Mixed Initial-Boundary Value Problem for the Capillary Wave Equation

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.


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: 3/2  = || 2 ,  > 0,  > 0; (, 0) =  0 () ,  > 0,  (0, ) +   (0, ) = ℎ () ,  > 0. ( Here |  | 3/2 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][2][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 |  | 3/2 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][10][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][13][14].In paper [15], it was shown that (1) with dispersive fractional derivative operator of order 1/2 admits global 2 Advances in Mathematical Physics 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 longtime behavior is not linear.In particular, a correction of logarithmic type (see (9)) is needed in order to obtain the  −1/2 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 halfline 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 () = || 3/2 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.

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 L and L −1 .The Fourier transform F and the inverse Fourier transform F −1 are defined as The usual Fourier sine transform F  and the Fourier cosine transform F  are given by Define the "distorted" Fourier sine transform K  and the inverse "distorted" Fourier sine transform K *  as follows: where For a detailed study of the properties of K   and K *  φ, see below Lemma 3.
Also we introduce the Green operator on a half-line as Advances in Mathematical Physics 3 Let () be a complex function, defined in Re  = 0, which obeys the Hölder condition for all finite  and tends to a definite limit (∞) as  → ∞.Then, Cauchy type integral () = (1/2) ∫ ∞ −∞ (()/( − ))  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 Re  = 0, 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 L  = { ∈ S  ; ‖‖ L  < ∞}, where the norm ,  > 0 is small.Different positive constants we denote by the same letter .We denote ⟨⟩ = 1+, {} = /⟨⟩.
Our main result is as follows.

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

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Now we consider (, −()) and (, −()) given by ( 22).Via (19) and Cauchy Theorem taking residue in the point  = 0, 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  1 in the form Here Since integrand is even function with respect to -variable, we get for the last summand of (52) Therefore, (, , ) =  1 (, , ) +  2 (, , ) takes the form Advances in Mathematical Physics 7 where . Therefore, we make the change of variables  → ||,  > 0, to get ) .
(57) Also since via Lemma 3 Thus, finally we obtain where operators K *  and K  are defined by (6).By the same way as in the proof of (59), we get Therefore, via (61) and (59), Proposition 2 follows.

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 K  , K  and K *  , K *  defined in (6).(64) Proof.Applying Sokhotski-Plemelj formula, we get Advances in Mathematical Physics Thus, for  > 0, and as a consequence By the definition, Ψ  (, )  () ; From (65) by Plancherel Theorem, Therefore, By the definition (6), Using analytic properties of integrand function, we can change the contour of integration such that Re  < 0 for all  ∈ C.
We introduce is valid for  > 0.
We introduce Lemma 5.The following asymptotics is valid: Advances in Mathematical Physics 11 Therefore, we rewrite K *   −() φ in the form where To obtain asymptotics for = 0 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 Hℎ given by Thus, the lemma is proved.
In the next lemma, we prove that K  -transform of the nonlinearity is decomposed into the resonant and remainder terms. Denote Then, the asymptotic formula for large time  holds:  Making the change of variables   () = ,  ∈ (0, +∞), we get where () = −()+  ().By the stationary phase method (see proof of Lemma 4), we get where (121) Using (120), we obtain where From (117), we have where by ( 6) Denote We rewrite () in the form where Now we estimate  1 ().
Now we prove the second estimate of Lemma 7. We have Firstly, we estimate Via (117), from which it follows that Applying Ψ  (0, ) = 0, we get where via (152) and ( 153) Therefore, making the change of variable  =   (), we get where Now we estimate ‖⟨⟩ −1    1 ()‖.By the definition (67), We estimate the first "more difficulty" term  +Γ(−) of Ψ  (, ).Another term can be estimated by the same way using Laplace method (see Lemma 4).
In this lemma, we estimate the Green operator: where Proof.Taking residue in the point  = 0, we have where Using the analytic properties of the integrand by Cauchy Theorem, we can change the contour of the integration to get By virtue of ( 186), (191), and (192) Lemma 10 is proved.
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) also for, || < 1, we get       Γ()      ≤ .
Lemma is proved.

Proof of Theorem 1
Via Proposition 2 by Duhamel principle, we have for solution of ( 1) where The local existence in the function space X  can be proved by a standard contraction mapping principle.We state it without a proof.
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  > 0 such that the a priori estimate ‖‖ X  < √ does not hold; namely, we have ‖‖ X  ≤ √.