Initial-Boundary Value Problems for Nonlinear Dispersive Equations of Higher Orders Posed on Bounded Intervals with General Boundary Conditions

: The present article concerns general mixed problems for nonlinear dispersive equations of any odd-orders posed on bounded intervals. The results on existence, uniqueness and exponential decay of solutions are presented.


Introduction
In this work, we formulate mixed problems with general boundary conditions for the following dispersive equation: where L is an arbitrary real positive number and l ∈ N. We propose Equation (1) because it includes classical models such as the Korteweg-de Vries (KdV) equation, when l = 1 [1][2][3][4] and the Kawahara equation, when l = 2 [5][6][7][8]. Dispersive equations posed on bounded and unbounded intervals with the Dirichlet type boundary conditions were studied in [9][10][11][12][13][14][15][16][17][18][19]. It is known that the KdV and Kawahara equations were deduced on the whole real line, however, approximating the line either by bounded or unbounded intervals, one needs to consider initial-boundary value problems posed either on finite or semi-finite intervals [2,4,[9][10][11][13][14][15][17][18][19][20][21][22]. Last years, publications on dispersive equations of higher orders appeared [14,16,[23][24][25][26]. Usually, Dirichlet conditions such as D i x u(t, 0) = D i x u(t, L) = D l x u(t, L) = 0, i = 0, . . . , l − 1; t > 0 were imposed for Equation (1), see [25,26]. In [27], general mixed problems for linear multi-dimensional (2b + 1)-hyperbolic equations were studied by means of functional analisys methods. In [28], we have studied boundary value problems for the following linear stationary dispersive equations on bounded intervals subject to general boundary conditions at the endpoints of intervals: where λ > 0 and f is a given function. Equation (2) appears while solving Equation (1) making use of either the semigroup theory or semi-discrete approaches [13]. We formulate well posed initial-boundary value problems to Equation (1) imposing the same boundary conditions as for Equation (2) [28]. Our goal is to prove the existence, uniqueness of local and global regular solutions for the formulated problems as well as exponential decay for small initial data. This article has the following structure: Section 2 contains notations and preliminaries. In Section 3, we formulate the initial-boundary value problems. In Section 4, we prove local existence and uniqueness of regular solutions as well as a "smoothing effect" of them similar to one established in [29] for the initial problem of the KdV equation. In Section 5, the global existence and uniqueness of regular solutions have been established for arbitrary initial data. In Section 6, the existence and uniqueness of small global regular solutions as well as their exponential decay have been established. Section 7 is a conclusion.

Notations and Auxiliary Facts
denote the partial derivatives of order i. By · ∞ we denote the norm in L ∞ (0, L). In what follows, we denote by (·, ·) and · as the inner product and the norm in L 2 (0, L) and · H m , m ∈ N stands for the norm in L 2 -based Sobolev spaces [30].
Lemma 1 (See [26], Lemma 2.2). Let u belong to H 1 0 (0, L), then the following inequality holds: Lemma 2 (See [31], p. 125). Suppose u and D m u, m ∈ N belong to L 2 (0, L). Then for the derivatives D i u, 0 ≤ i < m, the following inequality holds: where C 1 , C 2 are constants depending only on L, m, i.

Formulation of the Problem
Consider the following evolution equation: subject to initial data where u 0 is a given function. In [28], formulation of boundary value problems for the stationary linear equation Equation (2) on the interval (0, L) has been proposed. In the present work, we will use the same formulation for Equations (6) and (7): l = 1: where a ij , b ij are real constants. Assumptions on the coefficients imply that the L 2 -norm of the solutions of Equation (6) is decreasing. Multiplying Equation (6) by u and integrating over (0, L), we get Making use of integration by parts, finite induction and Young's inequality, we prove that the coefficients a ij , b ij satisfy the following conditions, see [28]: For l = 2: This implies that b 31 > 1 2 , a 31 < 1 2 , and |a 32 |, |b 21 | should be sufficiently small or zero. For l = 3: This implies that b 51 < − 1 2 , b 42 > 1 2 , a 51 > 1 2 , a 42 < 1 2 and the remaining coefficients in Inequality (13) should be sufficiently small or zero.
For l ≥ 4: It follows that and the remaining coefficients of the Inequality (14) should be sufficiently small or zero. Assuming these coefficients equal to zero in Inequalities (12)- (14), we get the following boundary conditions for all l ∈ N, [28]: , a 51 > 1 2 , a 42 < 1 2 for l = 3 and Inequality (15) for l ≥ 4.

Remark 2.
In this work, we will study the case l ≥ 2. For the case l = 1 see [26].

Lemma 4.
There is a real T * > 0 such that P(B R ) ⊂ B R .

Lemma 5.
There is a real T > 0 such that the mapping P is a contraction in B R . 0)) 2 ] + (D l z(t, 0)) 2 . Then z satisfies the equation boundary conditions Equation (16) and initial data z(0, ·) ≡ 0. Similar arguments used in the proof of Lemma 4 show that z E ≤ 1 2 w E . Therefore, P is a contraction in B R .
The existence part of Theorem 4 is proved.
Uniqueness part of Theorem 4 is thereby proved.
Similar arguments used in the proof of Lemma 6 with Inequality (77) instead of Inequality (59), show the uniqueness of the solution. The proof of Theorem 5 is complete.

Conclusions
Making use of the formulation of a linear stationary version of Equation (1) in [28], we prove in Theorem 3 local existence and uniqueness of regular solutions. In Theorem 4, we prove global in t ∈ (0, T) existence and uniqueness of regular solution for arbitrary smooth initial data and arbitrary T > 0. In Theorem 5, we prove global in t ∈ (0, +∞) existence and uniqueness of regular solutions as well as their exponential decay of u (t), u t (t) and u H 2l+1 (t) for small initial data. A smoothing effect has been established: if u 0 ∈ H 2l+1 (0, L), then u ∈ L 2 ((0, +∞); H (2l+1)+l (0, L)). Our results can be used for constructing of numerical schemes while studying various models of initial-boundary value problems for higher-order dispersive equations.