Advances in Mathematical Physics
Volume 2015 (2015), Article ID 269536, 13 pages
http://dx.doi.org/10.1155/2015/269536

Research Article

Kawahara-Burgers Equation on a Strip

N. A. Larkin

Departamento de Matemática, Universidade Estadual de Maringá, 87020-900 Maringá, PR, Brazil

Received 13 August 2015; Accepted 11 November 2015

Academic Editor: Mikhail Panfilov

Copyright © 2015 N. A. Larkin. 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.

Abstract.

An initial-boundary value problem for the 2D Kawahara-Burgers equation posed on a channel-type strip was considered. The existence and uniqueness results for regular and weak solutions in weighted spaces as well as exponential decay of small solutions without restrictions on the width of a strip were proven both for regular solutions in an elevated norm and for weak solutions in the -norm.

1. Introduction

We are concerned with an initial-boundary value problem (IBVP) for the two-dimensional Kawahara-Burgers (KB) equation

posed on a strip modeling an infinite channel . This equation is a two-dimensional analog of the Kawahara type equation

which includes dissipation and dispersion and has been studied intensively in the last years due to its applications in mechanics and physics [1–9]. The KB equation may be considered as a perturbation of the Zakharov-Kuznetsov-Burgers equation which describes the propagation of nonlinear ionic-sonic waves in a plasma submitted to a magnetic field directed along the -axis [10, 11]. Equations (1) and (2) are typical examples of the so-called dispersive equations which attract considerable attention of both pure and applied mathematicians in the past decades. The theory of the Cauchy problem for (2) and other dispersive equations like the KdV equation has been extensively studied and is considerably advanced today [1, 3–9, 12–17]. Results on IBVPs for one-dimensional dispersive equations both in bounded and in unbounded domains may be found in [4, 6, 18–22]. It was shown in [18, 19, 23–26] that the KdV and Kawahara equations have an implicit internal dissipation. This allowed the proof of exponential decay of small solutions in bounded domains without adding any artificial damping term [21, 23]. Later, this effect has been proven for a wide class of dispersive equations of any odd order with one space variable [27]. The control and stability for the linear KdV equation with the transport term may fail for critical domains, but it is possible to eliminate the term by simple scaling when the KdV and Kawahara equations are posed on the whole line. The same is true also for (1) posed on a strip [28]. Recently, interest in dispersive equations became to be extended to multidimensional models such as Kadomtsev-Petviashvili (KP), Zakharov-Kuznetsov (ZK) equations [11], and dispersive equations of higher orders [29]. As far as the ZK equation and its generalizations are concerned, the results on IVPs can be found in [30–34] and IBVPs were studied in [24, 35–39]. It was shown that IBVP for the ZK equation posed on a half-strip unbounded in direction with the Dirichlet conditions on the boundaries possesses regular solutions which decay exponentially as provided initial data are sufficiently small and the width of a half-strip is not too large [25, 38]. Similar result was established for the 2D Kawahara equation posed on a half-strip [24]. This means that a multidimensional dispersive equation may create an internal dissipative mechanism for some types of IBVPs. The goal of our paper is to prove that the KB equation on a strip also may create a dissipative effect without adding any artificial damping. We must mention that IBVP for the ZK equation on a strip has been studied in [39, 40] and IBVPs on a strip for the ZK equation and Zakharov-Kuznetsov-Burgers equation were considered in [28, 35] and for the ZK equation with some internal damping in [37]. In the domain , the term in (1) can be scaled out by a simple change of variables. Nevertheless, it can not be safely ignored for problems posed both on finite and on semi-infinite intervals as well as on infinite in direction bands without changes in the original domain [40, 41]. The main results of our paper are the existence and uniqueness of regular and weak global-in-time solutions for (1) posed on a strip with the Dirichlet boundary conditions and the exponential decay rate of these solutions as well as continuous dependence on initial data. To explore dissipativity of the term , we used exponential weight which was implied to define solutions of (1) as the product

We must mention that this idea has been proposed earlier in [12].

The paper has the following structure. Section 1 is Introduction. Section 2 contains formulation of the problem. In Section 3, we prove global existence and uniqueness theorems for regular solutions in some weighted spaces as well as continuous dependence on initial data. In Section 4, we prove exponential decay of small regular solutions in an elevated norm. In Section 5, we prove the existence, uniqueness, and continuous dependence on initial data for weak solutions as well as the exponential decay rate of the -norm for small solutions without limitations on the width of the strip.

2. Problem and Preliminaries

Let be finite positive numbers. Define ;; and

Hereafter subscripts , , and so forth denote the partial derivatives, as well as or when it is convenient. Operators and are the gradient and Laplacian acting over . By and we denote the inner product and the norm in , and stands for norms in the -based Sobolev spaces. We will use also the spaces , where ; see [12].

Consider the following IBVP:

3. Existence of Regular Solutions

Approximate Solutions. We will construct solutions to (4)–(6) by the Faedo-Galerkin method: the functions

where , are orthonormal in eigenfunctions of the following Dirichlet problem:

Define approximate solutions of (4)–(6) as follows:

where are solutions to the following Cauchy problem for the system of generalized Kawahara equations:

It can be shown that, for , the Cauchy problem (10) and (11) has a unique regular solution [1, 8, 12, 16]. To prove the existence of global solutions for (4)–(6), we will establish uniform in global in estimates of approximate solutions .

Estimate 1. Multiply the th equation of (10) by , sum up over , and integrate the result with respect to over to obtain

which implies

It follows from here that for sufficiently large and

In our calculations we will drop the index where it is not ambiguous.

Estimate 2. For some positive , multiply the th equation of (10) by , sum up over , and integrate the result with respect to over . Dropping the index , we get

In our calculations, we will frequently use the following multiplicative inequalities [42].

Proposition 1. (i) For all ,

(ii) For all ,

where the constant depends on a way of continuation of as such that .

Extending for a fixed into the exterior of by 0 and exploiting (16), we find

Substituting this into (15) and making use of the inequality

with an appropriate , when , we come to the inequality

By the Gronwall lemma,

Returning to (20) gives

It follows from this estimate and (13) that uniformly in and for any and

where does not depend on .

Estimates (22), (23) make it possible to prove the existence of a weak solution to (4)–(6) passing to the limit in (10) as . For details of passing to the limit in the nonlinear term, see [12].

We will need the following lemma.

Lemma 2. Let be such that

and for all there is some such that . Then

where are arbitrary positive numbers.

Proof. Denote . Then simple calculations give

Returning to the function , we prove Lemma 2.

Estimate 3. Multiplying the th equation of (10) by , and dropping the index , we come to the equality

Making use of Proposition 1, we estimate

Similarly,

Substituting into (27) with , we obtain

Estimate 4. Multiplying the th equation of (10) by , and dropping the index , we come to the equality

Making use of Proposition 1, we estimate

Taking , we transform (31) into the inequality

Making use of (14) and the Gronwall lemma, we get

This and (30) give

which imply that for all finite and all

Estimate 5. Multiplying the th equation of (10) by , and dropping the index , we come to the equality

Using (16), we estimate

Taking and substituting into (37), we obtain

Making use of (14), we find

Estimate 6. Differentiate (10) by and multiply the result by to obtain

Making use of (16), we estimate

Taking and substituting into (41), we get

This implies

where

Estimate 7. Multiplying the th equation of (10) by and dropping the index , we come to the equality

Using (16), we estimate

Taking , using (30)–(44) and substituting into (46), we get

Estimate 8. Multiplying the th equation of (10) by , we come, dropping the index , to the equality

Using Lemma 2 and (14), we estimate

Taking sufficiently small and positive and substituting into (49), we find

Consequently, it follows from the equalities

that

Jointly, estimates (30), (34), (35), (48), (51), and (53) read

In other words,

and these inclusions are uniform in .

Estimate 9. Multiplying the th equation of (10) by , we come, dropping the index , to the equality

We estimate

Choosing sufficiently small and positive, after integration, we transform (56) into the form

Acting similarly, we get from the scalar product

the estimate

Estimate 10. Differentiate the th equation of (10) two times with respect to and multiply the result by to get the equation

which in turn, dropping the index , can be rewritten in the form

Estimates (54), (55), (58), and (60) guarantee that

and inclusions do not depend on . These estimates show the smoothing effect of the Kawahara operator first observed for the KdV equation in [12] and for dispersive equations of any finite order in [27]. Independence of (14), (63) of allows us to pass to the limit in (10) and to prove the following result.

Theorem 3. Let be such that and for some satisfying the following inequality:

Then there exists a regular solution to (4)–(6), :

which for satisfies the identity

where is an arbitrary function from

Proof. Rewrite (10) in the form

where is an arbitrary function from the set of linear combinations and is an arbitrary function from . Taking into account estimates (14), (63) and fixing , we can easily pass to the limit as in linear terms of (67). To pass to the limit in the nonlinear term, we must use (36) and repeat arguments of [12]. Since linear combinations are dense in , we come to (66). Rewriting it in the form

and making use of (63), one can see that . The proof of Theorem 3 is complete.

Remark 4. Estimates (14), (63) are valid also for the limit function and (14) obtains its sharp form:

Uniqueness of a Regular Solution

Theorem 5. The regular solution from Theorem 3 is uniquely defined.

Proof. Let be two distinct regular solutions of (4)–(6); then satisfies the following initial-boundary value problem:

Multiplying (70) by , we get

We estimate

Substituting into (73) and taking sufficiently small, we find

Since

then, by the Gronwall lemma,

Hence, in

Remark 6. Changing initial condition (72) for , and repeating the proof of Theorem 5, we obtain from (75) that

This means continuous dependence of regular solutions on initial data.

4. Decay of Regular Solutions

In this section we will prove exponential decay of regular solutions in an elevated weighted norm. We start with the following theorem which is crucial for the main result.

Theorem 7. Let and be a regular solution of (4)–(6). Then for all finite the following inequalities are true:

where

Proof. Multiplying (4) by , we get the equality

Taking into account (16), we estimate

The following proposition is principal for our proof.

Proposition 8. The following inequality is true:

Proof. Since , fixing , we can use with respect to the following Steklov inequality: if , then

After a corresponding process of scaling we prove Proposition 8.

Making use of (84) and the fact that implies and substituting into (82), we come to the following inequality:

which can be rewritten as

where

Since we need , define

where . It implies with .

It is easy to see that

Solving (89), we find

and from (87) we get

The last inequality implies (79).

To prove (80), we return to (82) and multiply it by to obtain

We find

Substituting this into (93), we get

Integration and (79) imply

The proof of Theorem 7 is complete.

Observe that, differently from [24, 25, 38], we do not have any restrictions on the width of a strip .

The main result of this section is the following assertion.

Theorem 9. Let all the conditions of Theorem 7 be fulfilled. Then regular solutions of (4)–(6) satisfy the following inequality:

or

Proof. We start with the following lemma.

Lemma 10. Regular solutions of (4)–(6) satisfy the following equality:

Proof. First we transform the scalar product

into the following equality:

Multiplying this by and integrating, we prove (99).

Making use of Lemma 2, estimate separate terms in (99) as follows:

Choosing sufficiently small, substituting into (99), and taking into account (79), we prove that

Adding (79), we complete the proof of Theorem 9.

5. Weak Solutions

Here we will prove the existence and uniqueness and continuous dependence on initial data as well as exponential decay results for weak solutions of (4)–(6) when the initial function .

Theorem 11. Let . Then for all finite positive and there exists at least one function :

such that

and the following integral identity takes place:

where is an arbitrary function.

Proof. In order to justify our calculations, we must operate with sufficiently smooth solutions . With this purpose, we consider first initial functions , which satisfy conditions of Theorem 3, and obtain estimates (14), (36) for functions . This allows us to pass to the limit as in the identity

and come to (106).

Uniqueness of a Weak Solution

Theorem 12. A weak solution of Theorem 11 is uniquely defined.

Proof. Actually, this proof is provided by Theorem 5. It is sufficient to approximate the initial function by regular functions in the form:

where satisfies the conditions of Theorem 3. This guarantees the existence of the unique regular solution to (4)–(6) and allows us to repeat all the calculations which have been done during the proof of Theorem 5 and to come to the following inequality:

By the generalized Gronwall’s lemma,

Functions and for sufficiently large satisfy the estimate

Hence,

Since is a weak limit of regular solutions , then

This implies in The proof of Theorem 12 is complete.

Remark 13. Changing initial condition for , and repeating the proof of Theorem 12, we obtain that

This means continuous dependence of weak solutions on initial data.

Decay of Weak Solutions

Theorem 14. Let , and be a regular solution of (4)–(6). Then for all finite the following inequality is true:

where

Proof. Similar to the proof of the uniqueness result for a weak solution, we approximate by sufficiently smooth functions in order to work with regular solutions. Acting in the same manner as by the proof of Theorem 7, we come to the following inequality:

where

Since is a weak limit of regular solutions , then

The proof of Theorem 14 is complete.

We have in this theorem a more strict condition instead of in the case of decay for regular solution because for weak solutions we do not have the sharp estimate (69), but only (14).

Remark 15. We have established our results on the existence and uniqueness of regular and weak solutions as well as their exponential decay without restrictions on the width due to absence of the travel term . Of course, the presence of this term will cause restrictions on as in [38].

Remark 16. The rate of exponential decay in Theorem 7 and Theorem 14 is optimal for the fixed :

where

Remark 17. The results established in Theorems 7 and 14 show that a hot plasma simulated by system (4)–(6) can be stabilized by suitable boundary conditions provided perturbations of the initial state are sufficiently small.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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