\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 01, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2010/01\hfil Initial-boundary value problems]
{Initial-boundary value problems for quasilinear dispersive
equations posed on a bounded interval}

\author[A. V. Faminskii, N. A. Larkin\hfil EJDE-2010/01\hfilneg]
{Andrei V. Faminskii, Nikolai A. Larkin}  % in alphabetical order

\address{Andrei V. Faminskii \newline
Department of Mathematics, Peoples Friendship University of
Russia, Miklukho-Maklai str. 6, Moscow, 117198, Russia}
\email{andrei\_faminskii@mail.ru}

\address{Nikolai A. Larkin \newline
Departamento de Matem\'atica, Universidade Estadual de Maring\'a,
Av. Colombo 5790: Ag\^encia UEM, 87020-900, Maring\'a, PR, Brazil}
\email{nlarkine@uem.br}


\thanks{Submitted October 2, 2009. Published January 5, 2010.}
\thanks{A. V. Faminskii was supported by  grant 06-01-00253 from RFBR}
\subjclass[2000]{35M20, 35Q72}
\keywords{Nonlinear boundary value problems; odd-order
differential equations; \hfill\break\indent existence and uniqueness}

\begin{abstract}
 This paper studies nonhomogeneous initial-boundary value problems
 for quasilinear one-dimensional odd-order equations posed on a
 bounded interval. For reasonable initial and boundary conditions we
 prove existence and uniqueness of global weak and regular solutions.
 Also we show the exponential decay of the obtained solution with
 zero boundary conditions and right-hand side, and small initial data.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}

This work  concerns global well-posedness of nonhomogeneous
initial-boundary value problems for general odd-order quasilinear
partial differential equations
\begin{equation}\label{e1.1}
u_t+(-1)^{l+1}\partial_x^{2l+1}u+\sum_{j=0}^{2l}a_{j}\partial^j_x u
+uu_x=f(t,x),
\end{equation}
where $l\in \mathbb{N}$, $a_j$ are real constants. This class of equations
includes well-known Korteweg-de Vries and Kawahara equations which
model the dynamics of long small-amplitude waves in various media
\cite{benjamin,kawahara,topper}.

Our study is motivated by physics and numerics and our main goal
is to formulate a correct nonhomogeneous initial-boundary value
problem for \eqref{e1.1} in a bounded interval and to prove the
existence and uniqueness of global in time weak and regular
solutions in a large scale of Sobolev spaces as well as to study
decay of solutions while $t\rightarrow \infty$. Dispersive
equations such as 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 occasion some boundary conditions are needed to specify
the solution. Therefore, precise mathematical analysis of boundary
value problems in bounded domains for general dispersive equations
is welcome and attracts attention of specialists in the area of
dispersive equations, especially KdV and BBM equations,
\cite{gold1,bona,bona3,bona4,boutet,bubnov,bubnov1,cattabriga,colin,colin2,doronin,dzhuraev,
 faminski3,faminski0,faminski,faminski1,
gold2,gold3,goubet,holmer,khablov,kozhanov,larkin,larkin2,pyatkov,zhang,zhang2}.
Cauchy problem for dispersive equations of high orders was
successfully explored by various authors,
\cite{biagioni,cui,cui2,faminski3,kenig,saut,wang}. On the other
hand, we  know few published results on initial-boundary value
problems posed on a finite interval for general nonlinear odd-order
dispersive equations, such as the Kawahara equation, see
\cite{doronin2,doronin3,larkin3}.

In this paper we study an initial-boundary value problem for
 \eqref{e1.1} in a rectangle $Q_T=(0,T)\times (0,1)$ with
initial data
\begin{equation}\label{e1.2}
u(0,x)=u_0(x),\quad x\in(0,1),
\end{equation}
and boundary data
\begin{gather}\label{e1.3}
\partial_x^j u(t,0) =\mu_j(t),\quad j=0,\dots,l-1,\\
\label{e1.4} \partial_x^j u(t,1) =\nu_j(t),\quad j=0,\dots,l,\;
t\in(0,T).
\end{gather}
Well-posedness of such a problem for a linearized version of
\eqref{e1.1} with homogeneous initial and boundary data
\eqref{e1.2}--\eqref{e1.4} was established in \cite{ramazanov}. It
should be noted that imposed boundary conditions are reasonable at
least from mathematical point of view, see comments in
\cite{doronin2}.

The theory of global solvability of dispersive equations is based
on conservation laws, the first one -- in $L^2$. Let $u(t,x)$ be
a sufficiently smooth and decaying while $|x|\to \infty$ solution
of an initial value problem for \eqref{e1.1} (where $a_{2j}=0$,
$f\equiv 0$), then
\[
\int_{\mathbb{R}} u^2\,dx=\text{const}.
\]
The analogous equality can be written for problem
\eqref{e1.1}--\eqref{e1.4} in the case of zero boundary data. In
the general case one has to make this data zero with the help of a
certain auxiliary function. In the present paper we construct a
solution of an initial-boundary value problem for the linear
homogeneous equation
\begin{equation}\label{e1.5}
u_t+(-1)^{l+1}\partial_x^{2l+1}u=0
\end{equation}
with the same initial and boundary data \eqref{e1.2}--\eqref{e1.4}
and use it as such an auxiliary function. This idea gives us an
opportunity to establish our existence results for \eqref{e1.1}
under natural assumptions on boundary data (see Remark~\ref{R2.3}
below).

Another important fact is extra smoothing of solutions in comparison
with initial data. In a finite domain it was first established for
the KdV equation in \cite{khablov,bubnov,bubnov1}  based on
multiplication of the equation by $(1+x) u$ and consequent integration.
In our case, we also have an extra smoothing effect. Roughly
speaking, if $u_0 \in H^{(2l+1)k}(0,1)$, then $u\in
L^2(0,T;H^{(2l+1)k+l}(0,1))$.

 It has been shown in \cite{larkin,larkin2}
that the KdV equation is implicitly dissipative. This means that for
small initial data the energy decays exponentially as $t\to +\infty$
without any additional damping terms in the equation. Moreover, the
energy decays even for the modified KdV equation with a linear
source term, \cite{larkin2}. In the present paper we prove that this
phenomenon takes  place for general dispersive equations of
odd-orders.

 The paper has the following structure. Section 2
contains main notations and definitions. The main results of the
paper on well-posedness of the considered problem are also
formulated in this section. In Section 3 we study the aforementioned
initial-boundary value problem for equation \eqref{e1.5}. Section 4
is devoted to the corresponding problem for a complete linear
equation. In Section 5 local well-posedness of the original problem
is established. Section 6 contains global a priori estimates.
Finally, the decay of small solutions, while $t\to +\infty$, is
studied in Section 7.

\section{Notation and statement of main results}

For any space of functions, defined on the interval $(0,1)$, we omit
the symbol $(0,1)$, for example, $L^p=L^p(0,1)$, $H^k=H^k(0,1)$,
$C_0^\infty = C_0^\infty(0,1)$ etc.
Define linear differential operators in $L^2$ with constant
coefficients
\[
P_0\equiv \sum_{j=0}^{2l}a_{j}\partial^j_x,\quad
P\equiv (-1)^{l+1}\partial_x^{2l+1} +P_0.
\]
The main assumption on $P_0$ is the following.

\begin{definition}\label{D2.1} \rm
We say that the operator $P_0$ satisfies Assumption A if
either
\[
(-1)^ja_{2j}\geq 0, \quad j=1,\dots,l,
\]
or there is a natural number $m\leq l$ such, that
\[
(-1)^m a_{2m}>0\quad \text{and }\quad
a_{2j}=0, \quad j=m+1,\dots,l.
\]
\end{definition}

\begin{lemma}\label{L2.1}
Assumption A is equivalent to the following property:
There exists a constant $c_0\geq 0$ such that for any function $\varphi\in
H^{2l+1}$, $\varphi(0)=\dots=\varphi^{(l-1)}(0)=0$,
$\varphi(1)=\dots=\varphi^{(l-1)}(1)=0$,
\begin{equation}\label{e2.1}
(P_0\varphi,\varphi)\geq -c_0\|\varphi\|^2_{L^2}
\end{equation}
(here and further $(\cdot,\cdot)$ denotes scalar product in $L^2$).
\end{lemma}

\begin{proof}
Sufficiency of Assumption A is obvious (in the second case by virtue of
the Ehrling inequality, \cite{adams}).

In order to prove necessity, assume that there exists a natural
$m\leq l$ such that $a_{2j}=0, \quad j=m+1,\dots,l$.
Consider a set of functions
\[
\varphi_\lambda(x)\equiv \lambda^{1/2-m}\varphi(\lambda x)
\]
for certain $\varphi\in C_0^\infty$, $\varphi\not\equiv 0$, and
$\lambda\geq 1$ and write
down \eqref{e2.1} for $\varphi_\lambda$:
\[
(P_0\varphi_\lambda,\varphi_\lambda)=
\sum_{j=0}^m(-1)^ja_{2j}\lambda^{2(j-m)}
\|\varphi^{(j)}\|^2_{L^2}\geq -c_0\lambda^{-2m}\|\varphi\|^2_{L^2}.
\]
It follows for $\lambda\to +\infty$ that $(-1)^m a_{2m}\geq 0$.
\end{proof}

\begin{lemma}\label{L2.2}
If the operator $P_0$ satisfies Assumption A, then
for any function $\varphi$ as in Lemma~\ref{L2.1}
\begin{equation}\label{e2.3}
(P\varphi,\varphi)\geq -c_0\|\varphi\|^2_{L^2}-
\frac 12 \bigl(\varphi^{(l)}(1)\bigr)^2.
\end{equation}
Moreover, for certain positive constants $c_1$, $c_2$
\begin{equation}\label{e2.4}
(P\varphi,(1+x)\varphi)\geq c_1\|\varphi^{(l)}\|^2_{L^2}-
c_2\|\varphi\|^2_{L^2}-\bigl(\varphi^{(l)}(1)\bigr)^2.
\end{equation}
\end{lemma}

\begin{proof}
Inequality \eqref{e2.3} is obvious because
\[
(-1)^{l+1}(\varphi^{(2l+1)},\varphi)=-\frac 12
\bigl(\varphi^{(l)}\bigr)^2\big|^1_0.
\]
Then integration by parts yields
\begin{gather*}
(-1)^{l+1}(\varphi^{(2l+1)},(1+x)\varphi) = \frac {2l+1}2
\|\varphi^{(l)}\|^2_{L^2}
+\frac 12\bigl(\varphi^{(l)}(0)\bigr)^2
-\bigl(\varphi^{(l)}(1)\bigr)^2,\\
a_{2l}(\varphi^{(2l)},(1+x)\varphi) = (-1)^la_{2l}\|(1+x)^{1/2}
\varphi^{(l)}\|^2_{L^2}\geq 0;
\end{gather*}
and for $j\leq l-1$ again with the use of the Ehrling inequality
\begin{align*}
&a_{2j+1}(\varphi^{(2j+1)},(1+x)\varphi) + a_{2j}(\varphi^{(2j)},
(1+x)\varphi)\\
&= (-1)^{j+1}\frac {2j+1}2a_{2j+1}\|\varphi^{(j)}\|^2_{L^2}+
(-1)^{j}a_{2j}\|(1+x)^{1/2}\varphi^{(j)}\|^2_{L^2}\\
&\geq -\delta\|\varphi^{(l)}\|^2_{L^2}-c(\delta)\|\varphi\|^2_{L^2},
\end{align*}
where $\delta>0$ can be chosen arbitrary small, we obtain
\eqref{e2.4}.
\end{proof}

Let $ \mathcal{F}$ and $\mathcal{F}^{-1}$ be respectively the direct
and inverse Fourier transforms of a function $f$. For $s\in \mathbb{R}$
define the fractional order Sobolev space
\[
H^s(\mathbb{R})= \bigl\{f: \mathcal{F}^{-1}[(1+|\xi|^2)^{\frac{s}{2}}\widehat
f(\xi)] \in L_2(\mathbb{R})\bigr\}
\]
and for a certain interval $I\subset \mathbb{R}$ let $H^s(I)$ be a space of
restrictions on $I$ of
functions from $H^s(\mathbb{R})$. Define also
\[
H_0^s(I)=\bigl\{f\in H^s(\mathbb{R}): \mathop{\rm supp}
f\subset \overline{I}\bigr\}.
\]
If $\partial I$ is a finite part of the boundary of the interval $I$,
then for
$s\in (k+1/2,k+3/2)$, where $k\geq 0$ -- integer,
\[
H_0^s(I)=\bigl\{f\in H^s(I): f^{(j)}\bigl|_{\partial I}=0,\;
j=0,\dots,k\bigr\}.
\]
Note, that $H_0^s(I)=H^s(I)$ for $s\in [0,1/2)$.

If $\mathcal{X}$ is a certain Banach (or full countable-normed) space,
define by
$C_b(\overline{I};\mathcal{X})$ a space of continuous bounded mappings
from $\overline{I}$
to $\mathcal{X}$. Let
\begin{gather*}
C_b^k(\overline{I};\mathcal{X}) =\bigl\{f(t): \partial_t^j
f\in C_b(\overline{I};\mathcal{X}),
\; j=0,\dots,k\},\\
C_b^\infty(\overline{I};\mathcal{X}) =\bigl\{f(t): \partial_t^j
f\in C_b(\overline{I};\mathcal{X}),
\; \forall j\geq 0\}.
\end{gather*}
If $I$ is a bounded interval, the index $b$ is omitted.

The symbol $L^p(I;\mathcal{X})$ is used in the usual sense for the
space of Bochner measurable mappings from $I$ to $\mathcal{X}$, summable
with order $p$ (essentially bounded if $p=+\infty$).

Next we introduce some special functional spaces.

\begin{definition}\label{D2.2} \rm
For integer $k\geq 0$, $T>0$ and an interval (bounded or unbounded)
$I\subset \mathbb{R}$ define
\begin{gather*}
\begin{aligned}
X_k((0,T)\times I) &=\{u(t,x):\partial_t^n u\in C([0,T];
 H^{(2l+1)(k-n)}(I))\\
&\quad\cap L^2(0,T;H^{(2l+1)(k-n)+l}(I)),\; n=0,\dots,k\},
\end{aligned}
\\
\begin{aligned}
&M_k((0,T)\times I)\\
&=\{f(t,x): \partial_t^k f\in L^2(0,T;H^{-l}(I)),
\partial_t^n f\in C([0,T];H^{(2l+1)(k-n-1)}(I))\\
&\quad \cap L^2(0,T;H^{(2l+1)(k-n)-l-1}(I)),\; n=0,\dots,k-1\}.
\end{aligned}
\end{gather*}
\end{definition}

Obviously,
\begin{equation}\label{e2.5}
\|P_0u\|_{M_k((0,T)\times I))}\leq c\|u\|_{X_k((0,T)\times I))}.
\end{equation}
In fact, we construct solutions to  problem
\eqref{e1.1}--\eqref{e1.4} in the spaces $X_k(Q_T)$ for the right
parts of equation \eqref{e1.1} in the spaces $M_k(Q_T)$.
To describe properties of boundary functions $\mu_j$, $\nu_j$ we use
the following functional spaces.

\begin{definition}\label{D2.3} \rm
Let $s\geq 0$, $m=l-1$ or $m=l$,  define
\[
\mathcal{B}^m_s(0,T)=\prod_{j=0}^m H^{s+(l-j)/(2l+1)}(0,T).
\]
\end{definition}

Also we use auxiliary subsets of $\mathcal{B}^m_s(0,T)$:
\[
\mathcal{B}^m_{s0}(0,T)=\prod_{j=0}^m
H^{s+(l-j)/(2l+1)}_0(\mathbb{R}_+)\bigr|_{(0,T)},\quad
\mathbb{R}_+=(0,+\infty).
\]
For the study of properties of equation \eqref{e1.5} we need more
sophisticated spaces than $X_k$.

\begin{definition}\label{D2.4} \rm
For $s\geq 0$, $I\subset \mathbb R$ define
\begin{align*}
Y_s((0,T)\times I)=&\{u(t,x): \partial_t^n u\in
C([0,T];H^{(2l+1)(s-n)}(I)), \; n=0,\dots,[s],\\
&\partial_x^j u\in C_b(\overline{I}; H^{s+(l-j)/(2l+1)}(0,T)),\;
j=0,\dots,[(2l+1)s]+l\}.
\end{align*}
\end{definition}

Obviously, $Y_k(Q_T)\subset X_k(Q_T)$.
The spaces $Y_s$ originate from internal properties of the linear
operator $\partial_t+(-1)^{l+1}\partial_x^{2l+1}$. In fact, consider
an initial value problem in a strip $\Pi_T=(0,T)\times \mathbb{R}$ for
\eqref{e1.5} with the initial data \eqref{e1.2}. This problem was
studied in \cite{kenig}. In particular, if $u_0\in H^{(2l+1)s}(\mathbb{R})$,
then for any $T>0$ there exists a solution of \eqref{e1.5},
\eqref{e1.2}, $S(t,x;u_0)$, given by the formula
\begin{equation}\label{e2.6}
S(t,x;u_0)=\mathcal{F}_x^{-1}[e^{i\xi^{2l+1}t}\widehat
u_0(\xi)](x).
\end{equation}
For this solution for any $t\in \mathbb{R}$ and integer $0\leq n\leq s$,
$0\leq j\leq (2l+1)(s-n)$
\begin{equation}\label{e2.7}
\|\partial_t^n\partial_x^jS(t,\cdot;u_0)\|_{L^2(\mathbb{R})}=
\|u_0^{((2l+1)n+j)}\|_{L^2(\mathbb{R})},
\end{equation}
and for any $x\in \mathbb{R}$ and integer $0\leq j\leq (2l+1)s+l$
\begin{equation}\label{e2.8}
\|D_t^{s+(l-j)/(2l+1)}\partial_x^jS(\cdot,x;u_0)\|_{L^2(\mathbb{R})}=
c(l)\|D_x^{(2l+1)s}u_0\|_{L^2(\mathbb R)},
\end{equation}
where the symbol $D^s$ denotes the Riesz potential of the order
$-s$. In particular, the traces of $\partial_x^jS$ for $x=0$,
$j=0,\dots,m=l-1$, and $x=1$, $j=0,\dots,m=l$ lie in
$\mathcal{B}_s^m(0,T)$.
To formulate compatibility conditions for the original problem
we now introduce certain special functions.

\begin{definition}\label{D2.5} \rm
Let $\Phi_0(x)\equiv u_0(x)$ and for natural $n$
\[
\Phi_n(x)\equiv \partial_t^{n-1}f(0,x)-P\Phi_{n-1}(x)-
\sum_{m=0}^{n-1} \binom{n-1}{m} \Phi_m(x)\Phi'_{n-m-1}(x).
\]
\end{definition}

Now we can present the main results of this paper.

\begin{theorem}[local well-posedness]\label{T2.1}
Let the operator $P_0$ satisfy Assumption A. Let $u_0\in
H^{(2l+1)k}(0,1)$, $(\mu_0,\dots,\mu_{l-1})\in \mathcal{B}_k^{l-1}(0,T)$,
$(\nu_0,\dots,\nu_l)\in \mathcal{B}_k^l(0,T)$,
$f\in M_k(Q_T)$ for some $T>0$ and integer $k\geq 0$. Assume also
that $\mu_j^{(n)}(0)= \Phi_n^{(j)}(0)$, $j=0,\dots,l-1$,
$\nu_j^{(n)}(0)=\Phi_n^{(j)}(1)$, $j=0,\dots,l$, for $0\leq n\leq
k-1$. Then there exists $t_0\in (0,T]$ such that
\eqref{e1.1}--\eqref{e1.4} is well-posed in $X_k(Q_{t_0})$.
\end{theorem}

\begin{theorem}[global well-posedness]\label{T2.2}
Let the hypothesis of Theorem~\ref{T2.1} be satisfied and, in
addition, if $k=0$, then $f\in L^1(0,T;L^2)$, and if $l=1$, $k=0$,
then $\mu_0,\nu_0\in H^{1/3+\varepsilon}(0,T)$ for a certain
$\varepsilon>0$. Then  \eqref{e1.1}--\eqref{e1.4} is well-posed in
$X_k(Q_T)$.
\end{theorem}

\begin{remark}\label{R2.2} \rm
A problem is well-posed in the space $X_k$, if there
exists a unique solution $u(t,x)$ in this space and the map
$\left(u_0,(\mu_0,\dots,\mu_{l-1}), (\nu_0,\dots,\nu_l),
f\right)\mapsto u$ is Lipschitz continuous on any ball in the
corresponding norms.
\end{remark}

\begin{remark}\label{R2.3} \rm
Properties \eqref{e2.8} of the solution $S$ to the initial-value
problem \eqref{e1.5}, \eqref{e1.2} show that the smoothness conditions
on the boundary data in our results are natural (with the only
exception in the case $l=1$, $k=0$ for global results) because they
originate from the properties of the operator $\partial_t+
(-1)^{l+1}\partial_x^{2l+1}$.
\end{remark}

\begin{remark}\label{R2.4} \rm
All these well-posedness results can be easily generalized for an
equation of \eqref{e1.1} type with a nonlinear term $g(u)u_x$, where
a sufficiently smooth function $g$ has not more than a linear growth
rate.
\end{remark}

\section{Linear problem for a homogeneous equation}

The goal of this section is to construct solutions to an
initial-boundary value problem in $Q_T$ for  equation \eqref{e1.5}
with initial and boundary data \eqref{e1.2}--\eqref{e1.4} in the
spaces $Y_s(Q_T)$. Uniqueness will be discussed in the next section
for more general linear equations.

In what follows, we need simple properties of roots
$r_m(\lambda,\varepsilon)$, $m=0,\dots,2l$ of an algebraic equation
\begin{equation}\label{e3.1}
r^{2l+1}=(-1)^l(\varepsilon+i\lambda),\quad \varepsilon\geq 0,\; \lambda\in
\mathbb{R},\; (\lambda,\varepsilon)\ne (0,0).
\end{equation}
An enumeration of these roots can be chosen such that they are
continuous with respect to $(\lambda,\varepsilon)$,
$r_m(-\lambda,\varepsilon)=\overline{r_m(\lambda,\varepsilon)}$,
\begin{gather}\label{e3.2}
\mathop{\rm Re} r_m<0, \quad m=0,\dots,l-1;\quad \mathop{\rm Re} r_m>0, \quad m=l,\dots,2l-1;\\
\label{e3.3}
\mathop{\rm Re} r_{2l}>0, \quad \varepsilon>0;\quad \mathop{\rm Re} r_{2l}=0, \quad \varepsilon=0.
\end{gather}
It is obvious that for any $m$ and $j\ne m$
\begin{equation}\label{e3.4}
|r_m|=(\lambda^2+\varepsilon^2)^{1/(4l+2)},\quad
|r_m-r_j|= c(l,m,j)(\lambda^2+\varepsilon^2)^{1/(4l+2)}.
\end{equation}
Denote $r_m(\lambda)\equiv r_m(\lambda,0)$, then
\begin{equation}\label{e3.5}
|\mathop{\rm Re} r_m(\lambda)|= c(l,m)|\lambda|^{1/(2l+1)},\quad m=0,\dots,2l-1;
\quad r_{2l}(\lambda)=i\lambda^{1/(2l+1)}.
\end{equation}

To construct the desired solutions to the problem in a bounded
rectangle, we first consider corresponding problems in half-strips
and start with a problem in a right one:
$\Pi_T^+=(0,T)\times \mathbb R_+$.

\begin{lemma}\label{L3.1}
Let $u_0\in H^{(2l+1)s}(\mathbb{R}_+)$,
$(\mu_0,\dots,\mu_{l-1})\in \mathcal{B}_s^{l-1}(0,T)$
for some $T>0$ and $s\geq 0$ such that
$s+\frac {l-j}{2l+1}-\frac 12$ is non-integer for any $j=0,\dots,l-1$.
Assume also that $\mu_j^{(n)}(0)=(-1)^{nl}u_0^{((2l+1)n+j)}(0)$ for
$n=0,\dots,[s+\frac {l-j}{2l+1}-\frac 12]$, $j=0,\dots,l-1$. Then
there exists a solution to problem \eqref{e1.5}, \eqref{e1.2},
\eqref{e1.3} $u(t,x)\in Y_s(\Pi_T^+)$ such that
\begin{equation}\label{e3.6}
\|u\|_{Y_s(\Pi_T^+)}\leq c(T,l,s)\left(\|u_0\|_{H^{(2l+1)s}(\mathbb{R}_+)}+
\|(\mu_0,\dots,\mu_{l-1})\|_{\mathcal{B}_s^{l-1}(0,T)}\right).
\end{equation}
\end{lemma}

\begin{proof}
We construct the desired solution in the form
\begin{equation}\label{e3.7}
u(t,x)=S(t,x;u_0)+w(t,x),
\end{equation}
where $u_0$ is extended to the whole real line $\mathbb{R}$ in the same
class $H^{(2l+1)s}$ with an equivalent norm, the function $S$ is
defined by  formula (\ref{e2.6}) and $w(t,x)$ is a solution to the
problem in $\Pi_T^+$ for  equation \eqref{e1.5} with zero initial
data \eqref{e1.2} and boundary data
\begin{equation}\label{e3.8}
\partial_x^jw(t,0)=\sigma_j(t),\quad j=0,\dots,l-1,
\end{equation}
where $\sigma_j(t)\equiv \mu_j(t)-\partial_x^jS(t,0;u_0)$. Note that, by
virtue of \eqref{e2.8} and compatibility conditions,
$(\sigma_0,\dots,\sigma_{l-1}) \in \mathcal{B}_{s0}^{l-1}(0,T)$.

Assume at first that all functions $\sigma_j\in C_0^\infty(\mathbb{R}_+)$.
In this case, according to \cite{volevich}, there exists a solution
$w(t,x)$  and $w\in
C^\infty([0,T];H^\infty(\mathbb{R}_+))$ for any $T>0$. Moreover, if
$\sigma_j(t)=0$ for $t\geq T_0>0$ and all $j$, then it is easy to
show that for $t\geq T_0$ and all integer $n\geq 0$
\begin{equation}\label{e3.9}
\frac{d}{dt} \| \partial_t^n w\|_{L^2(\mathbb{R}_+)} \leq 0,
\end{equation}
whence with the use of   \eqref{e1.5} itself one can prove
 $w\in C_b^\infty(\overline{\mathbb{R}}_+;H^\infty(\mathbb{R}_+))$.

Therefore, for any $p=\varepsilon+i\lambda$, where $\varepsilon>0$, we can define
the Laplace transform
\begin{equation}\label{e3.10}
\widetilde w(p,x) \equiv \int_{\mathbb{R}_+} e^{-pt}w(t,x)\,dt.
\end{equation}
The function $\widetilde w(p,x)$ is a solution to the problem
\begin{gather}\label{e3.11}
p\widetilde w(p,x)+(-1)^{l+1}\partial_x^{2l+1}\widetilde w(p,x)=0,
\quad x\geq 0,\\
\label{e3.12}
\partial_x^j \widetilde w(p,0)=\widetilde \sigma_j(p)\equiv
\int_{\mathbb{R}_+} e^{-pt}\sigma_j(t)\,dt, \quad j=0,\dots,l-1.
\end{gather}
Since $\widetilde w(p,x)\to 0$ as $x\to +\infty$, it follows from
\eqref{e3.2}--\eqref{e3.4} that
\[
\widetilde w(p,x)=\sum_{m=0}^{l-1} \sum_{k=0}^{l-1}
c_{mk}(l)(\lambda^2+\varepsilon^2)^{-k/(4l+2)}e^{r_m(\lambda,\varepsilon)x}
\widetilde \sigma_k(\varepsilon+i\lambda).
\]
Using the formula of inversion of the Laplace transform and passing
to the limit as $\varepsilon\to+0$, we find
\begin{equation} \label{e3.13}
\begin{aligned}
w(t,x)&=\sum_{m=0}^{l-1} \sum_{k=0}^{l-1}
c_{mk}(l)\mathcal{F}_t^{-1}\left[|\lambda|^{-k/(2l+1)}
e^{r_m(\lambda)x}\widehat \sigma_k(\lambda)\right](t)\\
&\equiv \sum_{m=0}^{l-1} \sum_{k=0}^{l-1}
c_{mk}(l) w_{mk}(t,x).
\end{aligned}
\end{equation}
Now consider the  integral
\[
\mathcal I_m(t,x)\equiv \int_{\mathbb{R}} e^{i\lambda t +
r_m(\lambda)x}f(\lambda)\,d\lambda, \quad
m=0,\dots,l-1,
\]
and establish that, uniformly with respect to $t\in {\mathbb R}$,
\begin{equation}\label{e3.14}
\|\mathcal I_m(t,\cdot)\|_{L^2(\mathbb{R}_+)}
\leq c(l,m)\big\||\lambda|^{l/(2l+1)}f(\lambda)\big\|_{L^2(\mathbb{R})}.
\end{equation}
The proof of \eqref{e3.14} is based on the following fundamental
inequality from \cite{bona1}: If a continuous function $\gamma(\xi)$
satisfies an inequality $\mathop{\rm Re} \gamma(\xi)\leq -\varepsilon|\xi|$ for some
$\varepsilon>0$ and all $\xi\in \mathbb{R}$, then
\begin{equation}\label{e3.15}
\bigl\|\int_{\mathbb{R}} e^{\gamma(\xi)x} f(\xi)\,d\xi\bigr\|_
{L^2(\mathbb{R}_+)} \leq c(\varepsilon)\|f\|_{L^2(\mathbb{R})}.
\end{equation}
Changing variables $\xi=\lambda^{1/(2l+1)}$, we derive from
(\ref{e3.15}) (since $\mathop{\rm Re} r_m(\xi^{2l+1})=-c(l,m)|\xi|$)
\[
\|\mathcal I_m\|_{L^2(\mathbb{R}_+)} \leq c(l,m)\|\xi^{2l}
f(\xi^{2l+1})\|_{L^2(\mathbb R)}
\]
which proves \eqref{e3.14}.

Applying \eqref{e3.14} to (\ref{e3.13}) yields, by virtue of
(\ref{e3.3}), \eqref{e3.4}, that uniformly with respect to $t\in {\mathbb{R}}$
for $n=0,\dots,[s]$, $j=0,\dots,[(2l+1)(s-n)]$
\begin{equation}\label{e3.16}
\|\partial_t^n \partial_x^j w_{mk}(t,\cdot)\|_{L^2(\mathbb{R}_+)} \leq
c(l,m,s) \|\sigma_k\|_{H^{n+(l+j-k)/(2l+1)}(\mathbb{R})}.
\end{equation}
Next, let
\[
\sigma_{k0}\equiv \mathcal{F}^{-1}\left[\widehat\sigma_k(\lambda)
\chi(\lambda)\right], \quad
\sigma_{k1}\equiv \sigma_k-\sigma_{k0},
\]
where $\chi$ is the characteristic function of the interval $(-1,1)$,
and represent $w_{mk}$ as a sum of two corresponding functions
$w_{mk0}$ and $w_{mk1}$. Then, by virtue of \eqref{e3.16}, uniformly
with respect to $x\geq 0$ for $j=0,\dots,[(2l+1)s]+l$
\begin{equation} \label{e3.17}
\begin{aligned}
\|\partial_x^j w_{mk0}(\cdot,x)\|_{H^{s+(l-j)/2l+1)}(0,T)}
&\leq c(T)\|w_{mk0}\|_{C^{[s]+2}([0,T];H^{[(2l+1)s]+l+1}(\mathbb{R}_+))} \\
&\leq c(l,s,T) \|\sigma_{k0}\|_{H^{2s+4}(\mathbb{R})}\\
&\leq c_1(l,s,T) \|\sigma_{k}\|_{L^2(\mathbb{R})}
\end{aligned}
\end{equation}
and since $\mathop{\rm Re} r_m(\lambda)\leq 0$,
\begin{equation}\label{e3.18}
\|\partial_x^j w_{mk1}(\cdot,x)\|_{H^{s+(l-j)/(2l+1)}(\mathbb{R})}
\leq c(l,k)\|\sigma_k\|_{H^{(s+(l-k)/(2l+1)}(\mathbb{R})}.
\end{equation}
Combining \eqref{e2.7}, \eqref{e2.8}, (\ref{e3.7}), \eqref{e3.8},
(\ref{e3.13}), \eqref{e3.16}--(\ref{e3.18}) we derive  (\ref{e3.6}) in
the smooth case and  via closure in the general case.
\end{proof}

\begin{corollary}\label{C3.1}
Let $J(t,x;\sigma_0,\dots,\sigma_{l-1})$ denotes the solution to the
problem in $\Pi_T^+$ for  equation \eqref{e1.5} with zero initial
data and boundary data \eqref{e3.8} (where $w$ must be substituted by
$u$) constructed in Lemma~\ref{L3.1} . Then  $J$ is infinitely
differentiable for $x>0$; and for any $x_0>0$ and integer
$n, j \geq 0$
\begin{equation}\label{e3.19}
\sup_{x\geq x_0} |\partial^n_t\partial^j_x J(t,x)|\leq
c(l,n,j,x_0^{-1})
\sum_{m=0}^{l-1}\|\sigma_m\|_{L^2(0,T)}.
\end{equation}
\end{corollary}

\begin{proof}
 From representation (\ref{e3.13}) we obtain
\[
\partial_t^n\partial_x^j w_{mk}(t,x)= \mathcal{F}_t^{-1}
\big[(i\lambda)^nr_m^j(\lambda)
|\lambda|^{-k/(2l+1)}e^{r_m(\lambda)x}\widehat
\sigma_k(\lambda)\big](t),
\]
where, by virtue of \eqref{e3.2} and (\ref{e3.5}),
\[
\mathop{\rm Re} r_m(\lambda)x\leq -c(l,m)|\lambda|^{1/(2l+1)}x_0,\quad
|\lambda|^{n+\frac{j-k}{2l+1}}e^{-c(l,m)|\lambda|^{1/(2l+1)}x_0}\in
L^2(\mathbb{R}).
\]
\end{proof}

Now consider \eqref{e1.5}, \eqref{e1.2} in a half-strip
$\Pi_T^-=(0,T)\times \mathbb{R}_-$, $\mathbb{R}_-=(-\infty,0)$,
 with boundary data
\begin{equation}\label{e3.20}
\partial_x^j u(t,0)=\nu_j(t),\quad j=0,\dots,l.
\end{equation}

\begin{lemma}\label{L3.2}
Let $u_0\in H^{(2l+1)s}(\mathbb{R}_-)$, $(\nu_0,\dots,\nu_l)\in \mathcal{B}_s^l(0,T)$ for some $T>0$ and $s\geq 0$ such that $s+\frac
{l-j}{2l+1}-\frac 12$ is non-integer for any $j=0,\dots,l$. Assume
also that $\nu_j^{(n)}(0)=(-1)^{nl}u_0^{((2l+1)n+j)}(0)$ for
$n=0,\dots,[s+\frac {l-j}{2l+1}-\frac 12]$, $j=0,\dots,l$. Then
there exists a solution to  problem
\eqref{e1.5}, \eqref{e1.2}, \eqref{e3.20}, $u(t,x)\in Y_s(\Pi_T^-)$, such
that
\begin{equation}\label{e3.21}
\|u\|_{Y_s(\Pi_T^-)}\leq c(T,l,s)\left(\|u_0\|_{H^{(2l+1)s}(\mathbb{R}_-)}+
\|(\nu_0,\dots,\nu_l)\|_{\mathcal{B}_s^l(0,T)}\right).
\end{equation}
\end{lemma}

\begin{proof}
The scheme of the proof repeats the one of Lemma~\ref{L3.1}. The
desired solution is constructed in the form (\ref{e3.7}), where  $w$
is a solution to the problem in $\Pi_T^-$ for  equation \eqref{e1.5}
with zero initial data and boundary data
\begin{equation}\label{e3.22}
\partial_x^jw(t,0)=\sigma_j(t)\equiv \nu_j(t)-\partial_x^jS(t,0;u_0),
\quad j=0,\dots,l.
\end{equation}
By virtue of compatibility conditions, $(\sigma_0,\dots,\sigma_l)
\in \mathcal{B}_{s0}^l(0,T)$.

Assuming temporarily  that all functions $\sigma_j\in
C_0^\infty(\mathbb{R}_+)$, results of \cite{volevich} provide that there
exists a solution to this problem $w\in
C_b^\infty(\overline{\mathbb{R}}_+^t;H^\infty(\mathbb{R}_-))$ (where  inequality
(\ref{e3.9}) transforms into a corresponding equality). The Laplace
transform $\widetilde w(p,x)$, given by  formula (\ref{e3.10}),
satisfies (\ref{e3.11}) for $x\leq 0$ and $(l+1)$ boundary conditions
(\ref{e3.12}). Using the properties of the roots of (\ref{e3.1}), by
analogy with (\ref{e3.13}), one can easily derive
\begin{equation} \label{e3.23}
\begin{aligned}
w(t,x)&=\sum_{m=l}^{2l} \sum_{k=0}^l
c_{mk}(l)\mathcal{F}_t^{-1}\big[|\lambda|^{-k/(2l+1)}
e^{r_m(\lambda)x}\widehat \sigma_k(\lambda)\big](t)\\
&\equiv \sum_{m=l}^{2l} \sum_{k=0}^l
c_{mk}(l) w_{mk}(t,x).
\end{aligned}
\end{equation}
Similarly to \eqref{e3.16} for $m=l,\dots,2l-1$; $n=0,\dots,[s]$;
$j=0,\dots,[(2l+1)(s-n)]$; uniformly with respect to $t\in \mathbb{R}$
\[
\|\partial_t^n\partial_x^j w_{mk}(t,\cdot)\|_{L^2(\mathbb{R}_-)} \leq
c(l,m,s) \|\sigma_k\|_{H^{n+(l+j-k)/(2l+1)}(\mathbb{R})}.
\]
For $m=2l$, changing variables $\xi=\lambda^{1/(2l+1)}$ and using
(\ref{e2.6}) and (\ref{e3.5}), we find
\[
w_{(2l)k}=(2l+1) S(t,x;\mathcal{F}^{-1}_x\left[|\xi|^{2l-k}
\widehat\sigma_k(\xi^{2l+1})\right]),
\]
and, by virtue of \eqref{e2.7}, uniformly with respect to $t\in
{\mathbb R}$
\begin{align*}
\|\partial_t^n\partial_x^j w_{(2l)k}(t,\cdot)\|_{L^2(\mathbb{R})}
&= c(l)\||\xi|^{(2l+1)n+2l+j-k}\widehat\sigma_k(\xi^{2l+1})\|_{L^2(\mathbb{R})}\\
&\leq c_1(l) \|\sigma_k\|_{H^{n+(l+j-k)/(2l+1)}(\mathbb{R})}.
\end{align*}
Since $\mathop{\rm Re} r_k(\lambda)\geq 0$, $m=l,\dots,2l$, similarly to
(\ref{e3.17}), (\ref{e3.18}), one can obtain that for
$j=0,\dots,[(2l+1)s]+l$ uniformly with respect to $x\leq 0$
\[
\|\partial_x^j w_{mk}(\cdot,x)\|_{H^{s+(l-j)/(2l+1)}(0,T))}\leq
c(l,s,T,k)\|\sigma_k\|_{H^{s+(l-k)/(2l+1)}(\mathbb{R})}.
\]
The end of the proof is the same as in Lemma~\ref{L3.1}.
\end{proof}

Now we pass to a problem on a bounded rectangle.

\begin{lemma}\label{L3.3}
 Let $u_0\in H^{(2l+1)s}$, \
$(\mu_0,\dots,\mu_{l-1})\in \mathcal{B}_s^{l-1}(0,T)$, \
$(\nu_0,\dots,\nu_l)\in \mathcal{B}_s^l(0,T)$ for
some $T>0$ and $s\geq 0$ such that $s+\frac {l-j}{2l+1}-\frac 12$ is
non-integer for any $j=0,\dots,l$. Assume also that
$\mu_j^{(n)}(0)=(-1)^{nl}u_0^{((2l+1)n+j)}(0)$, $j=0,\dots,l-1$,
$\nu_j^{(n)}(0)=(-1)^{nl}u_0^{((2l+1)n+j)}(1)$, $j=0,\dots,l$, for
$n=0,\dots,[s+\frac {l-j}{2l+1}-\frac 12]$. Then there exists a
solution to  problem \eqref{e1.5}, \eqref{e1.2}--\eqref{e1.4},
$u(t,x)\in Y_s(Q_T)$, and the following inequality holds:
\begin{equation} \label{e3.24}
\begin{aligned}
\|u\|_{Y_s(Q_T)}
&\leq c(T,l,s)\Big(\|u_0\|_{H^{(2l+1)s}}\\
&\quad +
\|(\mu_0,\dots,\mu_{l-1})\|_{\mathcal{B}_s^{l-1}(0,T)}+
\|(\nu_0,\dots,\nu_l)\|_{\mathcal{B}_s^l(0,T)}\Big).
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
We construct the desired solution in the form
\begin{equation}\label{e3.25}
u(t,x)=w(t,x)+v(t,x),
\end{equation}
where $w(t,x)$ is a solution to an initial boundary-value problem in
$\Pi^-_{T,1}=(0,T)\times (-\infty,1)$ for  equation \eqref{e1.5} with
initial and boundary conditions \eqref{e1.2}, \eqref{e1.4} written for
the function $w$, ($u_0$ is extended here to the half-line
$(-\infty,1)$ in the same class $H^{(2l+1)s}$ with an equivalent
norm). According to Lemma~\ref{L3.2}, the solution $w\in
Y_s(\Pi_{T,1}^-)$ exists and
\begin{equation}\label{e3.26}
\|w\|_{Y_s(\Pi_{T,1}^-)} \leq c(T,l,s)
\bigl(\|u_0\|_{H^{(2l+1)s}} + \|(\nu_0,\dots,\nu_l)\|_{\mathcal{B}_s^l(0,T)}\bigr).
\end{equation}
Let
\[
\alpha_j(t)\equiv \mu_j(t)-\partial_x^jw(t,0),\quad j=0,\dots,l-1.
\]
It follows from (\ref{e3.26})
\begin{equation} \label{e3.27}
\begin{aligned}
\|(\alpha_0,\dots,\alpha_{l-1})\|_{\mathcal{B}_s^{l-1}(0,T)}
&\leq c(T,l,s)\Big(\|u_0\|_{H^{(2l+1)s}}+
\|(\mu_0,\dots,\mu_{l-1})\|_{\mathcal{B}_s^{l-1}(0,T)}\\
&\quad + \|(\nu_0,\dots,\nu_l)\|_{\mathcal{B}_s^l(0,T)}\Big)
\end{aligned}
\end{equation}
and $(\alpha_0,\dots,\alpha_{l-1})\in \mathcal{B}_{s0}^{l-1}(0,T)$
by virtue of  the compatibility conditions in the point $(0,0)$.

Consider the following problem for the function $v$, in $Q_T$,
\begin{gather}\label{e3.28}
v_t+(-1)^{l+1}\partial_x^{2l+1}v=0, \\
\label{e3.29}
v(0,x)=0,\quad x\in (0,1), \\
\label{e3.30}
\partial_x^j v(t,0)=\alpha_j(t),\quad j=0,\dots,l-1,\\
\label{e3.31} \partial_x^j v(t,1)=0,\quad j=0,\dots,l,\quad t\in(0,T).
\end{gather}
To construct a solution, we consider the
 function $J(t,x;\sigma_0,\dots,\sigma_{l-1})\in Y_s(\Pi_T^+)$
as in Corollary~\ref{C3.1} for a certain set of functions
$(\sigma_0,\dots,\sigma_{l-1})\in \mathcal{B}_{s0}^{l-1}(0,T)$. Let
\[
\beta_j(t)\equiv \partial_x^j J(t,1;\sigma_0,\dots,\sigma_{l-1}),\quad
j=0,\dots,l.
\]
Due to (\ref{e3.19}), for any $\delta\in (0,T]$
\[
\|(\beta_0,\dots,\beta_l)\|_{\mathcal{B}_s^l(0,\delta)} \leq
c(l,s)\delta^{1/2}\|(\sigma_0,\dots,
\sigma_{l-1})\|_{\mathcal{B}_s^{l-1}(0,\delta)}.
\]
Moreover, $(\beta_0,\dots,\beta_l)\in \mathcal{B}_{s0}^l(0,\delta)$.

Consider in the half-strip $\Pi_{\delta,1}^-$ a problem of the
\eqref{e1.5}, \eqref{e1.2}, \eqref{e1.4} type, where $u_0\equiv 0$,
$\nu_j\equiv -\beta_j$ for $j=0,\dots,l$. It follows again from
Lemma~\ref{L3.2} that a solution to this problem $V\in
Y_s(\Pi_{\delta,1}^-)$ exists and, in particular, if
\[
\gamma_j(t)\equiv \partial_x^j V(t,0),\quad j=0,\dots,l-1,
\]
then $(\gamma_0,\dots,\gamma_{l-1})\in \mathcal{B}_{s0}^{l-1}(0,\delta)$
and
\begin{equation} \label{e3.32}
\begin{aligned}
\|(\gamma_0,\dots,\gamma_{l-1})\|_{\mathcal{B}_s^{l-1}(0,\delta)}
&\leq c(T,l,s)\|(\beta_0,\dots,\beta_l)\|_{\mathcal{B}_s^l(0,\delta)}\\
&\leq c_1(T,l,s)\delta^{1/2}\|(\sigma_0,\dots,
\sigma_{l-1})\|_{\mathcal{B}_s^{l-1}(0,\delta)}.
\end{aligned}
\end{equation}
Consider a linear operator $\Gamma:
(\sigma_0,\dots,\sigma_{l-1})\mapsto (\gamma_0,\dots,\gamma_{l-1})$
in the space $\mathcal{B}_{s0}^{l-1}(0,\delta)$. For small
$\delta=\delta(T,l,s)$  estimate (\ref{e3.32}) provides that the
operator $(E+\Gamma)$ is invertible ($E$ is the identity operator)
and setting
\[
\sigma_j(t)\equiv (E+\Gamma)^{-1}\alpha_j(t),\quad j=0,\dots,l-1,
\]
we obtain the desired solution to  problem
(\ref{e3.28})--(\ref{e3.31}),
\[
v(t,x)\equiv J(t,x;\sigma_0,\dots,\sigma_{l-1}) + V(t,x),
\]
where
\begin{equation}\label{e3.33}
\|v\|_{Y_s(Q_\delta)} \leq c(T,l,s)\|(\alpha_0,\dots,
\alpha_{l-1})\|_{\mathcal{B}_s^{l-1}(0,T)}.
\end{equation}
Thus, the solution $u(t,x)$ to  problem \eqref{e1.5},
\eqref{e1.2}--\eqref{e1.4} in the rectangle $Q_\delta$ has been
constructed and, according to (\ref{e3.26}), (\ref{e3.27}),
(\ref{e3.33}), estimated in the space $Y_s(Q_\delta)$ by the right
part of (\ref{e3.24}). Moving step by step ($\delta$ is constant), we
obtain the desired solution in the whole rectangle $Q_T$.
\end{proof}

\begin{remark}\label{R3.1} \rm
The idea to construct solutions in a bounded rectangle from solutions
in half-strips for the linearized KdV equation goes back to the paper
\cite{holmer}, but the method of study of these
problems in  the infinite domains in \cite{holmer} differs from the one
used here. The method
of the present paper is analogous to \cite{faminski2}.
\end{remark}

\section{Complete linear problem}

In this section we consider an initial-boundary value problem in $Q_T$
for the equation
\begin{equation}\label{e4.1}
u_t+Pu=f(t,x)
\end{equation}
with initial and boundary conditions \eqref{e1.2}--\eqref{e1.4}.
First of all we introduce auxiliary functions necessary for
compatibility conditions.

\begin{definition}\label{D4.1} \rm
Let $\widetilde\Phi_0(x)\equiv u_0(x)$
and for natural $n$
\[
\widetilde\Phi_n(x)\equiv \partial_t^{n-1} f(0,x)-
P\widetilde\Phi_{n-1}(x).
\]
\end{definition}

\begin{remark}\label{R4.1} \rm
It is easy to see that
\[
\widetilde\Phi_n=(-1)^nP^nu_0+
\sum_{m=0}^{n-1}(-1)^{n-m-1}P^{n-m-1}\partial_t^m f\bigr|_{t=0}.
\]
\end{remark}

\begin{lemma}\label{L4.1}
Let the operator $P_0$ satisfies Assumption A. Let $u_0\in
H^{(2l+1)k}$, $(\mu_0,\dots,\mu_{l-1})\in \mathcal{B}_k^{l-1}(0,T)$,
$(\nu_0,\dots,\nu_l)\in \mathcal{B}_k^l(0,T)$, $f\in M_k(Q_T)$ for
some $T>0$ and integer $k\geq 0$. Assume also that $\mu_j^{(n)}(0)=
\widetilde\Phi_n^{(j)}(0)$, $j=0,\dots,l-1$,
$\nu_j^{(n)}(0)=\widetilde\Phi_n^{(j)}(1)$, $j=0,\dots,l$, for
$n=0,\dots,k-1$.  Then there exists a unique solution to  problem
\eqref{e4.1}, \eqref{e1.2}--\eqref{e1.4} $u(t,x)\in X_k(Q_T)$ and for
any $t_0\in (0,T]$
\begin{equation} \label{e4.2}
\begin{aligned}
\|u\|_{X_k(Q_{t_0})}
&\leq c(T,l,k)\Bigl(\|u_0\|_{H^{(2l+1)k}}+\|f\|_{M_k(Q_{t_0})}\\
&\quad +\|(\mu_0,\dots,\mu_{l-1})\|_{\mathcal{B}_k^{l-1}(0,T)}+
\|(\nu_0,\dots,\nu_l)\|_{\mathcal{B}_k^l(0,T)}\Bigr).
\end{aligned}
\end{equation}
For any natural $n\leq k$ the function $\partial_t^n u\in X_{k-n}(Q_T)$
is a solution to a problem of the \eqref{e4.1}, \eqref{e1.2}--\eqref{e1.4}
type, where $u_0$, $\mu_j$, $\nu_j$, $f$ are substituted by
$\widetilde\Phi_n$, $\mu_j^{(n)}$, $\nu_j^{(n)}$, $\partial_t^nf$.
\end{lemma}

\begin{proof}
It is sufficient to prove this lemma for $k=0$ and $k=1$. The cases
$k\geq 2$ are similar to $k=1$.
First consider the case $k=0$. Let $\psi(t,x)\in Y_0(Q_T)$ be a
solution to  problem \eqref{e1.5}, \eqref{e1.2}--\eqref{e1.4} for the
same $u_0$, $\mu_j$, $\nu_j$ constructed in Lemma~\ref{L3.3}.
Consider the initial-boundary value problem in $Q_T$ for the
equation
\begin{equation}\label{e4.3}
v_t+Pv=f-P_0\psi\equiv F
\end{equation}
with zero initial and boundary conditions of the
\eqref{e1.2}--\eqref{e1.4} type. By virtue of (\ref{e2.5}), $F\in
M_0(Q_T)$ with an appropriate estimate of its norm in $M_0(Q_{t_0})$
by the right part of (\ref{e4.2}).
Define
\[
w(t,x)\equiv v(t,x)e^{-\lambda t}, \quad \lambda\geq c_0,
\]
where $c_0$ is the constant from \eqref{e2.1}. Then (\ref{e4.3})
transforms into
\begin{equation}\label{e4.4}
w_t+(P+\lambda E)w= e^{-\lambda t}F\equiv F_1.
\end{equation}
Consider the corresponding initial-boundary value problem for the
function $w$ as an abstract Cauchy problem in $L^2$
\begin{equation}\label{e4.5}
w_t=\mathcal A w +F_1, \quad w\bigr|_{t=0}=0,
\end{equation}
where $\mathcal A=-(P+\lambda E)$ is the closed linear operator in $L^2$ with
the domain
\[
D(\mathcal A)=\big\{g\in H^{2l+1}: g^{(j)}(0)=g^{(j)}(1)=g^{(l)}(1)=0,
\; j=0,\dots,l-1\big\}.
\]
The adjoint operator $\mathcal A^*$ is defined as
\[
\mathcal A^*=-(P^*+\lambda E),
\]
where $P^*$ is the formally adjoint operator of $P$, with the domain
\[
D(\mathcal A^*)=\big\{g^*\in H^{2l+1}:
g^{*(j)}(0)=g^{*(j)}(1)=g^{*(l)}(0)=0,\quad j=0,\dots,l-1\big\}.
\]
It is easy to see that, by virtue of \eqref{e2.1}, both $\mathcal A$
and $\mathcal A^*$ are dissipative which means
\[
(\mathcal A g,g)\leq 0,\quad (\mathcal A^* g^*,g^*)\leq 0.
\]
Assume $F$ smooth, for example $F\in C^1([0,T];H^{2l+1})$.
By \cite[Corollaries 4.4, Chapter 1 and 2.10 of Chapter 4]{pazy},
$\mathcal A$ is the infinitesimal generator of a $C_0$-semigroup of
contraction in $L^2$ and the Cauchy problem (\ref{e4.5}) has a unique
strong solution $w\in C([0,T];H^{2l+1})\cap C^1([0,T];L^2)$.
Consequently, the initial-boundary value problem for (\ref{e4.3})
with zero initial and boundary conditions has a unique solution
$v(t,x)$ in the same class.

Multiplying (\ref{e4.3}) by $2(1+x) v(t,x)$ and integrating over $Q_t$,
$t\in (0,T]$, we find that by virtue of \eqref{e2.4}
\begin{equation}\label{e4.6}
\|v(t,\cdot)\|_{L^2}^2 + c_1\|\partial_x^l v\|^2_{L^2(Q_t)} \leq c
\|v\|^2_{L^2(Q_t)} +
c\|F\|^2_{L^2(0,t;H^{-l})}.
\end{equation}
This estimate  gives an opportunity to construct an
appropriate solution $v\in X_0(Q_T)$ to the considered problem in
the general case $F\in L^2(0,T;H^{-l})=M_0(Q_T)$ via closure and
then, by the formula
\[
u(t,x)=v(t,x)+\psi(t,x),
\]
a solution to  problem \eqref{e4.1}, \eqref{e1.2}--\eqref{e1.4} in the
same space $X_0(Q_T)$ with the estimate (\ref{e4.2}) for $k=0$.

For $k=1$, we consider the  initial-boundary value problem,
in $Q_T$,
\begin{gather}\label{e4.7}
z_t+Pz=f_t,\\
\label{e4.8}
z(0,x)=\widetilde\Phi_1(x),\quad x\in(0,1),\\
\label{e4.9}
\partial_x^j z(t,0)=\mu'_j(t),\quad j=0,\dots,l-1,\\
\label{e4.10} \partial_x^j z(t,1)=\nu'_j(t),\quad j=0,\dots,l,\quad
t\in(0,T).
\end{gather}
Note that $\widetilde\Phi_1\in L^2$, hence the hypothesis of Lemma
4.1 are satisfied for this problem in the case $k=0$. Consider the
solution $z\in X_0(Q_T)$ of (\ref{e4.7})--(\ref{e4.10}) and define
\[
u(t,x)\equiv u_0(x)+ \int_0^t z(\tau,x)\,d\tau.
\]
Using the compatibility conditions, it is easy to show that the
function $u(t,x)$ is a solution to the original problem \eqref{e4.1},
\eqref{e1.2}--\eqref{e1.4} and $u, u_t \in X_0(Q_T)$. Expressing from
the equation \eqref{e4.1} the derivative
\[
\partial_x^{2l+1} u= (-1)^{l+1}(f-u_t-P_0u)
\]
and using the Ehrling inequality, one can easily obtain that
$\partial_x^{2l+1} u\in X_0(Q_T)$, thus to construct the desired
solution $u\in X_1(Q_T)$ with  estimate (\ref{e4.2}) for $k=1$.

Uniqueness of the considered problem in $L^2(Q_T)$ can be proved via
the Holmgren principle from the existence in $X_1(Q_T)$ of a
solution to the adjoint problem
\begin{gather*}
\varphi_t-P^*\varphi=f\in C_0^\infty(Q_T),\\
\varphi\bigr|_{t=T}=0,\\
\partial_x^j \varphi\bigr|_{x=0}=0,\quad j=0,\dots,l,\\
\partial_x^j \varphi\bigr|_{x=1}=0,\quad j=0,\dots,l-1
\end{gather*}
which follows by simple change of variables from the already
established existence of a solution in the same space to the
original problem.
\end{proof}

\begin{corollary}\label{C4.2}
Let the hypothesis of Lemma~\ref{L4.1} be satisfied for $k=0$,
$\mu_j=\nu_j\equiv 0$ for $j\leq l-1$. Let $u\in X_0(Q_T)$ be a
solution to corresponding problem \eqref{e4.1},
\eqref{e1.2}--\eqref{e1.4}. Then for any $t\in (0,T]$
\begin{equation}\label{e4.11}
\int_0^1 u^2(t,x)\,dx \leq \int_0^1 u_0^2\,dx+
c\int_0^t\!\!\int_0^1 u^2\,dxd\tau +
2\int_0^t (f,u)\,d\tau+\int_0^t \nu_l^2\,d\tau.
\end{equation}
\end{corollary}

\begin{proof}
In the smooth case it follows from \eqref{e2.3} and in the general
case can be obtained via closure.
\end{proof}

\begin{remark}\label{R4.2} \rm
It was shown in \cite{ramazanov} that in the case of zero initial
and boundary data for $f\in L_2(Q_T)$ the solution to the problem
\eqref{e4.1}, \eqref{e1.2}--\eqref{e1.4}, $u(t,x):u\in
L^2(0,T;H^{2l})$.
\end{remark}

Properties of solutions to linear problems estimated in Lemma~\ref{L4.1}
are enough for our next purposes except the case $l=1$, $k=0$.

Consider now an algebraic equation related to the complete linear equation
in the case $l=1$
\begin{equation}\label{e4.12}
r^3+a_2r^2+a_1r+a_0=-i\lambda, \quad \lambda\in \mathbb R \setminus \{0\}.
\end{equation}
Then there exists $\lambda_0>0$ (without loss of genarality we assume
that $\lambda_0\geq 1$) such that for $|\lambda|\geq \lambda_0$ there
exist two roots $r_0(\lambda)$ and $r_1(\lambda)$ with properties similar
to \eqref{e3.2}--(\ref{e3.5}), namely, for certain constants $\widetilde c>0$,
$\widetilde c_1>0$ and $|\lambda|\geq \lambda_0\geq 1$
\begin{equation}\label{e4.13}
\mathop{\rm Re} r_0(\lambda)\leq -\widetilde c|\lambda|^{1/3},\quad
\mathop{\rm Re} r_1(\lambda)\geq \widetilde c|\lambda|^{1/3},\quad
|r_j(\lambda)|\leq \widetilde c_1|\lambda|^{1/3},\quad j=0,1.
\end{equation}
Let
\[
\mu_{00}(t)\equiv \mathcal{F}_t^{-1}
 \left[\chi_{\lambda_0}(\lambda)\widehat\mu_0(\lambda)\right](t), \quad
\nu_{00}(t)\equiv \mathcal{F}_t^{-1}
 \left[\chi_{\lambda_0}(\lambda)\widehat\nu_0(\lambda)\right](t),
\]
where $\chi_{\lambda_0}$ is the characteristic function of the interval
$(-\lambda_0,\lambda_0)$,
\[
\mu_{01}(t)\equiv \mu_0(t)-\mu_{00}(t), \quad
\nu_{01}(t)\equiv \nu_0(t)-\nu_{00}(t).
\]
Let
\begin{equation} \label{e4.14}
\begin{aligned}
\psi(t,x)&\equiv \left(\mu_{00}(t)+
\mathcal{F}_t^{-1}\bigl[e^{r_0(\lambda)x}\widehat\mu_{01}(\lambda)\bigr](t)
\right)\eta(1-x)\\
&\quad +\left(\nu_{00}(t)+
\mathcal{F}_t^{-1}\bigl[e^{r_1(\lambda)(x-1)}\widehat\nu_{01}
(\lambda)\bigr](t)\right) \eta(x),
\end{aligned}
\end{equation}
where $\eta$ is a certain smooth ``cut-off'' function, namely,
$\eta\geq 0$, $\eta'\geq 0$, $\eta(x)=0$ for $x\leq 1/4$, $\eta(x)=1$ for
$x\geq 3/4$. Note that $\psi(t,0)\equiv \mu(t)$, $\psi(t,1)\equiv \nu(t)$.

\begin{lemma}\label{L4.2}
Let $\mu_0, \nu_0 \in H^{1/3+\varepsilon}(0,T)$ for some $\varepsilon>0$.
Then
\[
\psi\in Y_0(Q_T),\quad
\psi_x\in L^2(0,T; L^\infty), \quad
\psi_t+P(\partial_x)\psi \in C^\infty(\overline{Q}_T)
\]
(with corresponding estimates).
\end{lemma}

\begin{proof}
The fact that $\psi\in Y_0(Q_T)$ is established similarly to
\eqref{e3.16}, (\ref{e3.18}) with the use of inequalities (\ref{e4.13})
(it is sufficient to assume here that $\mu_0, \nu_0 \in H^{1/3}(0,T)$).

Next, similarly to Corollary~\ref{C3.1} the function $J_0\equiv
\mathcal{F}_t^{-1}\bigl[e^{r_0x}\widehat\mu_{01}\bigr]$ is
infinitely differentiable for $x>0$ and by virtue of (\ref{e4.12})
satisfies the homogeneous equation \eqref{e4.1}. The same properties
are valid for $J_1\equiv
\mathcal{F}_t^{-1}\bigl[e^{r_1(x-1)}\widehat\nu_{01}\bigr]$ if
$x<1$. Therefore, $\psi_t+P(\partial_x)\psi \in
C^\infty(\overline{Q}_T)$ since $\mathop{\rm supp}\eta'\subset
[1/4,3/4]$ (here it is sufficient to assume that $\mu_0, \nu_0 \in
L^2(0,T)$).

Finally, for any integer $j\geq 0$
\begin{align*}
\|\partial_x^j J_0\|_{L^2(\mathbb R\times \mathbb R_+)}
&= \bigl\|r_0^j\widehat\mu_{01}\bigl(\int_{\mathbb R_+}
e^{2\mathop{\rm Re} r_0 x}\,dx\bigr)^{1/2}\bigr\|_{L^2(\mathbb R)} \\
&\leq c \||\lambda|^{j/3-1/6}\widehat\mu_{01}\|_{L^2(\mathbb R)}\\
&\leq c_1 \|\mu_0\|_{H^{j/3-1/6}(\mathbb R)},
\end{align*}
whence by interpolation it follows that for $s\geq 0$
\[
\|J_0\|_{L^2(\mathbb{R};H^s(\mathbb{R}))}\leq c\|\mu_0\|_{H^{s/3-1/6}(\mathbb R)}.
\]
Similar arguments can be applied to the function $J_1$ and so the
well-known embedding $H^{1/2+\varepsilon}\subset L^\infty$
provides the property
$\psi_x\in L^2(0,T; L^\infty)$.
\end{proof}

\section{Results for local solutions}

In this section local well-posedness for the original nonlinear problem
is proved under natural assumptions on initial and boundary data.

\begin{proof}[Proof of Theorem~\ref{T2.1}]
For $t_0\in (0,T]$ introduce a set of functions
\[
\widetilde X_k(Q_{t_0})= \{v(t,x)\in X_k(Q_{t_0}):
\partial_t^n v\bigr|_{t=0}=\Phi_n,\; n=0,\dots,k-1\}
\]
and define on this set a map $\Lambda$ in such a way: $u=\Lambda
v\in \widetilde X_k(Q_{t_0})$ is a solution in $Q_{t_0}$ to an
initial boundary value problem for the equation
\begin{equation}\label{e5.1}
u_t+Pu=f-vv_x
\end{equation}
with initial and boundary conditions \eqref{e1.2}--\eqref{e1.4}.
Making use of Lemma~\ref{L4.1}, we have to estimate
$\|vv_x\|_{M_k(Q_{t_0})}$.
Let $k=0$, then
\begin{equation} \label{e5.2}
\begin{aligned}
\|vv_x\|_{L^2(0,t_0;H^{-l})}
&\leq \frac 12\|v\|^2_{L^4(Q_{t_0})}\\
&\leq \frac 12 \sup_{t\in [0,t_0]} \|v\|_{L^2}
\Bigl(\int_0^{t_0} \sup_{x\in [0,1]} v^2\,dt\Bigr)^{1/2} \\
&\leq c \sup_{t\in [0,t_0]} \|v\|_{L^2}
\Bigl(\int_0^{t_0} \left(\|v_x\|_{L^2}\|v\|_{L^2}+
\|v\|_{L^2}^2\right)\,dt\Bigr)^{1/2}\\
&\leq c_1 t_0^{1/4}\|v\|^2_{X_0(Q_{t_0})}.
\end{aligned}
\end{equation}
Let $k=1$, then
\begin{align*}
&\|vv_x\|_{L^2(0,t_0;H^l)} \\
&\leq c\sum_{j=0}^{l+1}
\|\partial_x^jv\|^2_{L^4(Q_{t_0})} \\
&\leq c_1\sum_{j=0}^{l+1} \sup_{t\in [0,t_0]}
\|\partial_x^jv\|_{L^2} \Bigl(\int_0^{t_0}
\left(\|\partial_x^{j+1}v\|_{L^2}\|\partial_x^jv\|_{L^2}+
\|\partial_x^jv\|_{L^2}^2\right)\,dt\Bigr)^{1/2} \\
&\leq c_2 t_0^{1/2}\|v\|^2_{X_1(Q_{t_0})};
\end{align*}
similarly to (\ref{e5.2}),
\begin{equation} \label{e5.3}
\begin{aligned}
\|(vv_x)_t\|_{L^2(0,t_0;H^{-l})}
&\leq \|vv_t\|_{L^2(Q_{t_0})}\\
&\leq \sup_{t\in [0,t_0]} \|v_t\|_{L^2}
\Bigl(\int_0^{t_0} \sup_{x\in [0,1]} v^2\,dt\Bigr)^{1/2} \\
&\leq ct_0^{1/4} \|v_t\|_{X_0(Q_{t_0})}\|v\|_{X_0(Q_{t_0})}\\
&\leq ct_0^{1/4} \|v\|_{X_1(Q_{t_0})}^2
\end{aligned}
\end{equation}
and
\begin{align*}
\|vv_x\|_{C([0,t_0];L^2)}
&\leq \|u_0u_0'\|_{L_2}+ \|(vv_x)_t\|_{L^1(0,t_0;L^2)}\\
&\leq c\|u_0\|^2_{H^{2l+1}}+ct_0^{1/2}\|(vv_x)_t\|_{L^2(Q_{t_0})} \\
&\leq c\|u_0\|^2_{H^{2l+1}}+ct_0^{1/2}\|v\|_{C([0,t_0];C^1)}
\|v_t\|_{L^2(0,t_0;H^1)}\\
&\leq  c\|u_0\|^2_{H^{2l+1}}+c_1 t_0^{1/2}\|v\|^2_{X_1(Q_{t_0})}.
\end{align*}
The cases $k\geq 2$ can be handled in the same manner as the case
$k=1$.

As a result, the map $\Lambda$ exists and it follows from
(\ref{e4.2}) that
\begin{equation}\label{e5.4}
\|\Lambda v\|_{X_k(Q_{t_0})} \leq c\left(1+t_0^{1/4}
\|v\|^2_{X_k(Q_{t_0})}\right).
\end{equation}
By  standard arguments, see \cite{kenig2}, it can be derived from
(\ref{e5.4}) that for small $t_0$ the map $\Lambda$ transforms a
certain large ball in $\widetilde X_k(Q_{t_0})$ into itself.
Similarly to (\ref{e5.4}), one can obtain for two functions $v,
\widetilde v\in \widetilde X_k(Q_{t_0})$:
\[
\|\Lambda v-\Lambda\widetilde v\|_{X_k(Q_{t_0})} \leq c
\left(\|v\|_{X_k(Q_{t_0})},\|\widetilde v\|_{X_k(Q_{t_0})}\right)
t_0^{1/4}\|v-\widetilde v\|_{X_k(Q_{t_0})},
\]
and  $t_0$ can be reduced in  such a way that $\Lambda$ becomes a
contraction on this ball in $\widetilde X_k(Q_{t_0})$, therefore,
there exists a unique solution $u\in X_k(Q_{t_0})$ to  problem
\eqref{e1.1}--\eqref{e1.4}.
Continuous dependence can be established by analogous arguments.
\end{proof}

\begin{remark}\label{R5.1} \rm
The idea to use the contraction principle to establish local
well-posedness for KdV-like equations goes back to \cite{kenig2}.
\end{remark}

\section{Results for global solutions}

Theorem~\ref{T2.2} succeeds from the already established local
well-posedness and the following global a priori estimates.

\begin{lemma}\label{L6.1}
Let the hypothesis of Theorem~\ref{T2.2} be satisfied and
$u(t,x)\in X_k(Q_{T'})$ be a solution to
\eqref{e1.1}--\eqref{e1.4} for certain $T'\in (0,T]$. Then
\begin{equation}\label{e6.1}
\|u\|_{C([0,T'];H^{(2l+1)k})}\leq c,
\end{equation}
where the constant $c$ depends upon $T$, $l$, $k$;
properties of the operator $P_0$ and the norms of initial data,
boundary data and the right-hand side of \eqref{e1.1} in the spaces
from the hypothesis of the theorem hold, but do not depend on $T'$.
\end{lemma}

\begin{proof}
First put $k=0$. Let for $l=1$ a function $\psi$ is given by formula
(\ref{e4.14}) and for $l\geq 2$ a function $\psi\in X_0(Q_T)$ be
a solution to  \eqref{e4.1}, \eqref{e1.2}--\eqref{e1.4} for the
same $u_0$, $\mu_j$, $\nu_j$ and $f$. Then
\[
\psi \in L^2(0,T;H^l)\cap C([0,T];L^2).
\]
Define
\begin{gather*}
U(t,x)\equiv u(t,x)-\psi(t,x),\\
F\equiv f-uu_x-\psi_t-P(\partial_x)\psi\quad \text{for } l=1 \quad \text{and }\quad
F\equiv -uu_x\quad \text{for } l\geq 2.
\end{gather*}
Then the function $U$ satisfies
\[
U_t+PU=F\in L^1(0,T;L^2)\cap L^2(0,T;H^{-l}),
\]
\[
U\big|_{x=0}=U\big|_{x=1}=0,\quad
U_x\big|_{x=1}=\nu_1-\psi_x\big|_{x=1}\in L^2(0,T)\quad \text{for } l=1,
\]
zero boundary conditions of  \eqref{e1.3}, \eqref{e1.4} type for
$l\geq 2$ and the initial condition
\[
U(0,x)=U_0(x),
\]
where $U_0\equiv u_0-\psi|_{t=0}\in L^2$ for $l=1$ and $U_0\equiv 0$ for
$l\geq 2$. Write down  inequality \eqref{e4.11} for the function $U$.
Since
\[
2\int_0^1 uu_xU\,dx = \int_0^1 \psi_x(U^2+2U\psi)\,dx,
\]
where $\psi_x\in L^2(0,T;L^\infty)$,  estimate \eqref{e6.1} for $k=0$
(and, consequently, desired global well-posedness) follows.

Next let $k=1$, then a function $u_1\equiv u_t\in X_0(Q_{T'})$ is a
solution to an initial boundary value problem for the equation
\[
u_{1t}+Pu_1= (f-uu_x)_t
\]
with initial data $\Phi_1=f\bigr|_{t=0}-Pu_0-u_0u'_0\in L^2$ and
boundary data $(\mu'_0,\dots,\mu'_{l-1})\in \mathcal{B}_0^{l-1}(0,T)$, $(\nu'_0,\dots,\nu'_l)\in \mathcal{B}_0^l(0,T)$. We
use for the function $u_1$ the inequality (\ref{e4.2}) in the case
$k=0$. Note that, by virtue of (\ref{e5.3}) (where $v$ is substituted
by $u$), for any $t_0\in (0,T']$
\[
\|(uu_x)_t\|_{L^2(0,t_0;H^{-l})} \leq ct_0^{1/4}\|u\|_{X_0(Q_{T'})}
\|u_t\|_{X_0(Q_{t_0})}.
\]
Since for $k=0$ global well-posedness is already established and,
consequently, $\|u\|_{X_0(Q_{T'})}\leq c$, we first derive from
inequality (\ref{e4.2}) that
\[
\|u_t\|_{X_0(Q_{t_0})}\leq c\left(1+\|u_t\bigr|_{t=0}\|_{L^2}+
t_0^{1/4}\|u_t\|_{X_0(Q_{t_0})}\right)
\]
and, finally, by standard arguments the estimate
\begin{equation}\label{e6.3}
\|u_t\|_{C([0,T'];L^2)}\leq c.
\end{equation}
Using the equality
\[
\partial_x^{2l+1}u=(-1)^{l+1}(f-u_t-P_0u-uu_x),
\]
the Ehrling inequality, \eqref{e6.1} for $k=0$, and
(\ref{e6.3}), one can obtain  estimate \eqref{e6.1} for $k=1$.
The cases $k\geq 2$ are handled similarly to the case $k=1$.
\end{proof}

\section{Decay of small solutions}

Consider \eqref{e1.1}--\eqref{e1.4} with zero boundary
data, $f\equiv 0$ and small initial data $u_0$. Define
\begin{equation}\label{e7.1}
A_j=(-1)^{j+1}(2j+1)a_{2j+1}+(-1)^j\sigma a_{2j}, \quad j=0,
\dots,l,
\end{equation}
where $\sigma=2$ if $(-1)^j a_{2j}\geq 0$, $\sigma=4$ if $(-1)^j
a_{2j}< 0;\;(-1)^{l+1}a_{2l+1}=1$.

\begin{theorem}\label{T7.1}
Let $u_0\in L^2$, $\mu_j=\nu_j=\nu_l\equiv 0$ for $j=0,\dots,l-1$,
$f\equiv 0$ and Assumption A is satisfied. Let
\begin{gather}\label{e7.2}
A_l+\sum_{j: A_j <0} 2^{3(j-l)}A_j=2K>0 \\
\label{e7.3}
\|(1+x)^{\frac{1}{2}}u_0\|_{L^2} < 3\times 2^{3(l-1)} K.
\end{gather}
Then a unique solution $u(t,x)$ to
\eqref{e1.1}--\eqref{e1.4}, such that $u\in X_0(Q_T)$ for any
$T>0$, satisfies the following inequality, for all $t\geq 0$,
\begin{equation}\label{e7.4}
\|u(t,\cdot)\|^2_{L^2}\leq 2e^{-\kappa t}\|u_0\|^2_{L^2},
\end{equation}
where
\[
\kappa= 2^{3l}K+\sum_{j: A_j \geq 0} 2^{3j}A_j.
\]
\end{theorem}

\begin{proof}
First of all note that the hypothesis of Theorem~\ref{T2.2} are
satisfied, hence such a unique solution exists. By Assumption A,
$(-1)^la_{2l}\geq 0$, hence $A_l\geq 2l+1> 0.$ Multiplying
\eqref{e1.1} by $2(1+x) u(t,x)$ and integrating, we find
\begin{equation} \label{e7.5}
\begin{aligned}
&\frac{d}{dt}\int_0^1 (1+x) u^2(t,x)\,dx \\
&+\sum_{j=0}^{l} \int_0^1
\bigl[(-1)^{j+1}(2j+1)a_{2j+1}+
(-1)^j2a_{2j}(1+x)\bigr](\partial_x^j u)^2\,dx
-\frac 23  \int_0^1 u^3\,dx= 0.
\end{aligned}
\end{equation}
(In fact, such a calculation must be first performed for smooth
solutions and the general case can be obtained via closure). We use
the Friedrichs inequality as follows: for any $\varphi\in H_0^l$
\begin{equation}\label{e7.6}
\|\varphi\|_{L^\infty}\leq 2^{1-3l/2}\|\varphi^{(l)}\|_{L^2},\quad
\|\varphi\|_{L^2}\leq 2^{-3l/2}\|\varphi^{(l)}\|_{L^2}.
\end{equation}
Then
\[
\Bigl|\int_0^1 u^3\,dx\Bigr| \leq \|u\|_{L^\infty}^2\|u\|_{L^2}\leq
2^{2-3l}\|u(t,\cdot)\|_{L^2}\|\partial_x^l u\|^2_{L^2}.
\]
This and \eqref{e7.1}, \eqref{e7.2} allow us to rewrite \eqref{e7.5} as
\[
 \frac{d}{dt}\int_0^1 (1+x) u^2(t,x)\,dx+ \int_0^1
 \bigl[2K-\frac{1}{3}2^{3(1-l)}\|(1+x)^{\frac{1}{2}}u(t,\cdot)\|_{L^2}\bigr]
 (\partial^l_x u)^2\,dx\leq 0.
 \]
Taking into account \eqref{e7.3} and exploiting  standard arguments, one
can prove that
\[
\|(1+x)^{\frac{1}{2}}u(t,\cdot)\|_{L^2} < 3\times 2^{3(l-1)} K
\mbox{ for all}\; t\geq 0.
\]
Returning to \eqref{e7.5}, we rewrite it as
\[
\frac{d}{dt}\int_0^1 (1+x) u^2(t,x)\,dx+ \int_0^1 2^{3l} K u^2\,dx
+\int_0^1\sum_{j: A_j \geq 0} 2^{3j}A_j u^2\,dx \leq 0,
\]
whence
\[
\frac{d}{dt}\int_0^1 (1+x) u^2(t,x)\,dx+ \kappa\int_0^1 (1+x) u^2\,dx
\leq 0.
\]
 From here follows \eqref{e7.4}.
\end{proof}

\begin{remark}\label{R7.1} \rm
Zero boundary data are chosen here for simplicity in order to show
the idea of the method. In the general case similar results can be
established for a difference between the solution to problem
\eqref{e1.1}--\eqref{e1.4} and a solution to a certain linear
problem with the same boundary data under suitable assumptions on
behavior of the solution of this linear problem as $t\to +\infty$.
\end{remark}

\begin{remark}\label{R7.2} \rm
In \cite{goubet} a non-trivial stationary solution to the
initial-boundary value problem for the homogeneous KdV equation
under zero boundary data \eqref{e1.3}, \eqref{e1.4} (here $l=1$) was
constructed. Therefore certain assumptions on the initial data $u_0$
are necessary for the decay of the corresponding solution as $t\to
+\infty$.
\end{remark}

\begin{thebibliography}{00}

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