . Consequently by (51), passing to the limit as , we find Thus . Hence, in implies in . This proves that is continuous.

Finally, we must show that the set is bounded in . Assume such that then that is,and satisfies boundary conditions (5).

5.1. A Priori Estimates

Estimate IV. Multiplying (57) by and integrating over , we have Integrating by parts and using boundary conditions (5), we get Hence, similar to (28), we obtain

Estimate V. Multiplying (57) by and integrating over , we have It is easy to verify that Integrating by parts, using boundary conditions (5) and Lemma 2, we get By the Young inequality, with , , and , we obtain where .

Moreover, by the Young inequality with arbitrary , we get therefore Since it follows from (28) that Taking , we concludewhere

Remark 6. Note that estimate (69) does not depend on . This estimate may be used to prove the existence of a weak solution, .

Estimate VI. Rewriting (57) in the form we estimateBy (69),Acting in the same way as we have proved (42),Substituting (73), (74) into (72), we obtain Setting , we concludewhere depends only on , , , and .

Making use of (60), (76), and Theorem 3, we obtainTaking into account (69), (76), and (77), we finally concludewith depending only on , , , and .

Applying Theorem 4, we complete the proof of the existence part of Theorem 1.

6. Uniqueness and Continuous Dependence

We separated two cases.

(1) Case . Let and be two distinct solutions of (4)-(5). Then the difference satisfies the equationand boundary conditions (5).

Multiplying (79) by and integrating over , we haveIntegrating by parts and using boundary conditions (5), we get Similarly, We reduce (80) to the inequalityFor , we have and . By Lemma 2 and estimate (69), we obtain with Therefore (83) can be rewritten asSince it follows that if , for fixed , then and consequently Hence, (86) implies and uniqueness is proved for .

To show continuous dependence of solutions of perturbations of , let such thatConsider and solutions of (4)-(5) with the right-hand sides and , respectively. Then, similar to (83), satisfies the following inequality: which can be rewritten as where Making use of (90), we obtain with . Hence implies . This proves the continuous dependence for .

(2) Case . For , problem (4)-(5) becomesLet and be two distinct solutions of (95)-(96). Then the difference satisfies the equationand boundary conditions (96).

Multiplying (97) by and integrating over , we haveIntegrating by parts and using the boundary conditions (96), we get and (98) becomes By (60), (69), it follows thatAccording to Theorem 3 and (60), (101), we estimate for Suppose , then ; therefore Hence we can rewrite (100) as follows:For a fixed assume that Then and hence (104) implies . Assuming that, for , we complete the proof of uniqueness. The continuous dependence for this case follows in the same manner as it has been done for the case provided is sufficiently small.

This completes the proof of the uniqueness and continuous dependence part of Theorem 1.

Conflicts of Interest

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

Acknowledgments

N. A. Larkin was supported by Fundação Araucária, Estado do Paraná, Brazil.