Higher-order stationary dispersive equations on bounded intervals: a relation between the order of an equation and the growth of its convective term

A boundary value problem for a stationary nonlinear dispersive equation of order $2l+1\;\; l\in \mathbb{N}$ with a convective term in the form $u^ku_x\;\; k\in \mathbb{N}$ was considered on an interval $(0,L)$. The existence, uniqueness and continuous dependence of a regular solution as well as a relation between $l$ and critical values of $k$ have been established.


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 au + l j=1 (−1) j+1 D 2j+1 x u + u k u x = f (x), l, k ∈ N, where a is a positive constant. This class of stationary equations appears naturally while one wants to solve the corresponding evolution equation making use of an implicit semi-discretization scheme: where h > 0, [37]. Comparing (1.3) with (1.1), it is clear that a = 1 h > 0 and f (x) = u n−1 h . The case k = 1 has been studied in [27]. For l = 1, we have the well-known generalized Korteweg-de Vries (KdV) equation which has been studied intensively for critical and supercritical values of k. In [12,29,30,31] it was proved that a supercritical equation does not have global solutions and a critical one has a global solution for "small" initial data and the right-hand side. For l = 2, k = 2 the generalized Kawahara equation has been studied in [2]. Initial value problems for the Kawahara equation, l = 2, which had been derived in [19] as a perturbation of the KdV equation, have been considered in [3,8,12,14,16,18,20,21,34,35] 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 [13,15,17]. On the other hand, last years appeared publications on solvability of initial-boundary value problems for various dispersive equations (which included the KdV and Kawahara equations) in bounded and unbounded domains [2,4,5,7,11,22,23,26,27,28]. 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 [6]. 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 [2,4,5,7,9,11,26].
As a rule, simple boundary conditions at x = 0 and x = 1 such as u = u x = 0| x=0 , u = u x = u xx = 0| x=1 for the Kawahara equation were imposed. Different kind of boundary conditions was considered in [7,25]. Obviously, boundary conditions for (1.1) are the same as for (1.2). Because of that, study of boundary value problems for (1.1) helps to understand solvability of initial-boundary value problems for (1.2).
Last years, publications on dispersive equations of higher orders appeared [11,14,20,21,36]. Here, we propose (1.1) as a stationary analog of (1.2) because the last equation includes classical models such as the generalized KdV and Kawahara equations.
The goal of our work is to formulate a correct boundary value problem for (1.1) and to prove the existence, uniqueness and continuous dependence on perturbations of f (x) for regular solutions as well as to study relation between the term l of equation and the critical values of k.
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. In Section 4 the existence of a regular solutions for the problem is proved. Here, a connection between the order of the equation and the growth of its convective term is established. Finally, in Section 5 uniqueness is proved provided certain restriction on f as well as continuous dependence of solutions.

Formulation of the Problem and Main Results
For real a > 0, consider the following one-dimensional stationary higher order equation: subject to boundary conditions: where 0 < L < ∞, l, k ∈ N with 1 ≤ k ≤ 4l, D i = d i /dx i , D 1 ≡ D are the derivatives of order i ∈ N, and f ∈ L 2 (0, L) is the given function. Throughout this paper we adopt the usual notation (·, ·) for the inner product in L 2 (0, L) and · , · ∞ and · H i , i ∈ N for the norm in L 2 (0, L), L ∞ (0, L) and H i (0, L), respectively [1]. Symbols C * , C 0 , C i , K i , i ∈ N, mean positive constants appearing during the text. Definition 2.1. For a fixed l ∈ N, equation (2.1) is a regular one for k < 4l and is critical when k = 4l.
The main results of this article are the following theorems: with the constant C depending only on L, l, k, a and ((1 + x), f 2 ).
In the critical case, k = 4l, let f be such that with C * an absolute constant. Then problem (2.1)-(2.2) admits at least one regular solution u ∈ H 2l+1 (0, L) such that with the constant C ′ depending only on L, l, a and ((1 + x), f 2 ). Theorem 2.3. Let l, k ∈ N 1 ≤ k ≤ 4l and let ((1 + x), f 2 ) be sufficiently small. Then the solution from Theorem 2.2 is unique and continuously depends on perturbations of f .

Preliminary Results
From this, (3.1) follows immediately.
Theorem 3.2. Let u belong to H l 0 (0, L), then the following inequality holds: with C * an absolute constant.
is bounded. Then B has a fixed point.
We start with the linearized version of (2.1) subject to boundary conditions (2.2).
with the constant C 0 depending only on L and a.
By Theorem 4.1, let w ∈ H 2l+1 (0, L) be a unique solution of the linear equation subject to boundary conditions (2.2). By (4.2)-(4.3), We will write henceforth Bu = w whenever w is derived from u via and the boundary conditions (2.2). Multiplying (4.6) by v n and integrating by parts over (0, L), we obtain According to (3.1), On the other hand, let g ∈ C 1 (R) be such that g(y) = y k . By the Mean Value Theorem, for arbitrary y, z ∈ R there is ξ ∈ (y, z) such that , we conclude that v n → 0. Multiplying (4.6) by (1 + x)v n and integrating over (0, L), we obtain Integrating by parts and using (2.2) it follow that and and u satisfies the boundary conditions (2.2).
Returning to (4.13), we find we transform (4.12) as follows For fixed l, a and f ∈ L 2 (0, L) such that Retunrning to (4.9) and acting as in the regular case with (4.23), we conclude (2.5), that is with C ′ depending only on L, l, a and ((1 + x), f 2 ).
Applying Theorem 3.4, we complete the proof of the Theorem 2.2.
We separated two cases: l ≥ 2 and l = 1. For l ≥ 2, let u 1 and u 2 be two distinct solutions of (2.1)-(2.2). Then the difference w = u 1 − u 2 satisfies the equation and the boundary conditions (2.2). Multiplying (5.1) by w and integrating over (0, L), we obtain a w 2 + 1 2 (D l w(0)) 2 + (u k 1 Dw, w) Integrating by parts and using (2.2),(3.1), we get By (3.1),(4.8), we have Substituting I 1 , I 2 into (5.2), we reduce it to the inequality Making use of (4.15), we can estimate (5.3) as where with β = min{ a 2 , 1} and C 3 depending only on l, k and a. For fixed l, k and a, assume that Hence (5.4) implies w = 0 and uniqueness is proved for l ≥ 2 and 1 ≤ k < 4l.
Rewrite (5.3) in the form: Making use of (4.23), we obtain For fixed l and a, suppose that Thus w = 0 and uniqueness is proved for l ≥ 2 and k = 4l. The case l = 1.
By (4.11), (4.19), Returning to (5.11) and using (4.15), we find For fixed k and a assume that This implies w = 0 and uniqueness is proved for l = 1 and k < 4.
This completes the proof of the uniqueness part of Theorem 2.3.