Advances in Mathematical Physics
Volume 2016 (2016), Article ID 7475061, 21 pages
http://dx.doi.org/10.1155/2016/7475061

Research Article

Mixed Initial-Boundary Value Problem for the Capillary Wave Equation

B. Juarez Campos,¹ Elena Kaikina,² and Hector F. Ruiz Paredes¹

¹Instituto Tecnologico de Morelia, Avenida Tecnologico No. 1500, Lomas de Santiaguito, 58120 Morelia, MICH, Mexico
²Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), 58089 Morelia, MICH, Mexico

Received 3 November 2015; Accepted 18 April 2016

Academic Editor: Pavel Kurasov

Copyright © 2016 B. Juarez Campos 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.

Abstract.

We study the mixed initial-boundary value problem for the capillary wave equation: , where . We prove the global in-time existence of solutions of IBV problem for nonlinear capillary equation with inhomogeneous Robin boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions.

1. Introduction

This paper is concerned with the initial-boundary value problem (IBV problem) for the nonlinear capillary wave equation with mixed (Robin) boundary conditions posed on a half-unbounded domain:

Here is a fractional derivative defined by

Mixed boundary value problems arise in a variety of applied mathematics, engineering, and physics, such as gas dynamics, nuclear physics, chemical reaction, studies of atomic structures, and atomic calculations. Therefore, the mixed problems have attracted much attention and have been studied by many authors. For detailed description of the mixed boundary conditions, see [1–3] and the references cited therein. This paper is the first attempt to investigate the inhomogeneous mixed initial-boundary value problem for the dispersive fractional nonlinear equation, considering as an example the famous capillary water wave equation (1). Fractional differential equations appear in many applications of the applied sciences, such as the fractional diffusion and wave equations [4], subdiffusion and superdiffusion equations [5], electrical systems [6], viscoelasticity theory [6], control systems [6], bioengineering [7], and finance [8]. Many articles have appeared in the literature, where fractional derivatives are used for a better description of certain material properties. Thus, for example, the fractional NLS model (1) comes from the study of the long-time behavior of solutions to the water waves equations [9]. The operator corresponds to the dispersive relation of the linearized gravity water wave equations for one-dimensional interfaces with surface tension. Furthermore, thanks to the absence of resonances at the quadratic level, one expects the nonlinear dynamics of water waves to be governed by nonlinearity of cubic type like those appearing in (1). Papers [9–11] addressed some other applications of fractional Schrödinger equations. Works concerning the Cauchy problem for fractional type Schrödinger equations, which address the existence of small solutions, and in particular the question of modified scattering, include [12–14]. In paper [15], it was shown that (1) with dispersive fractional derivative operator of order admits global solutions whose long-time behavior is not linear. Global existence results and asymptotic behavior of small solutions of Cauchy problem for capillary water wave equation were obtained in [9]. The initial-boundary problems have been much less studied than the Cauchy problems in spite of their importance. The inhomogeneous problems are often called forced ones, when an external force is applied to a system. Frequently the forcing is putted as the inhomogeneous boundary condition. In the case of the initial-boundary value problems there appear new difficulties comparing with the Cauchy problems due to the boundary. For example, in the case of the initial-boundary value problems it is not clear how many of the boundary conditions are required for the well-posedness of the initial-boundary value problem. The answer to this question relies on the construction of the Green function for the linear capillary water wave equation that is interesting on its own. Also it is necessary to take into account the boundary effects which affect the behavior of the solutions. Also usually we ask as less as possible regularity on the initial and boundary data, since the regularity of the solution implies the compatibility conditions on the initial and boundary data. Observe that for the Cauchy problem there is no such complication, in general, and we can ask more regularity on the initial data.

It is well-known that boundary value problems with homogeneous boundary conditions are easier than the corresponding homogeneous problems. However, we present in this paper remarkable results, such as global in-time existence of solutions and its large time asymptotic behavior. In book [16], it was proved that in the case of mixed problem for dissipative equations the solutions obtain more rapid time decay comparing with the case of the Cauchy problem. This phenomenon was also observed for some dispersive equations, such as intermediate-long wave and Benjamin-Ono equations, posed on the positive half-line [17, 18]. However, there are several important examples of equations whose small solutions do not behave like linear ones, as it is the case for the Schrödinger equation [19]. As we will show below, the same happens for the case of the capillary wave equation (1) with mixed boundary conditions: the cubic nonlinearity is also critical with respect to the large time decay. Theorem 1 below shows that (1) admits global solutions whose long-time behavior is not linear. In particular, a correction of logarithmic type (see (9)) is needed in order to obtain the decay and the scattering of solutions. Our approach is based on the estimates of the integral equation in the Sobolev spaces. To construct the Green operator on a half-line we adopt the analytic continuation method proposed in the paper [20], based on the Riemann theory. To get smooth solutions we use a method of the decomposition of the critical cubic nonlinearity. A major complication of IBV problem for nonlocal equation (1) on a half-line is that its symbol is nonanalytic; therefore, we can not apply the Laplace theory directly. The construction of the Green operator involves the solution of the inhomogeneous Riemann problem with discontinuous coefficients. Another difficulty is that the symbol is dispersive. To get the asymptotics of solution, we need to solve the nonlinear singular integrodifferential equation with Hilbert kernel. We believe that the method developed in this paper could be applied to a wide class of dispersive nonlinear nonlocal equations.

2. Main Result and Notation

To state the results of the present paper, we give some notations. The usual direct and inverse Laplace transformations we denote by and . The Fourier transform and the inverse Fourier transform are defined as

The usual Fourier sine transform and the Fourier cosine transform are given by

Define the “distorted” Fourier sine transform and the inverse “distorted” Fourier sine transform as follows:

where

For a detailed study of the properties of and , see below Lemma 3.

Also we introduce the Green operator on a half-line as

Let be a complex function, defined in , which obeys the Hölder condition for all finite and tends to a definite limit as . Then, Cauchy type integral constitutes a function which is analytic in the left and right complex semiplanes. Here and below these functions will be denoted by and , respectively. These functions have the limiting values and at all points of imaginary axis , on approaching the contour from the left and from the right, respectively. These limiting values are expressed by Sokhotzki-Plemelj formula (see [20] for these definitions and a discussion of related issues):

The usual Lebesgue space , where the norm if and if . Weighted Lebesgue space is , where the norm .

Weighted Sobolev space is , where . The usual Sobolev space is , so the index we omit if it does not cause a confusion. Also ,   and  .

Let , where is small. Different positive constants we denote by the same letter . We denote ,.

Our main result is as follows.

Theorem 1. Let ,  and   and the norm . Then, there exists such that for all the initial-boundary value problem (1) has a unique global solution . Furthermore, there exists a unique final state such that the large time asymptotics

is true, where , and

3. Sketch of Proof

Firstly, we construct the Green operator defined by the linear IBV problem posed on a half-line:

To derive an integral representation for the solution of problem (11), we adopt the analytic continuation method proposed in the paper [20]. The Laplace transform of the fractional operator with respect to space and time variables is

where , , and are Laplace transforms of ,, and , respectively.

Thus, since

applying the Laplace transforms to (11) with respect to space and time variables we obtain

Let be some complex function analytic in the left-half complex plane and , , such that

Thus, from (14), we obtain

Note that there are three unknown types of data, ,  , and in (16). To find some of these data we use analytic properties of , which is analytic in by the construction. Also, as we see in the left-hand side of (16), is not analytic. We define the “analyticity switching” functions . Denote . Note that for and the equality has only one root such that in . We make a cut along the negative imaginary axis. Let

We prove that for

Therefore, via Index Zero Theorem, we have

where

Via Sokhotski-Plemelj formula, we have

where

Applying (19) and (21) into (16), we obtain

For the analyticity of the function in the right-half complex plane, we need to put the following conditions:

Thus, we must put in the problem (11) only one type of boundary data. Another type of unknown boundary data can be find from relation (24). For example, if we consider Robin boundary condition, , boundary data and are completely determined by

where is Laplace transform of . We will prove that is analytic in under condition . Finally substituting (25) and (24) into (23) and applying inverse Laplace transform, we will prove that the solution (11) is represented by , where

This fact is exploited in Proposition 2 below. Therefore, the IBV problem (1) can be rewritten as

The estimates of the integral equation (27) in the Sobolev spaces yield an a priori estimate of . To get the -estimate of , we will prove the asymptotics

We introduce the new function as

Then, we obtain from (1) and (27)

In Lemma 7 below, we will prove that the nonlinearity is decomposed into the resonant term and the remainder such that

where and . Also we prove

Then, the estimates of and follow. Therefore, along with (28) and (29) we prove the -estimate of . Also we obtain asymptotics of the solution. We construct the Green operator for the initial-boundary value problem in the next section. Section 5 is devoted to several lemmas involved in the proof of the main result. In Section 6, we prove our main result.

4. Green Operator

We consider linearized version of problem (1). We prove the following proposition.

Proposition 2. Let ,  and   Then, the solution of the initial-boundary value problem (11) has the following integral representation:

where , and operators ,  and   were given by (6).

Proof. Substituting (25) and (24) into (23) and applying inverse Laplace transform we prove that the solution (11) is represented:

where

Denote

where was introduced by (20). Taking the Laplace transform of (34) with respect to and variables, we get

where

with

Now we consider given by (39). Using analytic properties of integrand functions for , we rewrite the function in the following form:

where

where

Since , we get

which implies . We make the changes of the variables and in the first and second terms of (43) correspondingly. Using that, by the definitions and , we obtain

Now we consider and given by (22). Via (19) and Cauchy Theorem taking residue in the point , we get

and by the same way

As a consequence of (47) and (48), from (35) it follows that

Substituting (49) into (46), we attain

where

By Jordan Lemma taking residue in the point , we rewrite in the form

Here

Since integrand is even function with respect to -variable, we get for the last summand of (52)

Therefore, takes the form

where

Via Lemma 11  . Therefore, we make the change of variables , to get

Also since via Lemma 3  ,

Thus, finally we obtain

where operators and are defined by (6). By the same way as in the proof of (59), we get

Applying this relation into , we obtain

Therefore, via (61) and (59), Proposition 2 follows.

5. Lemmas

Section 5 is devoted to several lemmas involved in the proof of the main result. Via Proposition 2 by Duhamel principle we have for solution of (1)

where

Firstly, we prove the main properties of the operators , and defined in (6).

Lemma 3. The following estimates are valid:

Proof. Applying Sokhotski-Plemelj formula, we get

Thus, for ,

and as a consequence

By the definition,

From (65) by Plancherel Theorem,

By the same way, we can prove . Since via (68) , after integrating by part we have

Therefore,

By the definition (6), , where

Using analytic properties of integrand function, we can change the contour of integration such that for all . Since , we have

As a consequence via Plancherel Theorem and estimates (70) and (72), we get

Since by the definition applying we get

thus via (70), (72), (75), and (76) the lemma is proved.

We introduce

Lemma 4. Let be an analytic in the right-half complex plane, except when . Then, the asymptotics

is valid for .

Proof. We have, for , ,

Indeed, by the direct calculation, it follows that

Denote . Therefore, changing , we get

Using analytic properties of function , we can change the contour of integration such that

and using

we get

from which it follows (79). We use the stationary phase method. Let .
We define such that . Note that .
Denote ,  ,  and  .
We rewrite as

where

Via (79),

Since and   after changing the variables , we get for

To estimate

we use analytic properties of the function . Since for , we can to change the contour of the integration to obtain

Therefore,

Integrating by part, we get

where

We have . Also for ,

and for

Therefore,

Collecting estimates (91) and (96) from (89), we get

Since for using analytic properties of , we can the change of the contour of the integration to get

Thus, by the same way as in the estimation of , we obtain

Finally collecting the estimates (87), (88), (97), and (99) from (85), we obtain

Lemma is proved.

We introduce

Lemma 5. The following asymptotics is valid:

Proof. By the definition (6), , where

Therefore, we rewrite in the form

where

Applying Lemma 4,

To obtain asymptotics for , we observe that along line of the integration. Therefore, applying standard Laplace method after integrating by part, we can prove

Also using decay properties of the integrand function, we prove that

The lemma is proved.

In the next lemma, we obtain asymptotics of boundary operator given by

Lemma 6. The asymptotics

are valid, provided the right-hand sides are finite.

Proof. From Lemma 5,

Thus, the lemma is proved.

In the next lemma, we prove that -transform of the nonlinearity is decomposed into the resonant and remainder terms.

Denote

Lemma 7. Let

Then, the asymptotic formula for large time holds:

Moreover,

Proof. We introduce a new function such that

Note that in view of Lemmas 4 and 6,

Making the change of variables ,  , we get

where . By the stationary phase method (see proof of Lemma 4), we get

where

Using (120), we obtain

where

From (117), we have

where by (6)

with

Denote

We rewrite in the form

where

Now we estimate .
Integrating by the part, we have , where

We have

where

Since is analytic for all except for , we can change the contour of the integration such that

Applying (133) by Young inequality, we have

Since , we have

and as a consequence

Thus, from (134),

Using the same approach, we obtain

Thus, from (137), (138), and (131), we obtain

By analogy, we can estimate and :

which imply the following estimate for :

Note that by the same way we can prove that

Now we estimate

We have

Thus,

Since by the construction , therefore we get

Therefore, from (141) and (142), it follows that

Also

Therefore,

Also we can prove

Substituting estimates (149) and (150) into (123), we get the first estimate of Lemma 7.
Now we prove the second estimate of Lemma 7. We have

Firstly, we estimate . Denote .
Via (117),

from which it follows that

Applying , we get

where via (152) and (153)

Therefore, making the change of variable , we get

where

Now we estimate . By the definition (67),

We estimate the first “more difficulty” term of . Another term can be estimated by the same way using Laplace method (see Lemma 4).
As in the proof of (131) integrating by part, we obtain

where and were defined by (124) and (130) with . By the same way as in the proof of estimate (141), we obtain

By the direct calculation,

Therefore, as in the proof of estimate (141), we get

Also we have

Applying (163) and (160) into (156), we obtain

Note that, using the same procedure, we can prove

Therefore, via (165) and (166) from (151), we obtain

For small using estimate (130) with , we get

Thus, applying (141) and (118) along with (168) and (167), we have

By the same way, we can prove

The lemma is proved.

In this lemma, we estimate the Green operator:

where

Lemma 8. The following estimate is valid:

Proof. Via Lemma 3,   and therefore

Denote by

Integrating by the part, we get for

Therefore, we rewrite operator in the form

where

where

Since , we get

Thus, via (174), we get the estimate of the lemma. The lemma is proved.

Now we estimate the nonlinear term of (62).

Lemma 9. The following estimate is valid:

Proof. From Lemma 3,  . Therefore, we have

Thus, using , we get

The lemma is proved.

Lemma 10. Let

then, the following estimate holds:

Proof. Taking residue in the point , we have

where

Using the analytic properties of the integrand by Cauchy Theorem, we can change the contour of the integration to get

Therefore,

Since via Lemma 3

from (189) we get

Also via (180),

By virtue of (186), (191), and (192) Lemma 10 is proved.

In the next lemma, we consider “analyticity switching” functions , where

Lemma 11. The following formula is valid for :

where

Moreover,

Proof. Using analytic properties of the integrand function after integrating by part and changing variables , we get

Also taking residue in the point , we get

Thus, substituting (198) into (197), we obtain

and as a consequence

Also we have (we make a cut along the negative axis)

Since for and we get

also for, ,

we get

Lemma is proved.

6. Proof of Theorem 1

Via Proposition 2 by Duhamel principle, we have for solution of (1)

where

We introduce , where

The local existence in the function space can be proved by a standard contraction mapping principle. We state it without a proof.

Theorem 12. Let ,  , and the norm . Then, there exist and such that for all the initial value problem (1) has a unique local solution with the estimate .

Let us prove that the existence time can be extended to infinity which then yields the result of Theorem 1. By contradiction, we assume that there exists a minimal time such that the a priori estimate does not hold; namely, we have .

Applying Lemmas 8 and 9 from (205) and Theorem 12, we get

From Lemma 10 and Theorem 12, we obtain

where

To estimate -norm of the solution, we find the solution in the form

Via Proposition 2, we obtain for the new function the following ordinary differential equation for depending on a parameter :

from which it follows that

By the direct calculation,

Thus, using , we get

From Lemma 7, the asymptotic formula for large time holds:

where and

Firstly, we consider the case of . Multiplying both sides of (216) by , from (215) we get

Denote

Then, from (218) we have

Therefore, from (220) we get

Now we consider the case of . We multiply both sides of (215) by to get

Via Lemma 3, we have . Therefore, integrating by part, we obtain

Also since

from (222) we obtain and as a consequence

Thus, along with (221), applying we get

Via Lemmas 4 and 6, we have

Thus, from estimate (226) along with Theorem 12 it follows that

Now we estimate . Via (212),

From estimate (116) of Lemma 7, we have

From Lemma 3 we have and therefore

Applying (230) and (231) into (229), we get

Also, we have

We have

Integrating by part, we attain for

From estimates (221) and (232) along with Theorem 12, it follows that

Via (208), (226), and (236), we get that

which implies the desired contradiction. Thus, there exists a unique global solution of (27) with the time decay estimate .

We now prove the asymptotics of solutions. Denote

where

By (215), we have

with and

Thus, we see that there exists a unique final state such that We consider the asymptotics of the phase function given by (241). By a direct calculation, we have

where . Hence,

from which it follows that there exists a unique real valued function such that

Therefore, we have the asymptotics of the phase function

where . We also have

Collecting these estimates, we find

Via Lemma 4, estimate (247) means that

where . Theorem 1 is now proved.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is partially supported by CONACYT and PAPIIT IN100114.

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