\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 127, pp. 1--23.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/127\hfil Well-posed initial-boundary value problems]
{Well-posed initial-boundary value problems for the
Zakharov-Kuznetsov equation}

\author[A. V. Faminskii\hfil EJDE-2008/127\hfilneg]
{Andrei V. Faminskii}

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

\thanks{Submitted February 23, 2008. Published September 10, 2008.}
\thanks{Supported by grant 06-01-00253 from the RFBR}
\subjclass[2000]{35Q72}
\keywords{Zakharov-Kuznetsov equation; boundary value problems;
 \hfill\break\indent well-posedness}

\begin{abstract}
 This paper deals with non-homogeneous initial-boundary value problems
 for the Zakharov-Kuznetsov equation, which is one of the variants of
 multidimensional generalizations of the Korteweg~de~Vries equation.
 Results on local and global well-posedness are established in a scale
 of Sobolev-type spaces under natural assumptions on initial and boundary
 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}
\allowdisplaybreaks

\section{Introduction}\label{S1}

The goal of the present paper is to study initial-boundary value problems
for the Zakharov--Kuznetsov (ZK) equation
\begin{equation}\label{1.1}
u_t+u_{xxx}+u_{xyy}+uu_x=f(t,x,y)
\end{equation}
($u=u(t,x,y)$) in three domains:
\begin{gather*}
 \Pi_T^+=\{(t,x,y): t\in (0,T),x>0,y\in \mathbb{R}\}\equiv (0,T)\times\mathbb{R}^2_+,\\
 \Pi_T^-=\{(t,x,y): t\in (0,T),x<0,y\in \mathbb{R}\}\equiv (0,T)\times\mathbb{R}^2_-,\\
 Q_T=\{(t,x,y): t\in (0,T),x\in(0,1),y\in \mathbb{R}\}\equiv
(0,T)\times\Sigma,
\end{gather*}
where $T>0$. In all three cases we set an initial condition
\begin{equation}\label{1.2}
u(0,x,y)=u_0(x,y)
\end{equation}
(where respectively $(x,y)\in \mathbb{R}^2_+$, $(x,y)\in
\mathbb{R}^2_-$, $(x,y)\in \Sigma$) and the following boundary
conditions for $(t,y)\in B_T=(0,T)\times\mathbb{R}$:

\noindent (1) for the problem in $\Pi_T^+$ one condition:
\begin{equation}\label{1.3}
u(t,0,y)=u_1(t,y),
\end{equation}
(2) for the problem in $\Pi_T^-$ two conditions:
\begin{equation}\label{1.4}
u(t,0,y)=u_2(t,y),\quad u_x(t,0,y)=u_3(t,y),
\end{equation}
(3) for the problem in $Q_T$ three conditions:
\begin{equation}\label{1.5}
u(t,0,y)=u_1(t,y),\quad u(t,1,y)=u_2(t,y),\quad u_x(t,1,y)=u_3(t,y).
\end{equation}

The ZK equation is one of the variants of multidimensional generalizations
of the famous Korteweg-de~Vries equation (KdV)
\begin{equation}\label{1.6}
u_t+u_{xxx}+uu_x=f(t,x).
\end{equation}
It describes nonlinear wave processes in dispersive media, when waves
propagate in the $x$-direction and are deformated in the transverse
$y$-direction. In particular, it is a model equation for ion-acoustic
waves in magnetized plasma, \cite{zakharov}.

Initial-boundary value problems for the KdV equation in domains similar
to $\Pi_T^+$, $\Pi_T^-$ and $Q_T$ (without the variable $y$) have been
intensively studied in the recent years (see \cite{colliander,
bona02,bona03,faminskii04,holmer,faminskii07} for the last results
and references there). In the first approximation the scheme of these
investigations is similar and consists of 1) a proof of local well-posedness
based on the contraction principle, where solutions are constructed
as fixed points of mappings $u=\Lambda v$ such that $u$ is a solution to
a corresponding initial-boundary value problem for an equation
$u_t+u_{xxx}=f-vv_x$, and 2) global a priori estimates based on
conservation laws for the initial value problem for KdV ($f\equiv 0$):
\begin{equation}\label{1.7}
\int_\mathbb{R}u^2\,dx=\text{const},\ \int_\mathbb{R}(u_x^2-\frac
13 u^3)\,dx=\text{const},\ \int_\mathbb{R}(u_{xx}^2+\frac 56
u^2u_{xx}+\frac 5{36} u^4)\,dx=\text{const}.
\end{equation}
For the initial value problem for KdV this scheme was for the first time
implemented in \cite{kenig}.

One of the common features of the aforementioned papers is an idea
to establish well-posedness under natural assumptions on initial
and boundary data, namely, $u_0\in H^s$, $u_1,u_2\in H^{(s+1)/3}$,
$u_3\in H^{s/3}$. Such assumptions originate from internal
properties of the operator $\partial_t+\partial_x^3$. In fact, if
$u(t,x)\in C(\mathbb{R}^t;H^s(\mathbb{R}^x))$ is a solution to the
initial value problem
\begin{equation}\label{1.8}
u_t+u_{xxx}=0,\quad u\big|_{x=0}=u_0(x)\in H^s(\mathbb{R}),\quad
s\in\mathbb{R},
\end{equation}
then for any $x\in\mathbb{R}$
\begin{equation}\label{1.9}
\|D_t^{1/3}u(\cdot,x)\|_{H^{s/3}{(\mathbb{R}^t)}}=
\|u_x(\cdot,x)\|_{H^{s/3}{(\mathbb{R}^t)}}=c(s)
\|u_0\|_{H^s{(\mathbb{R})}}
\end{equation}
(see, for example, \cite{kenig}).

The pointed out approach requires the necessity of study of the
corresponding initial-boundary value problems for the linearized
KdV equation. In \cite{faminskii04,faminskii07} solutions to such
problems are constructed via combination of solutions to the
initial value problem and solutions to the initial-boundary value
problems for the homogeneous linearized equation (\ref{1.8}) with
zero initial data, which can be referred as "boundary potentials".
For the problem in a right half-strip $(0,T)\times \mathbb{R}_+$
such a boundary potential $J$ was for the first time introduced in
\cite{cattabriga} with the use of the Airy function. Alternative
representations for this function $J$ were obtained in the papers
\cite{bona02,faminskii04}. For example, in \cite{faminskii04} the
following formula was derived:
\begin{equation}\label{1.10}
J(t,x;u_1)=\EuScript{F}_t^{-1}\big[e^{r(\lambda)x}\widehat
u_1(\lambda)\big](t)
\end{equation}
for $x\geq 0$, where
$r(\lambda)=-\frac12(\sqrt3|\lambda|^{1/3}+i\lambda^{1/3})$ is the
unique root of the algebraic equation $r^3+i\lambda=0$,
$\lambda\ne 0$, with the negative real part. Similar boundary
potentials for the problem in a left half-strip $(0,T)\times
\mathbb{R}_-$ were constructed in \cite{faminskii07}. All these
boundary potentials were also used in that paper for the problem
in a bounded rectangle $(0,T)\times(0,1)$.

As a result, local well-posedness under natural assumptions on initial and
boundary data was established for all three initial-boundary value problems
for the KdV equation if $s>-3/4$, $s\ne 3m+1/2$, $s\ne 3m+3/2$
(for the last two problems), $m\geq 0$ -- integer,
\cite{faminskii04,holmer,faminskii07}. Solutions to these problems were
constructed, in particular, in functional spaces of Bourgain type, first
introduced in \cite{bourgain} for the initial value problem and modified
in \cite{colliander} for initial-boundary value problems.

In comparison with the initial value problem presence of boundary
conditions produces additional difficulties for global a priori
estimates in the case of initial-boundary value problems.
Consider, for example, an estimate in $L_2$. Let $I$ be either
$\mathbb{R}$ or $\mathbb{R}_+$ or $\mathbb{R}_-$ or $(0,1)$ and
let $\partial I$ denotes the finite part of its boundary. Let
$u(t,x)$ be a solution of the equation (\ref{1.6}), where $f\equiv
0$, in $(0,T)\times I$ sufficiently smooth and decaying at
infinity. Multiplying (\ref{1.6}) by $2u$ and integrating over $I$
one obtains an equality
\begin{equation}\label{1.11}
\frac d{dt}\int_I u^2\,dx+ \bigl(2uu_{xx}-u_x^2+
\frac 23 u^3\bigr)\bigr|_{\partial I}=0.
\end{equation}
For $I=\mathbb{R}$ this equality coincides with the first
conservation law (\ref{1.7}). For the initial-boundary value
problems in the case $u|_{\partial I}=0$ an estimate on the
solution $u$ in $L_2(I)$ uniform with respect to $t\geq 0$ also
succeeds from (\ref{1.11}). But in the case of non-homogeneous
boundary data the presence of the term $uu_{xx}|_{\partial I}$
makes it impossible to derive such an estimate directly from
(\ref{1.11}). Then it is quite natural to introduce an auxiliary
function $\varphi(t,x)$ such that $\varphi|_{\partial
I}=u|_{\partial I}$, define a new function $U(t,x)\equiv
u(t,x)-\varphi(t,x)$ and try to obtain the desired estimate first
for the function $U$. This function satisfies a more complicated
equation, so this approach implies, that the function $\varphi$
can be chosen such that its properties ensure such a possibility.
In the papers \cite{faminskii04, faminskii07} the function
$\varphi$ was constructed on the base of the boundary potential
$J$ and the estimates in $L_2$ were obtained under
$\varepsilon$-close to natural $u_1,u_2\in H^{1/3+\varepsilon}$,
$u_3\in L_2$ assumptions on the boundary data.

Further obstacles appear for estimates in more smooth spaces, e.g. in
$H^1$ and $H^2$. The difficulties on this way can be shown even in the linear
case and zero boundary data. Multiplying the equation (\ref{1.8}) by
$-2u_{xx}(t,x)$ and integrating over $I$ one derives an equality
\begin{equation}\label{1.12}
\frac d{dt}\int_I u_x^2\,dx - u_{xx}^2\bigr|_{\partial I}=0,
\end{equation}
so an estimate on $u_x$ in $L_2(I)$ can be obtained only for the problem
in the right half-strip. Next, multiplying this equation by $2u_{xxxx}(t,x)$
and integrating over $I$ one derives an equality
\begin{equation}\label{1.13}
\frac d{dt}\int_I u_{xx}^2\,dx - 2u_{tx}u_{xx}\bigr|_{\partial I}=0
\end{equation}
and here the estimate on $u_{xx}$ in $L_2(I)$ can be obtained only for the
problem in the left half-strip.

Note that differentiation with respect to $t$ leads to the initial-boundary
value problem of the same type for the derivative $u_t$. Therefore, for example,
an estimate on the solution $u$ in $H^3(I)$ can be obtained from an estimate
for $u_t$ in $L_2(I)$ via expressing the third derivative $u_{xxx}$ from the
equation (\ref{1.6}) itself.

On this way estimates on solutions in $H^{3k}$, $H^{3k+1}$, $k\geq 0$ --
integer, to the problem in the right half-strip, in $H^{3k}$, $H^{3k+2}$ to the
problem in the left half-strip and in $H^{3k}$ to the problem in the bounded
rectangle were established in \cite{faminskii04,faminskii07} and for
intermediate orders of smoothness, following the approach from \cite{bona02},
nonlinear interpolation was used. As a result, global well-posedness of
all three considered initial-boundary value problems for the KdV equation
was established in these papers under natural assumptions on initial and
boundary data for $s>0$, $s\ne 3m+1/2$, $s\ne 3m+3/2$
(for the last two problems), $m\geq 0$ -- integer, and
$\varepsilon$-close to natural for $s=0$.

The study of the ZK equation in comparison with KdV besides
traditional difficulties originating from the transfer from the
line to the plane has some additional obstacles. First of all,
Bourgain-type spaces, which turned out to be very useful for KdV,
are not found yet for this equation. Next, in contrast to
(\ref{1.7}) only two conservation laws are known for \eqref{1.1},
$f\equiv 0$:
\begin{equation}\label{1.14}
\iint_{\mathbb{R}^2}u^2\,dx\,dy=\text{const},\quad
\iint_{\mathbb{R}^2}(u_x^2+u_y^2-\frac 13 u^3)\,dx\,dy=\text{const}.
\end{equation}
Note that first global existence result (without uniqueness) for
the initial value problem for ZK in the space
$L_\infty(0,T;H^1(\mathbb{R}^2))$ in the case $u_0\in
H^1(\mathbb{R}^2)$ was, in particular, established in \cite{saut}
just on the base of these conservation laws.

On the other hand, the so-called local smoothing effect is valid
for this equation as for KdV. Let $u(t,x,y)$ be a smooth and
decaying at infinity solution to the initial value problem
\eqref{1.1}, \eqref{1.2}, where $f\equiv 0$. Multiplying
\eqref{1.1} by $2u(t,x,y)\rho(x)$ for certain smooth, non-negative
and non-decreasing function $\rho$ one can easily derive after
integration that
\begin{equation}\label{1.15}
\frac d{dt}\iint_{\mathbb{R}^2} u^2\rho\,dx\,dy+
\iint_{\mathbb{R}^2} (3u_x^2+u_y^2)\rho'\,dx\,dy-
\iint_{\mathbb{R}^2} \bigl(u^2\rho'''+ \frac 23
u^3\rho'\bigr)\,dx\,dy=0,
\end{equation}
and after an appropriate choice of $\rho$, making use of the first
conservation law (\ref{1.14}), establish an estimate
\begin{equation}\label{1.16}
\lambda(u;T)=\sup_{m\in \mathbb Z}
\int_0^T\!\!\int_m^{m+1}\!\!\int_\mathbb{R} (u_x^2+u_y^2)\,dy\,dx\,dt
\leq c(T,\|u_0\|_{L_2(\mathbb{R}^2)}).
\end{equation}
The estimate (\ref{1.16}) gave an opportunity in the paper
\cite{faminskii89} to establish global existence result for the
problem \eqref{1.1}, \eqref{1.2} in the class $\{u: u\in
L_\infty(0,T;L_2(\mathbb{R}^2)), \lambda(u;T)<\infty\}$ for
$u_0\in L_2(\mathbb{R}^2)$ (in fact, in \cite{saut,faminskii89}
more general quasilinear evolution equations of an arbitrary high
odd order in the multidimensional case were considered). Moreover,
if, in addition, $xu_0\in L_2(\mathbb{R}_+^2)$, a class of global
well-posedness was constructed for this problem in
\cite{faminskii89}.

In the paper \cite{faminskii95} results from \cite{kenig} on
global well-posedness of the initial value problem for KdV were
transferred to ZK, namely, classes of global well-posedness for
the problem \eqref{1.1}, \eqref{1.2} were constructed for $u_0\in
H^k(\mathbb{R}^2)$, $k$ -- natural (see Remark~\ref{R2.4} below).

In \cite{levandosky} gain of regularity for solutions to the initial value
problem for ZK under decaying at infinity initial data was established.

The study of initial-boundary value problems for the ZK equation
started only in recent years (with the only exception in
\cite{pyatkov}, where one problem in a bounded domain for an
equation, which can be reduced to ZK by a simple transformation,
was considered). Certain results in the case $u_0\in L_2$ on
global existence and uniqueness of weak solutions to the problem
\eqref{1.1}--\eqref{1.3} in $\Pi_T^+$ were obtained in
\cite{faminskii02} and similar results on global existence to the
problem \eqref{1.1}, \eqref{1.2}, \eqref{1.4} in $\Pi_T^-$ -- in
\cite{baykova}. These results are as in \cite{faminskii89} based
on the first conservation law (\ref{1.14}) and the local smoothing
effect (\ref{1.15}), (\ref{1.16}) (in more details they are
discussed further in Section~\ref{S6}).

The approach of the present paper repeats the one from \cite{faminskii04,
faminskii07} (besides the use of Bourgain-type spaces, of course). Special
solutions of the "boundary potential" type are constructed and studied
for a linearized ZK equation and further used for linear
problems in $\Pi_T^+$, $\Pi_T^-$ and $Q_T$. Solutions to the corresponding
linear initial value problem, which was previously studied in
\cite{faminskii95}, are also used here. Moreover, properties of this
initial value problem show, that by analogy with (\ref{1.9}) smoothness
assumptions $u_0\in H^k$, $u_1,u_2\in H^{(k+1)/3,k+1}_{t,y}$,
$u_3\in H^{k/3,k}_{t,y}$, where $H^{s_1,s_2}$ are anysotropic Sobolev spaces,
are natural for the considered initial-boundary value problems
(see Remark~\ref{R3.1} below).

Then local well-posedness of all three considered problems for the ZK equation
under natural assumptions on boundary data is established for $u_0\in H^k$
via the contraction principle.

Global a priori estimates in $L_2$ for the considered problems are obtained
by methods similar to ones for KdV (see (\ref{1.11}) and subsequent arguments).
Obstacles similar to (\ref{1.12}), (\ref{1.13}) also appear for the ZK equation,
so an estimate in $H^1$ is established (by methods similar to ones for the
second conservation law (\ref{1.14})) for the problem in $\Pi_T^+$. The
absence of an analogue for ZK of the third conservation law (\ref{1.7})
has not allowed to establish a global estimate in $H^2$ for the problem in
$\Pi_T^-$.

Global a priori estimate in $H^3$ for the problem in $Q_T$ are
obtained via differentiation of the equation with respect to $t$
and to $y$. Note that in comparison with KdV an extra obstacle to
establish such an estimate is that one can express from the
equation \eqref{1.1} not a single derivative of the third order
but the term $(u_{xxx}+u_{xyy})$.

Other global estimates in more smooth classes are obtained on the basis
of the aforementioned ones.

As a result, global well-posedness is established for the problem
\eqref{1.1}--\eqref{1.3} in $\Pi_T^+$ for $u_0\in
H^k(\mathbb{R}_+^2)$, $k$ -- natural, and for the problem
\eqref{1.1}, \eqref{1.2}, \eqref{1.5} in $Q_T$ for $u_0\in
H^k(\Sigma)$, $k\geq 3$ -- natural, under natural assumptions on
the boundary data (the result for the problem in $\Pi_T^+$ in the
case $k=1$ was previously published in \cite{faminskii06}). Global
well-posedness for the problem \eqref{1.1}, \eqref{1.2},
\eqref{1.4} in $\Pi_T^-$ is an open problem.

The paper is organized as follows. Section~\ref{S2} contains main notation
and a statement of the main result on local and global well-posedness of the
considered problems. In Section~\ref{S3} potentials for the linearized ZK
equation are studied. Section~\ref{S4} is devoted to the corresponding
initial-boundary value problems for this linear equation. The proof of the
main result is accomplished in Section~\ref{S5}. Certain remarks on global
weak solutions to the considered problems can be found in Section~\ref{S6}.

\section{Notation and Statement of the main result}\label{S2}

In what follows (if there are no other conditions) in introduced
notation we use a symbol $I$ for an arbitrary interval (bounded or
unbounded) on the real axis, $\Omega$ -- for a domain in
$\mathbb{R}^n$, $k$, $l$, $m$, $n$, $j$ -- non-negative integers,
$p\in [1,+\infty]$, $s\in \mathbb{R}$.

Let $[s]$ be the integer part of $s$ ($s-[s]\in [0,1)$).

Let $C_b^k(\overline{\Omega})$ be a space of functions with all derivatives up to
the order $k$ continuous and bounded in $\overline{\Omega}$. Define
$C_b(\overline{\Omega})=C_b^0(\overline{\Omega})$.
If $\Omega$ is bounded, the index $b$ is omitted.

Let $\widehat f\equiv \EuScript{F}[f]$ and $\EuScript{F}^{-1}[f]$
be respectively the direct and inverse Fourier transforms of a
function $f$, considered as operations in $L_2(\mathbb{R}^n)$. In
particular, for $f\in \EuScript S(\mathbb{R})$
$$
\widehat f(\xi)=\int_\mathbb{R} e^{-i\xi x} f(x)\,dx,\quad
\EuScript{F}^{-1}[f](x)=\frac 1 {2\pi} \int_\mathbb{R} e^{i\xi x}
f(\xi)\,d\xi.
$$

Define the fractional order Sobolev space
$$
H^s(\mathbb{R}^n)= \bigl\{f: \EuScript{F}^{-1}[(1+|\xi|)^s\widehat
f(\xi)] \in L_2(\mathbb{R}^n)\bigr\}
$$
and let $H^s(\Omega)$ be a space of restrictions on $\Omega$ of
functions from $H^s(\mathbb{R}^n)$. Note that
$H^k(\Omega)=W_2^k(\Omega)$. Define
$$
H_0^s(\Omega)=\bigl\{f\in H^s(\mathbb{R}^n): \mathop{\rm supp.} f\subset
\overline{\Omega}\bigr\}.
$$
Properties of the spaces $H^s$ and $H^s_0$ can be
found, for example, in \cite{lions}.

For domains $\Omega\subset \mathbb{R}^2$ with regular boundaries
(in particular, for $\Omega=\mathbb{R}_+^2$ and $\Omega=\Sigma$)
the following interpolation inequality is valid:
\begin{equation}\label{2.1}
\|f\|_{L_p(\Omega)}\leq c(p,\Omega)\left[ \|\nabla f\|_{L_2(\Omega)}^{(p-2)/p}
\|f\|_{L_2(\Omega)}^{2/p}+\|f\|_{L_2(\Omega)}\right],
\end{equation}
where $2\leq p<+\infty$ (see, e.g. \cite{besov}).

For description of properties of boundary data we also use the anysotropic
Sobolev spaces for $s_1,s_2\geq 0$:
$$
H^{s_1,s_2}(\mathbb{R}^2)=H^{s_1,s_2}_{t,y}(\mathbb{R}^2)=
\bigl\{\mu(t,y):
\EuScript{F}^{-1}[(1+|\lambda|^{s_1}+|\eta|^{s_2})\widehat
\mu(\lambda,\eta)] \in L_2(\mathbb{R}^2)\bigr\}.
$$
For $\Omega\subset \mathbb{R}^2$ a symbol $H^{s_1,s_2}(\Omega)$ is
also used for a space of corresponding restrictions.

If $\EuScript{B}$ is a certain Banach space, define by
$C_b(\overline{I};\EuScript{B})$ a space of continuous bounded mappings from
$\overline{I}$ to $\EuScript{B}$ (for bounded $I$ the index $b$ is omitted).
The symbol $L_p(I;\EuScript{B})$ is used in the conventional sense.

Solutions to the considered problems are constructed in special
functional spaces $X_k$.

\begin{definition}\label{D2.1} \rm
For any $T>0$ let $X_k((0,T)\times I\times\mathbb{R})$ be a space
of functions $u(t,x,y)$ such that
\begin{gather}\label{2.2}
\partial_t^m u \in C\bigl([0,T];H^{k-3m}(I\times\mathbb{R})\bigr),\quad
m\leq [k/3],\\
\label{2.3}
\partial_x^l u \in C_b\bigl(\overline{I};H^{(k-l+1)/3,k-l+1}(B_T)\bigr), \quad
l\leq k+1, \\
\label{2.4}
\partial^m_t\partial^l_x\partial^j_y u \in L_2\bigl(0,T;C_b(\overline{I}\times\mathbb{R})\bigr),
\quad 3m+l+j \leq k,\\
\label{2.5}
\partial^m_t\partial^l_x\partial^j_y u \in L_2\bigl(I;C_b(\overline{B}_T)\bigr), \quad
k\geq 1,\quad 3m+l+j \leq k-1.
\end{gather}
\end{definition}

\begin{remark}\label{R2.1} \rm
For small $k$ such solutions are interpreted in a weak (distributional) sense
(see, e.g. \cite{faminskii04,faminskii07} for corresponding definitions
in similar situations for KdV).
\end{remark}

For description of properties of the right part of the equation introduce the following
spaces $M_k$.

\begin{definition}\label{D2.2} \rm
For any $T>0$ let $M_k((0,T)\times I\times\mathbb{R})$ be a space
of functions $f(t,x,y)$ such that
$$
\partial_t^m f \in L_2\bigl(0,T;H^{k-3m}(I\times\mathbb{R})\bigr),\quad
m\leq m_0=[(k+1)/3].
$$
\end{definition}

For simplicity we often use shortened symbols $X_k$ and $M_k$. Let
$\partial_{x,y}$ denotes either $\partial_x$ or $\partial_y$.

\begin{lemma}\label{L2.1}
For any $T>0$ and $I\subset \mathbb{R}$
\begin{gather}\label{2.6}
\|u\partial_{x,y}v\|_{M_0}+\|v\partial_{x,y}u\|_{M_0}\leq c\|u\|_{X_1}\|v\|_{X_0},\\
\label{2.7}
\|u\partial_{x,y}v\|_{M_k}\leq c(k)\|u\|_{X_k}\|v\|_{X_k},\quad k\geq 1,\\
\label{2.8} \|u\partial_{x,y}u\|_{M_k}\leq
c(k)\|u\|_{X_{k-1}}\|u\|_{X_k},\quad k\geq 2.
\end{gather}
\end{lemma}

\begin{proof}
Note that $M_0=L_2$, so (\ref{2.6}) follows from obvious inequalities
\begin{gather}\label{2.9}
\|u\partial_{x,y}v\|_{L_2((0,T)\times I\times \mathbb{R})}\leq
\|u\|_{L_2(I;C_b(\overline{B}_T))}
\|\partial_{x,y}v\|_{C_b(\overline{I};L_2(B_T))},\\
\label{2.10} \|v\partial_{x,y}u\|_{L_2((0,T)\times I\times
\mathbb{R})}\leq \|u\|_{C([0,T];H^1(I\times\mathbb{R}))}
\|v\|_{L_2(0,T;C_b(\overline{I}\times\mathbb{R}))}.
\end{gather}
Let in (\ref{2.7}) and (\ref{2.8}) $k=3n+j$, $0\leq j\leq 2$. If $j\leq 1$,
then $m_0=n$ and these inequalities can be derived similarly to (\ref{2.9}),
(\ref{2.10}). Let $j=2$, then $m_0=n+1$ and in addition to the previous cases
we must evaluate $\partial_t^{m_0}(u\partial_{x,y}v)$ in $L_2(0,T;H^{-1})$. Here
$$
u\partial_t^{m_0}\partial_{x,y}v=\partial_{x,y}(u\partial_t^{m_0}v)-
\partial_{x,y}u\partial_t^{m_0}v
$$
and similarly to (\ref{2.9})
\begin{gather}\label{2.11}
\|\partial_{x,y}(u\partial_t^{m_0}v)\|_{L_2(0,T;H^{-1}(I\times
\mathbb{R}))}\leq \|u\partial_t^{m_0}v\|_{L_2((0,T)\times I\times
\mathbb{R})}\leq
\|u\|_{X_1}\|v\|_{X_k},\\
\|\partial_{x,y}u\partial_t^{m_0}v\|_{L_2(0,T;H^{-1}(I\times
\mathbb{R}))}\leq
\|\partial_{x,y}u\partial_t^{m_0}v\|_{L_2((0,T)\times I\times
\mathbb{R})}\leq \|u\|_{X_2}\|v\|_{X_k}. \notag
\end{gather}
Thus (\ref{2.8}) for $k\geq 3$ and (\ref{2.7}) are established. Finally, note that
if $k=2$, then $\partial_t(u\partial_{x,y}u)=\partial_{x,y}(uu_t)$, and the inequality
(\ref{2.8}) in this case follows from (\ref{2.11}).
\end{proof}

In order to describe properties of boundary data we introduce some special
notation common for all three considered problems.

\begin{definition}\label{D2.3} \rm
Let $n=1$, $I=\mathbb{R}_+=(0,+\infty)$ for the problem in
$\Pi_T^+$; $n=2$, $I=\mathbb{R}_-=(-\infty,0)$ for the problem in
$\Pi_T^-$; $n=3$, $I=(0,1)$ for the problem in $Q_T$. Let
$\EuScript{B}_n^k(T)$ be a space of ordered assemblies $\EuScript{U}^n$, where
$$
\EuScript{U}^1=(u_1),\quad \EuScript{U}^2=(u_2,u_3),\quad
\EuScript{U}^3=(u_1,u_2,u_3),
$$
such that
$$
u_1,u_2\in H^{(k+1)/3,k+1}(B_T),\quad u_3\in H^{k/3,k}(B_T),
$$
with the natural norm.
\end{definition}

We also need to formulate compatibility conditions for the considered problems.

\begin{definition}\label{D2.4} \rm
Let $\Phi_0(x,y)\equiv u_0(x,y)$ and for $m\geq 1$
\begin{equation}\label{2.12}
\begin{aligned}
\Phi_m(x,y) &\equiv \partial_t^{m-1} f(0,x,y) -
(\partial_x^3+\partial_x\partial_y^2)\Phi_{m-1}(x,y)\\
&\quad - \sum_{l=0}^{m-1} {\binom {m-1}l}
\Phi_l(x,y)\partial_x\Phi_{m-l-1}(x,y).
\end{aligned}
\end{equation}
We say that the compatibility conditions of the order $k$ are satisfied if
\begin{enumerate}
\item  $\partial_t^mu_1(0,y)\equiv \Phi_m(0,y)$ for $m<k/3$ in the
case of the problem in $\Pi_T^+$;

\item $\partial_t^mu_2(0,y)\equiv \Phi_m(0,y)$ for $m<k/3$,
$\partial_t^mu_3(0,y)\equiv \partial_x\Phi_m(0,y)$ for $m<(k-1)/3$
in the case of the problem in $\Pi_T^-$;

\item  $\partial_t^mu_1(0,y)\equiv \Phi_m(0,y)$,
$\partial_t^mu_2(0,y)\equiv \Phi_m(1,y)$ for $m<k/3$,
$\partial_t^mu_3(0,y)\equiv \partial_x\Phi_m(1,y)$ for $m<(k-1)/3$
in the case of the problem in $Q_T$.
\end{enumerate}
\end{definition}

Now we can present the main result of the paper.

\begin{theorem}\label{T2.1}
Let either $n=1$, $I=\mathbb{R}_+$ for the problem in $\Pi_T^+$ or
$n=2$, $I=\mathbb{R}_-$ for the problem in $\Pi_T^-$ or $n=3$,
$I=(0,1)$ for the problem in $Q_T$. Let $u_0\in
H^k(I\times\mathbb{R})$, $\EuScript{U}^n\in \EuScript{B}_n^k(T)$,
$f\in M_k((0,T)\times I\times\mathbb{R})$ for certain $T>0$,
$k\geq 1$. Assume also that the compatibility conditions of the
order $k$ are satisfied for the considered problem. Then
respectively
\begin{enumerate}
\item  the problem \eqref{1.1}--\eqref{1.3} is well-posed in
$X_k(\Pi_T^+)$;

\item  there exists $t_0\in (0,T]$ such that the problem
\eqref{1.1}, \eqref{1.2}, \eqref{1.4} is well-posed in
$X_k(\Pi_{t_0}^-)$;

\item  the problem \eqref{1.1}, \eqref{1.2}, \eqref{1.5} is
well-posed in $X_k(Q_T)$ if $k\geq 3$ and there exists $t_0\in
(0,T]$ such that this problem is well-posed in $X_k(Q_{t_0})$ if
$k=1$ or $k=2$.
\end{enumerate}
\end{theorem}

\begin{remark}\label{R2.2} \rm
We mean that the problem is well-posed in the space $X_k$, if
there exists a unique solution $u(t,x,y)$ in this space and the
map $(u_0,\EuScript{U}^n,f)\mapsto u$ is Lipschitz continuous on
any ball in the norm of the map $H^k(I\times\mathbb{R})\times
\EuScript{B}_n^k(T) \times M_k((0,T)\times I\times\mathbb{R})$ into
$X_k$.
\end{remark}

\begin{remark}\label{R2.3} \rm
All these well-posedness results can be easily generalized for an
equation of the \eqref{1.1} type with a nonlinear term $g(u)u_x$,
where the sufficiently smooth function $g$ has not more than
linear rate of growth (more precisely, $g'$ is bounded on
$\mathbb{R}$) and $g(0)=0$.
\end{remark}

\begin{remark}\label{R2.4} \rm
In the paper \cite{faminskii95} global well-posedness of the
initial value problem \eqref{1.1}, \eqref{1.2} was established
under assumptions $u_0\in H^k(\mathbb{R}^2)$, $f\in
L_1(0,T;H^k(\mathbb{R}^2))$, $k\geq 1$, in the classes similar to
$X_k$ but without smoothness properties with respect to $t$.
\end{remark}

\section{Potentials}\label{S3}

Consider a linear equation
\begin{equation}\label{3.1}
u_t+u_{xxx}+u_{xyy}=f(t,x,y).
\end{equation}
Solution to the initial value problem in a domain
$\Pi_T=(0,T)\times \mathbb{R}^2$ with the initial profile
\eqref{1.2} can be constructed in a form (see \cite{faminskii95})
\begin{equation}\label{3.2}
u(t,x,y)=S(t,x,y;u_0)+K(t,x,y;f),
\end{equation}
where potentials $S$ and $K$ are given by formulae
\begin{equation} \label{3.3}
\begin{gathered}
S(t,x,y;u_0)\equiv
\EuScript{F}^{-1}_{x,y}\big[e^{it(\xi^3+\xi\eta^2)}
\widehat u_0(\xi,\eta)\big](x,y),\\
K(t,x,y;f)\equiv\int^t_0 S(t-\tau,x,y;f(\tau,\cdot,\cdot))\,d\tau.
\end{gathered}
\end{equation}
 By analogy with (\ref{2.12}) let $\widetilde \Phi_0(x,y)
\equiv u_0(x,y)$ and for $m\geq 1$
\begin{equation}\label{3.4}
\widetilde\Phi_m(x,y) \equiv \partial^{m-1}_t f(0,x,y)-
(\partial_x^3+\partial_x\partial_y^2)\widetilde\Phi_{m-1}(x,y).
\end{equation}

\begin{lemma}\label{L3.1}
If $u_0\in H^k(\mathbb{R}^2)$, $f\in M_k(\Pi_T)$ for some $T>0$
and $k\geq 0$, then a unique solution $u(t,x,y)$ to the problem
\eqref{3.1}, \eqref{1.2} exists and for any $t_0\in (0,T]$
\begin{equation} \label{3.5}
\begin{aligned}
&\|u\|_{X_k(\Pi_{t_0})}\\
&\leq c(T,k)\Bigl(\|u_0\|_{H^k(\mathbb{R}^2)}+
t_0^{1/6}\|f\|_{M_k(\Pi_{t_0})}
 +\sum_{m=0}^{m_0-1} \|\partial_t^m
f\big|_{t=0}\|_{H^{k-3(m+1)}(\mathbb{R}^2)}\Bigr).
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
First of all note that
\begin{equation}\label{3.6}
\partial_t^m S(t,x,y;u_0)+\partial_t^m K(t,x,y;f)=
S(t,x,y;\widetilde\Phi_m)+ K(t,x,y;\partial_t^mf).
\end{equation}
For $m=0$ corresponding estimates on the solution $u$ in the norms
(\ref{2.2}), (\ref{2.4}), (\ref{2.5}) (where $I=\mathbb{R}$) by
$\|u_0\|_{H^k(\mathbb{R}^2)}$ and
$\|f\|_{L_1(0,t_0;H^k(\mathbb{R}^2))}$ are established in
\cite{faminskii95} (moreover, in (\ref{2.4}) $L_2$ with respect to
$t$ can be enlarged to $L_3$).

In \cite{faminskii95} it was also proved that
\begin{equation}\label{3.7}
\|\nabla_{x,y} u\|_{C_b(\mathbb{R};L_2(B_{t_0}))}\leq
c(T,k)\left(\|u_0\|_{H^1(\mathbb{R}^2)}
+\|f\|_{L_1(0,t_0;H^1(\mathbb{R}^2))}\right).
\end{equation}
Similarly to (\ref{3.7}) a corresponding estimate on $\partial_x^l
S$ in $C_b(\mathbb{R};H^{(k-l+1)/3,k-l+1}(B_{t_0}))$ by
$\|u_0\|_{H^k(\mathbb{R}^2)}$ for $l\leq k+1$ can be also derived.

For the potential $K$ first of all we show that for $s\in[0,1]$
\begin{equation}\label{3.8}
\|K(\cdot,\cdot,\cdot;f)\|_{C_b(\mathbb{R};H^{s,3s}(B_{t_0}))}
\leq c(T)t_0^{(1-s)/2} \|f\|_{L_2(0,t_0;H^{3s-1}(\mathbb{R}^2))}.
\end{equation}
In fact, if $s=0$ then this inequality is similar to (\ref{3.7}), if $s=1$
it succeeds from an equality
$$
K_t(t,x,y;f)=f(t,x,y)-\int_0^t(\partial_x^3+\partial_x\partial_y^2)
S(t-\tau,x,y;f(\tau,\cdot,\cdot))\,d\tau
$$
and the already established estimates on the potential $S$, for
intermediate values of $s$ (\ref{3.8}) is obtained via interpolation.

Finally, it is suffice to note that if one applies (\ref{3.8}) to
$K(t,x;\partial_t^mf)$, where $m=[(k-l+1)/3]$, $s=(k-l+1)/3-m$, the
minimal value $1/6$ for the degree $(1-s)/2$ is achieved if $k-l+1=3m+2$.
\end{proof}

\begin{remark}\label{R3.1} \rm
By the methods from \cite{faminskii95} it is easy to show that for
the function $S=S(t,x,y;u_0)$, where $u_0\in H^s(\mathbb{R}^2)$,
uniformly with respect to $x\in \mathbb{R}$
$$
\bigl\|D_t^{1/3}S\bigr\|_{H_{t,y}^{s/3,s}(\mathbb{R}^2)}^2+
\bigl\|\partial_xS\bigr\|_{H_{t,y}^{s/3,s}(\mathbb{R}^2)}^2+
\bigl\|\partial_yS\bigr\|_{H_{t,y}^{s/3,s}(\mathbb{R}^2)}^2 \sim
\|u_0\|_{H^s(\mathbb{R}^2)}^2
$$
(here $D^\alpha$ denotes the Riesz potential of the order $-\alpha$).
\end{remark}

In what follows we need simple properties of solutions to an algebraic
equation
\begin{equation}\label{3.9}
r^3-r\eta^2+i\lambda=0, \quad (\lambda,\eta)\ne (0,0).
\end{equation}
This equation has one root $r_0(\lambda,\eta)$ with the negative real part,
one root $r_1(\lambda,\eta)$ with the positive real part and one pure
imaginary root $r_2(\lambda,\eta)$. These roots can be written in a form
\begin{equation}\label{3.10}
r_0=-p(\lambda,\eta)+iq(\lambda,\eta), \quad
r_1=p(\lambda,\eta)+iq(\lambda,\eta), \quad
r_2= i\kappa(\lambda,\eta),
\end{equation}
where $p>0$, $q\in \mathbb{R}$ and the function $\kappa$ for a
fixed $\eta$ is the inverse function to $\varphi(\xi)\equiv
\xi^3+\xi\eta^2$. Moreover, for certain positive constants $c$,
$c_1$ and any $(\lambda,\eta)$
\begin{gather}\label{3.11}
p(\lambda,\eta)\geq c(|\lambda|^{1/3}+|\eta|),\\
\label{3.12}
|r_j(\lambda,\eta)|\leq c_1(|\lambda|^{1/3}+|\eta|) \quad \forall j,\\
\label{3.13}
|r_j(\lambda,\eta)-r_k(\lambda,\eta)|\geq c(|\lambda|^{1/3}+|\eta|), \quad
j\ne k.
\end{gather}

Now we can introduce boundary potentials for the homogeneous
equation \eqref{3.1}.

\begin{definition}\label{D3.1} \rm
Let $\mu,\nu\in L_2(\mathbb{R}^2)$. Define for $x\geq 0$
\begin{equation}\label{3.14}
J_+(t,x,y;\mu)\equiv
\EuScript{F}_{t,y}^{-1}\left[e^{r_0x}\widehat\mu(\lambda,\eta)\right](t,y)
\end{equation}
and for $x\leq 0$,
\begin{equation}\label{3.15}
J_-(t,x,y;\mu,\nu)\equiv \EuScript{F}^{-1}_{t,y}
\Bigl[\frac{r_1e^{r_2x}
-r_2e^{r_1x}}{r_1-r_2}\widehat\mu(\lambda,\eta)
+\frac{e^{r_1x}-e^{r_2x}}{r_1-r_2}\widehat\nu(\lambda,\eta)\Bigr](t,y),
\end{equation}
where $r_j=r_j(\lambda,\eta)$ are the aforementioned roots of the equation
(\ref{3.9}).
\end{definition}

\begin{lemma}\label{L3.2}
Let $\mu\in H^{(k+1)/3,k+1}(\mathbb{R}^2)$ for some $k\geq 0$,
then for any $T>0$
\begin{equation}\label{3.16}
\|J_+(\cdot,\cdot,\cdot;\mu)\|_{X_k(\Pi_T^+)} \leq c(T,k)
\|\mu\|_{H^{(k+1)/3,k+1}(\mathbb{R}^2)}.
\end{equation}
\end{lemma}

\begin{proof}
In order to obtain an estimate in the norm (\ref{2.2}) we use the
following fundamental inequality from \cite{bona02}: if certain
continuous function $\gamma(\theta)$ satisfies an inequality $\mathop{\rm Re}
\gamma(\theta)\leq -\varepsilon|\theta|$ for some $\varepsilon>0$
and all $\theta\in \mathbb{R}$, then
\begin{equation}\label{3.17}
\Bigl\|\int_\mathbb{R} e^{\gamma(\theta)x}
f(\theta)\,d\theta\Bigr\|_ {L_2(\mathbb{R}^x_+)} \leq
c(\varepsilon)\|f\|_{L_2(\mathbb{R})}.
\end{equation}
Therefore, changing variables $\lambda=\theta^3$ we derive from
(\ref{3.14}) with the use of (\ref{3.11}) and (\ref{3.12}) that
for $3m+l+j\leq k$ uniformly with respect to $t\in \mathbb{R}$,
\begin{equation}\label{3.18}
\begin{aligned}
\|\partial_t^m\partial_x^l\partial_y^j
 J_+(t,\cdot,\cdot;\mu)\|_{L_2(\mathbb{R}_+^2)}
&\leq c\left\|\theta^{3m+2}\eta^j(|\theta|^l+|\eta|^l)
 \widehat\mu(\theta^3,\eta)\right\|_{L_2(\mathbb{R}^2)} \\
&\leq c_1 \bigl\|\lambda^{m+1/3}\eta^j(|\lambda|^{l/3}+|\eta|^l)
\widehat\mu(\lambda,\eta)\bigr\|_{L_2(\mathbb{R}^2)} \\
& \leq c_2\|\mu\|_{H^{(k+1)/3,k+1}(\mathbb{R}^2)}.
\end{aligned}
\end{equation}
Similarly to (\ref{3.18}), for $3m+l+j\leq k-1$,
\begin{equation} \label{3.19}
\begin{aligned}
&\bigl\|\sup_{(t,y)\in\mathbb{R}^2}
|\partial_t^m\partial_x^l\partial_y^jJ_+(\cdot,t,y;\mu)|\bigr\|_{L_2(\mathbb{R}_+^x)}
\\
&\leq c\int_\mathbb{R}\bigl\|\int_\mathbb{R}
e^{-p(\lambda,\eta)x}
|\lambda^m\eta^j|(|\lambda|^{l/3}+|\eta|^l)|\widehat\mu(\lambda,\eta)|\,
d\lambda\bigr\|_{L_2(\mathbb{R}_+^x)}d\eta \\
&\leq c_1\int_\mathbb{R}
|\eta^j|\bigl\|\theta^{3m+2}(|\theta|^l+|\eta|^l)
\widehat\mu(\theta^3,\eta)\bigr\|_{L_2(\mathbb{R}^\theta)}d\eta \\
&\leq c_2 \|\lambda^{m+1/3}\eta^{j+1}(|\lambda|^{l/3}+|\eta|^l)
\widehat\mu(\lambda,\eta)\|_{L_2(\mathbb{R}^2)}\\
& \leq c_3 \|\mu\|_{H^{(k+1)/3,k+1}(\mathbb{R}^2)}
\end{aligned}
\end{equation}
and we obtain the desired estimate in the norm (\ref{2.5}).

The estimate in the norm (\ref{2.3}) simply follows from the
equality (\ref{3.14})
since $\mathop{\rm Re} r_0\leq 0$.

Finally, the estimate in the norm (\ref{2.4}) succeeds by virtue
of the well-known embedding $H^{1+\varepsilon}(\Omega)\subset
C_b(\overline{\Omega})$ for domains $\Omega\subset\mathbb{R}^2$
from the following inequality: for $s\geq 0$
\begin{equation}\label{3.20}
\|\partial_t^m
J_+(\cdot,\cdot,\cdot;\mu)\|_{L_2(0,T;H^{s-3m}(\mathbb{R}_+^2))}
\leq c(T,s)\|\mu\|_{H^{(2s-1)/6,s-1/2}(\mathbb{R}^2)}.
\end{equation}
It is suffice to prove (\ref{3.20}) for $m=0$. Let
\begin{equation}\label{3.21}
\mu_0(t,y)\equiv \EuScript{F}_{t,y}^{-1}\left[\chi(\lambda,\eta)
\widehat\mu(\lambda,\eta)\right](t,y), \quad \mu_1(t,y)\equiv
\mu(t,y)-\mu_0(t,y),
\end{equation}
where $\chi$ denotes the characteristic function of the unit circle
$\{(\lambda,\eta):\lambda^2+\eta^2<1\}$. Then it follows from the already
established estimate (\ref{3.18}) that for any $s\geq 0$
\begin{align*}
\|J_+(\cdot,\cdot,\cdot;\mu_0)\|_{L_2(0,T;H^s(\mathbb{R}_+^2))}
&\leq T^{1/2}\sup_{t\in [0,T]}
 \|J_+(t,\cdot,\cdot;\mu_0)\|_{H^{[s]+1}(\mathbb{R}_+^2)} \\
&\leq c(T,s)\|\mu\|_{H^{-1}(\mathbb{R}^2)}.
\end{align*}
Next,
\begin{align*}
&\|\partial_x^l\partial_y^j
J_+(\cdot,\cdot,\cdot;\mu_1)\|_{L_2(\mathbb{R}^t\times \mathbb{R}_+^2)}\\
&= \Bigl\|r_0^l\eta^j\widehat\mu_1(\lambda,\eta)\Bigl(\int_{\mathbb{R}_+}
e^{-2p(\lambda,\eta)x}\,dx\Bigr)^{1/2}\Bigr\|_{L_2(\mathbb{R}^2)}\\
&\leq c\bigl\|(|\lambda|^{l/3}+|\eta|^l)\eta^jp^{-1/2}(\lambda,\eta)
\widehat\mu(\lambda,\eta)\bigl(1-\chi(\lambda,\eta)\bigr)
 \bigr\|_{L_2(\mathbb{R}^2)}\\
&\leq c_1 \|\mu\|_{H^{(2(l+j)-1)/6,l+j-1/2}(\mathbb{R}^2)}
\end{align*}
and using interpolation we complete the proof of (\ref{3.20}).
\end{proof}

\begin{remark}\label{R3.2} \rm
It follows from the proof of Lemma~\ref{L3.2} that the estimates on $J_+$
are valid in norms of the (\ref{2.2}), (\ref{2.3}), (\ref{2.5}) type,
where the domain of the variable $t$ is the whole real axis.
\end{remark}

The potential $J_+$ possesses also certain additional properties.

\begin{lemma}\label{L3.3}
Let $\mu\in L_2(\mathbb{R}^2)$ and $\mu(t,y)=0$ for $t<0$. Then
the function $J_+(t,x,y;\mu)$ is infinitely differentiable for
$x>0$, $J_+(t,x,y;\mu)=0$ for $t\leq 0$ and for any $T>0$,
$x_0>0$, $\beta\geq 0$ and $m,l,j$
\begin{equation}\label{3.22}
\sup_{t\in [0,T],x\geq x_0}
(1+x)^\beta\|\partial_t^m\partial_x^l
J_+(t,x,\cdot;\mu)\|_{H^j(\mathbb{R})} \leq
c(T,x_0,\beta,m,l,j)\|\mu\|_{L_2(\mathbb{R}^2)}.
\end{equation}
\end{lemma}

\begin{proof}
These properties succeed from the following representation of the function
$J_+$ for $x>0$:
\begin{gather}\label{3.23}
J_+(t,x,y;\mu)= \int_{-\infty}^t\!\!\int_\mathbb{R}
(3\partial_x^2+\partial_y^2)G(t-\tau,x,y-z)\mu(\tau,z)\,dzd\tau,
\\ \label{3.24}
G(t,x,y)\equiv \frac 1 {t^{2/3}} A\left(\frac x {t^{1/3}},\frac
y{t^{1/3}}\right), \quad A(x,y)\equiv
\EuScript{F}^{-1}_{x,y}\bigl[e^{i(\xi^3+\xi\eta^2)}\bigr](x,y).
\end{gather}
This formula was proved in \cite{faminskii06}. We reproduce here the scheme
of the proof. Changing variables $\xi=\kappa(\lambda,\eta)$, where $\kappa$
is the function from (\ref{3.10}), we can write an equality
$$
G(t,x,y)=\EuScript{F}^{-1}_{t,y}\bigl[\partial_\lambda\kappa(\lambda,\eta)
e^{i\kappa(\lambda,\eta)x}\bigr](t,y).
$$
So if we denote by $J$ the right part of (\ref{3.23}) then
\begin{align*}
\EuScript{F}_{t,y}[J](\lambda,\eta)
&= \EuScript{F}_{t,y}
\bigl[(3\partial_x^2+\partial_y^2)G(t,x,y)\vartheta(t)\bigr]
(\lambda,\eta)\widehat\mu(\lambda,\eta) \\
&=-\frac 1 {4\pi^2}\Bigl(e^{i\kappa(\lambda,\eta)x}*\bigl(\widehat\vartheta(\lambda)\times
\delta(\eta)\bigl)\Bigl)\widehat\mu(\lambda,\eta) \\
&=-\Big(\frac 1 2 e^{i\kappa(\lambda,\eta)x}+\frac i {2\pi} \mathop{\rm v.p.}
\int_\mathbb{R} \frac {e^{i\kappa(\zeta,\eta)x}}{\zeta-\lambda}
d\zeta\Big)\widehat\mu(\lambda,\eta),
\end{align*}
where $\vartheta$ is the Heaviside function. The last integral can be easily
calculated:
$$
\mathop{\rm v.p.} \int_\mathbb{R} \frac
{e^{i\kappa(\zeta,\eta)x}}{\zeta-\lambda} d\zeta= \mathop{\rm v.p.}
\int_\mathbb{R} \frac
{e^{izx}(3z^2+\eta^2)}{z^3+z\eta^2-\lambda}dz = 2\pi i
e^{r_0(\lambda,\eta)x}+\pi i e^{i\kappa(\lambda,\eta)x}
$$
and, consequently, $J=J_+$.

The function $A$ was studied in \cite{faminskii02} (in fact, more
general one). In particular, it was proved that $A\in \EuScript
S(\overline{\mathbb{R}}_+^2)$ -- the space of restrictions on
$\overline{\mathbb{R}}_+^2$ of functions from $\EuScript
S(\mathbb{R}^2)$. This property applied to (\ref{3.23}) provides
the assertion of the lemma (see \cite{faminskii02} for more
details).
\end{proof}

Now we consider properties of the potential $J_-$.

\begin{lemma}\label{L3.4}
Let $\mu\in H^{(k+1)/3,k+1}(\mathbb{R}^2)$, $\nu\in
H^{k/3,k}(\mathbb{R}^2)$ for some $k\geq 0$, then for any $T>0$
\begin{equation}\label{3.25}
\|J_-(\cdot,\cdot,\cdot;\mu,\nu)\|_{X_k(\Pi_T^-)} \leq c(T,k)
\bigl(\|\mu\|_{H^{(k+1)/3,k+1}(\mathbb{R}^2)}
 +\|\nu\|_{H^{k/3,k}(\mathbb{R}^2)}\bigr).
\end{equation}
\end{lemma}

\begin{proof}
First consider the part of $J_-$ containing the term $e^{r_1x}$.
Since the root $r_1$ has the properties similar to $r_0$, taking
into account the inequality (\ref{3.13}) one can derive
corresponding analogues of (\ref{3.18})--(\ref{3.20}) for this
part by the same methods as for $J_+$ (the analogues of
(\ref{3.18}), (\ref{3.19}) are supplemented with
$\|\nu\|_{H^{k/3,k}(\mathbb{R}^2)}$, of (\ref{3.20}) -- with
$\|\nu\|_{H^{(2s-3)/6,s-3/2}(\mathbb{R}^2)}$).

The estimate in the norm (\ref{2.3}) is obvious just as for $J_+$
except the case $l=0$ because of the denominator near the point $(0,0)$
in the part containing $\widehat\nu$, but here one can use the
partition of $\nu$ similar to (\ref{3.21}) and for $\nu_0$ apply
the already established estimate in the (\ref{2.2}) norm.

In order to evaluate the part of $J_-$ containing the term $e^{r_2x}$
consider an expression
$$
\EuScript{I}(t,x,y)\equiv \EuScript{F}_{t,y}^{-1}
\bigl[e^{r_2(\lambda,\eta)x} f(\lambda,\eta)\bigr](t,y).
$$
By virtue of (\ref{3.3}), (\ref{3.10}) and the change of variables $\lambda=
\xi^3+\xi\eta^2$ it can be written in a form
$$
\EuScript{I}=S\bigl(t,x,y;\EuScript{F}^{-1}\bigl[(3\xi^2+\eta^2)
f(\xi^3+\xi\eta^2,\eta)\bigr]\bigl).
$$
It is easy to see that
$$
\bigl\|\EuScript{F}^{-1}\bigl[(3\xi^2+\eta^2)
f(\xi^3+\xi\eta^2,\eta)\bigr]\bigr\|_{H^k(\mathbb{R}^2)} \leq c
\|(|\lambda|^{1/3}+|\eta|)f(\lambda,\eta)\|_{H^{k/3,k}(\mathbb{R}^2)}
$$
and so the desired estimates on the rest part of $J_-$ succeed from
Lemma~\ref{L3.1}.
\end{proof}


\section{Linear problems}\label{S4}

Consider for the equation \eqref{3.1} initial-boundary value
problems in the domains $\Pi_T^+$, $\Pi_T^-$, $Q_T$ with the
initial data \eqref{1.2} and the boundary data \eqref{1.3},
\eqref{1.4} or \eqref{1.5} respectively. First we establish one
auxiliary lemma for the first two problems. Note that solutions to
these problems are unique in the spaces $L_2(\Pi_T^+)$ and
$L_2(\Pi_T^-)$ respectively because of the already proved
solvability in smooth classes (see \cite{volevich}, these problems
are, in fact, adjoint to each other).

\begin{lemma}\label{L4.1}
Let $u_0\equiv 0$, $f\equiv 0$, $u_1,u_2\in
H^{1/3,1}(\mathbb{R}^2)$, $u_3\in L_2(\mathbb{R}^2)$ and
$u_1(t,y)=u_2(t,y)=u_3(t,y)=0$ for $t<0$. Then
$J_+(t,x,y;u_1)$ and $J_-(t,x,y;u_2,u_3)$ are respectively
(unique) solutions to the problems \eqref{3.1}, \eqref{1.2},
\eqref{1.3} in $\Pi_T^+$ or \eqref{3.1}, \eqref{1.2}, \eqref{1.4}
in $\Pi_T^-$ for any $T>0$ in the classes $X_0(\Pi_T^+)$ or
$X_0(\Pi_T^-)$.
\end{lemma}

\begin{proof}
By virtue of Lemmas \ref{L3.2} and \ref{L3.4} without loss of
generality one can assume that $u_j\in
C_0^\infty(\mathbb{R}_+^t\times\mathbb{R}^y)$.

It is obvious that the functions $J_+$ and $J_-$ satisfy the
homogeneous equation \eqref{3.1} if $x\geq 0$ or $x\leq 0$
respectively and satisfy the corresponding boundary conditions
\eqref{1.3} or \eqref{1.4}. Lemma~\ref{L3.3} provides also that the
function $J_+$ satisfies the zero initial condition \eqref{1.2},
so for the problem in $\Pi_T^+$ the proof is complete.

Since for the function $J_-$ we don't have an equality of the
(\ref{3.23}) type, we choose in this case an indirect way and
prove that a solution to the problem in $\Pi_T^-$ coincides with
$J_-$. According to \cite{volevich} (see also \cite{faminskii02})
there exists a solution $u(t,x,y)$ to the problem \eqref{3.1},
\eqref{1.2}, \eqref{1.4} and $\partial_t^mu\in
C([0,T];H^l(\mathbb{R}_-^2))$ for any $T>0$, $m,l\geq 0$.
Moreover, if $u_2(t,y)=u_3(t,y)=0$ for $t\geq T_0>0$ then
multiplying \eqref{3.1} by $2u(t,x,y)$ and integrating over
$\mathbb{R}^2_-$ one can easily derive that for $t\geq T_0$
\begin{equation}\label{4.1}
\frac d{dt} \|u(t,\cdot,\cdot)\|_{L_2(\mathbb{R}_-^2)}=0.
\end{equation}
Obviously similar equality can be obtained for any derivative
$\partial_t^m\partial_y^j u$. Derivatives with respect to $x$ can
be expressed from the equation \eqref{3.1} itself (see for more
details \cite{faminskii02} or the following arguments in the proof
of Lemma~\ref{L5.4}). Finaly, we obtain that $\partial_t^m u\in
C_b(\overline{\mathbb{R}}_+^t;H^l(\mathbb{R}_-^2))$ for any
$m,l\geq 0$.

Therefore for any $p=\varepsilon+i\lambda$, where $\varepsilon>0$,
and $\eta\in \mathbb{R}$ we can define the Laplace transform with
respect to $t$ and the Fourier transform with respect to $y$:
$$
\widetilde u(p,x,\eta) \equiv \iint_{\mathbb{R}_+^2} e^{-pt-i\eta
y} u(t,x,y)\,dtdy.
$$
The function $\widetilde u$ solves a problem
\begin{gather}\label{4.2}
p\widetilde u(p,x,\eta)+\widetilde u_{xxx}(p,x,\eta)-
\eta^2\widetilde u_x(p,x,\eta)=0, \quad x\leq 0,\\
\widetilde u(p,0,\eta)=\widetilde u_2(p,\eta), \quad \widetilde
u_x(p,0,\eta)=\widetilde u_3(p,\eta),\notag
\end{gather}
where $\widetilde u_2$, $\widetilde u_3$ are the similar Laplace--Fourier
transforms of $u_2$, $u_3$.
The corresponding characteristic equation for (\ref{4.2})
$r^3-\eta^2r+i\lambda+\varepsilon=0$ has exactly two roots $r_1(\lambda,\eta,
\varepsilon)$ and $r_2(\lambda,\eta,\varepsilon)$ with the positive real parts
(and one root with the negative one), so
since $\widetilde u(p,x,\eta)\to 0$ as $x\to -\infty$ it follows that
$$
\widetilde u(p,x,\eta)=\frac {r_1e^{r_2x}-r_2e^{r_1x}}
{r_1-r_2} \widetilde u_2(p,\eta)
+ \frac {e^{r_1x}-e^{r_2x}} {r_1-r_2} \widetilde u_3(p,\eta).
$$
Applying the formulae of invertion of the Laplace and the Fourier
transforms and passing to the limit as $\varepsilon\to+0$ we derive
that $u\equiv J_-$.
\end{proof}

For the considered initial-boundary value problems for the equation \eqref{3.1}
introduce the notion of compatibility conditions of the order $k$
similar to Definition~\ref{D2.4}, where only $\Phi_m$ must be
substituted by $\widetilde\Phi_m$ (see (\ref{3.4})).

Now we can establish the main lemma for the linear problems.

\begin{lemma}\label{L4.2}
Let $n=1$, $I=\mathbb{R}_+$ for the problem in $\Pi_T^+$; $n=2$,
$I=\mathbb{R}_-$ for the problem in $\Pi_T^-$; $n=3$, $I=(0,1)$
for the problem in $Q_T$. Let $u_0\in H^k(I\times\mathbb{R})$,
$\EuScript{U}^n\in \EuScript{B}_n^k(T)$, $f\in M_k((0,T)\times
I\times\mathbb{R})$ for certain $T>0$, $k\geq 0$. Assume also that
the compatibility conditions of the order $k$ are satisfied for
each of the considered problems. Then there exists a unique
solution $u(t,x,y)$ to each problem in the space $X_k((0,T)\times
I\times\mathbb{R})$ and for any $t_0\in (0,T]$
\begin{equation}\label{4.3}
\begin{aligned}
\|u\|_{X_k((0,t_0)\times I\times\mathbb{R})}
&\leq c(T,k) \Bigl(\|u_0\|_{H^k(I\times\mathbb{R})}
 + \|\EuScript{U}^n\|_{\EuScript{B}_n^k(T)}
 + t_0^{1/6}\|f\|_{M_k((0,t_0)\times I\times\mathbb{R})}\\
&\quad +\sum_{m=0}^{m_0-1} \|\partial_t^m
f\big|_{t=0}\|_{H^{k-3(m+1)} (I\times\mathbb{R})}\Bigr).
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Consider first the problems in $\Pi_T^+$ and $\Pi_T^-$. Extend
$u_0$ and $f$ to the whole real axis with respect to $x$ in the
classes $H^k(\mathbb{R}^2)$ and $M_k(\Pi_T)$ respectively and
consider a solution $U(t,x,y)$ to the initial value problem
\eqref{3.1}, \eqref{1.2} in the class $X_k(\Pi_T)$ given by
Lemma~\ref{L3.1}. Note that by virtue of the compatibility
conditions
$$
u_1-U|_{x=0},\ u_2-U|_{x=0}\in
H_0^{(k+1)/3,k+1}(\mathbb{R}_+^2)\big|_{B_T},\quad
u_3-U_x|_{x=0}\in H_0^{k/3,k}(\mathbb{R}_+^2)\big|_{B_T},
$$
so these functions can be extended by zero to the whole plane
$\mathbb{R}^2$ in the same classes. Then the desired result
succeeds from Lemmas~\ref{L4.1},~\ref{L3.2} and~\ref{L3.4}.

Solutions to the last problem (similarly to the corresponding problem
for KdV in \cite{faminskii07}) are constructed with the help of
solutions to the first two in the form
\begin{equation}\label{4.4}
u(t,x,y)=w(t,x,y)+v(t,x,y),
\end{equation}
where $w(t,x,y)$ is a solution to an initial-boundary value
problem in $\Pi^-_{T,1}=(0,T)\times (-\infty,1)$ for the equation
\eqref{3.1} with the initial and boundary conditions \eqref{1.2}
and \eqref{1.4} (for $x=1$) in the class $X_k(\Pi^-_{T,1})$. Then
\begin{equation}\label{4.5}
\begin{aligned}
\|w\|_{X_k(\Pi_{T,1}^-)}
& \leq c(T,k) \bigl(\|u_0\|_{H^k(\Sigma)}
 + \|\EuScript{U}^2\|_{\EuScript{B}_2^k(T)} \\
&\quad +t_0^{1/6}\|f\|_{M_k(Q_{t_0})}
+\sum_{m=0}^{m_0-1} \|\partial_t^m f\big|_{t=0}\|_{H^{k-3(m+1)}(\Sigma)}\bigr).
\end{aligned}
\end{equation}
Moreover, by virtue of the compatibility conditions on the line $(0,0,y)$
$$
v_1(t,y)\equiv u_1(t,y)-w(t,0,y)\in
H_0^{(k+1)/3,k+1}(\mathbb{R}_+^2)\bigr|_{B_T}
$$
and
\begin{equation}\label{4.6}
\begin{aligned}
\|v_1\|_{H^{(k+1)/3,k+1}(B_T)}
&\leq c(T,k) \bigl(\|u_0\|_{H^k(\Sigma)}
 + \|\EuScript{U}^3\|_{\EuScript{B}_3^k(T)} \\
&\quad + t_0^{1/6}\|f\|_{M_k(Q_{t_0})}
+\sum_{m=0}^{m_0-1} \|\partial_t^m f\big|_{t=0}\|_{H^{k-3(m+1)}(\Sigma)}\bigr).
\end{aligned}
\end{equation}

Consider in $Q_T$ a problem for the function $v$:
\begin{gather}\label{4.7}
v_t+v_{xxx}+v_{xyy}=0, \\
\label{4.8} v\big|_{t=0}=0, \quad v\big|_{x=0}=v_1, \quad
v\big|_{x=1}=v_x\big|_{x=1}=0.
\end{gather}
In order to construct a solution to this problem we consider the
boundary potential $J_+(t,x,y;\mu)$ for an arbitrary function
$\mu\in H_0^{(k+1)/3,k+1}(\mathbb{R}_+^2)\bigr|_{B_T}$. According
to Lemma~\ref{L3.3} for any $\delta\in (0,T]$
\begin{equation}\label{4.9}
\|J_+(\cdot,1,\cdot;\mu)\|_{H^{(k+1)/3,k+1}(B_\delta)}+
\|\partial_x J_+(\cdot,1,\cdot;\mu)\|_{H^{k/3,k}(B_\delta)}  \leq
c(T,k)\delta^{1/2}\|\mu\|_{L_2(B_\delta)}.
\end{equation}
Moreover, $J_+(\cdot,1,\cdot;\mu)\in
H_0^{(k+1)/3,k+1}(\mathbb{R}_+^2)|_{B_T}$, $\partial_x
J_+(\cdot,1,\cdot;\mu)\in H_0^{k/3,k}(\mathbb{R}_+^2)|_{B_T}$.

Consider in the domain $\Pi_{\delta,1}^-$ the problem of the
\eqref{3.1}, \eqref{1.2}, \eqref{1.4} (for $x=1$) type, where
$u_0\equiv 0$, $f\equiv 0$, $u_2\equiv -J_+(\cdot,1,\cdot;\mu)$,
$u_3\equiv -\partial_x J_+(\cdot,1,\cdot;\mu)$. A solution to this
problem $V\in X_k(\Pi_{\delta,1}^-)$ exists and, in particular,
\begin{equation}\label{4.10}
\begin{aligned}
&\|V(\cdot,0,\cdot)\|_{H^{(k+1)/3,k+1}(B_\delta)}  \\
&\leq c(T,k) \left(\|J_+(\cdot,1,\cdot;\mu)\|_{H^{(k+1)/3,k+1}(B_\delta)}+
\|\partial_xJ_+(\cdot,1,\cdot;\mu)\|_{H^{k/3,k}(B_\delta)}\right).
\end{aligned}
\end{equation}
Moreover, it is obvious that $V(\cdot,0,\cdot)\in
H_0^{(k+1)/3,k+1}(\mathbb{R}_+^2)|_{B_\delta}$.

Consider a linear operator $\Gamma: \mu \mapsto V(\cdot,0,\cdot)$
in the space $H_0^{(k+1)/3,k+1}(\mathbb{R}_+^2)|_{B_\delta}$. For
small $\delta=\delta(T,k)$ the estimates (\ref{4.9}) and
(\ref{4.10}) provide that the operator $(E+\Gamma)$ is invertible
($E$ is the identity operator) and setting $\mu\equiv
(E+\Gamma)^{-1}v_1$ we obtain the desired solution to the problem
(\ref{4.7}), (\ref{4.8})
$$
v(t,x,y)\equiv J_+(t,x,y;\mu) + V(t,x,y),
$$
where
\begin{equation}\label{4.11}
\|v\|_{X_k(Q_\delta)} \leq c(T,k)\|v_1\|_{H^{(k+1)/3,k+1}(B_T)}.
\end{equation}

Thus the solution $u(t,x,y)$ to the problem \eqref{3.1},
\eqref{1.2}, \eqref{1.5} in the domain $Q_\delta$ is constructed
and according to (\ref{4.4})--(\ref{4.6}) and (\ref{4.11})  is
evaluated in the space $X_k(Q_\delta)$ by the right part of
(\ref{4.3}). Moving step by step ($\delta$ is constant) we obtain
the desired solution in the whole domain $Q_T$.

Uniqueness of weak solutions to the problem \eqref{3.1},
\eqref{1.2}, \eqref{1.5} in $L_2(Q_T)$ succeeds from existence of
smooth solutions to the adjoint problem
\begin{gather*}
\phi_t+\phi_{xxx}+\phi_{xyy}=f\in C_0^\infty(Q_T), \\
\phi\big|_{t=T}=0, \quad
\phi\big|_{x=0}=\phi_x\big|_{x=0}=\phi\big|_{x=1}=0,
\end{gather*}
which after simple change of variables transforms to the original one.
\end{proof}

For global a priori estimates for solutions to nonlinear problems
we also need certain integral inequalities.

\begin{lemma}\label{L4.3}
Let the hypothesis of Lemma~\ref{L4.2} be satisfied for $n=1$,
$I=\mathbb{R}_+$, $k=0$ and, in addition, $u_1\equiv 0$. Consider
a solution to the problem \eqref{3.1}, \eqref{1.2}, \eqref{1.3} in
the class $X_0(\Pi_T^+)$. Then for any $t\in (0,T]$
\begin{equation} \label{4.12}
\begin{aligned}
&\iint_{\mathbb{R}_+^2} u^2(t,x,y)\,dx dy +
\iint_{B_t} u_x^2(\tau,0,y)\,dy d\tau  \\
&= \iint_{\mathbb{R}_+^2} u_0^2\,dx dy + 2\iiint_{\Pi_t^+} fu\,dx
dy\,d\tau.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
For smooth solutions (\ref{4.12}) is obtained obviously by multiplication
of the equation \eqref{3.1} by $2u(t,x,y)$ and consequent integration
(compare with (\ref{4.1})) and then for weak ones via closure.
\end{proof}

\begin{lemma}\label{L4.4}
Let the hypothesis of Lemma~\ref{L4.2} be satisfied for $n=1$,
$I=\mathbb{R}_+$, $k=1$ and, in addition, $u_1\equiv 0$. Consider
a solution to the problem \eqref{3.1}, \eqref{1.2}, \eqref{1.3} in
the class $X_1(\Pi_T^+)$. Then for any $t\in (0,T]$
\begin{equation} \label{4.13}
\begin{aligned}
&\iint_{\mathbb{R}_+^2} \Bigl(u_x^2+u_y^2-\frac 1 3
u^3\Bigr)\rho(x)\,dx dy +
\frac 12  \iiint_{\Pi_t^+} (u_{xx}^2+u_{xy}^2+u_{yy}^2) \rho'(x)\,dx\,dy\,d\tau
\\
&+2 \iiint_{\Pi_t^+} uu_x (u_{xx}+u_{yy}) \rho\,dx\,dy\,d\tau\\
&\leq
\iint_{\mathbb{R}_+^2} \Bigl(u_{0x}^2+u_{0y}^2-\frac 1 3 u_0^3\Bigr)
 \rho\,dx\, dy
 +2 \iiint_{\Pi_t^+} (f_xu_x+f_yu_y)\rho\,dx\,dy\,d\tau\\
&\quad - \iiint_{\Pi_t^+} fu^2\rho\,dx\,dy\,d\tau
+c\iint_{B_t} (f^2+u_x^2)\Bigr|_{x=0}dy d\tau \\
&\quad +c\left(1+\|u\|^2_{C([0,t];L_2(\mathbb{R}^2_+))}\right)
\iiint_{\Pi_t^+} (u_x^2+u_y^2+u^2)\rho\,dx\,dy\,d\tau,
\end{aligned}
\end{equation}
where $\rho(x)\equiv 2-(1+x)^{-1/2}$.
\end{lemma}

\begin{proof}
This lemma was proved in \cite{faminskii06}. We represent here the
slightly modified version of the proof.

As in the previous lemma it it sufficient to consider smooth
solutions. Multiplying \eqref{3.1} by
$-\bigl(2(u_x(t,x,y)\rho(x))_x+2u_{yy}(t,x,y)\rho(x)
+u^2(t,x,y)\rho(x)\bigr)$ and integrating over $\mathbb{R}_+^2$ we
derive an equality
\begin{equation}\label{4.14}
\begin{aligned}
&\frac d{dt} \iint_{\mathbb{R}_+^2} (u_x^2+u_y^2-\frac 1 3
u^3)\rho\,dx dy +
\iint_{\mathbb{R}_+^2} (3u_{xx}^2+4u_{xy}^2+u_{yy}^2)\rho'\,dx dy\\
&+\int_\mathbb{R} \bigl(u_{xx}^2\rho
+2u_{xx}u_x\rho'-u_x^2\rho''\bigr)\bigl|_{x=0}dy-
\iint_{\mathbb{R}_+^2} (u_x^2+u_y^2)\rho'''\,dx dy\\
&+2\iint_{\mathbb{R}_+^2} uu_x(u_{xx}+u_{yy})\rho\,dx dy+
\iint_{\mathbb{R}_+^2} u^2(u_{xx}+u_{yy})\rho'\,dx dy\\
&=2\iint_{\mathbb{R}_+^2} (f_xu_x+f_yu_y)\rho\,dx dy +
2\int_\mathbb{R} (fu_x\rho)\bigl|_{x=0}dy-\iint_{\mathbb{R}_+^2}
fu^2\rho\,dx dy.
\end{aligned}
\end{equation}
Applying the interpolational inequality (\ref{2.1}) in the case $p=4$
we find that
\begin{align*}
&\Bigl|\iint_{\mathbb{R}_+^2} u^2(u_{xx}+u_{yy})\rho'\,dx dy\Bigr|\\
&\leq \frac 1 2 \iint_{\mathbb{R}_+^2} (u_{xx}^2+u_{yy}^2)\rho'\,
dx dy + c \iint_{\mathbb{R}_+^2} (u_x^2+u_y^2+u^2)\rho\, dx dy
\iint_{\mathbb{R}_+^2} u^2\, dx dy
\end{align*}
and derive (\ref{4.13}) from (\ref{4.14}).
\end{proof}

\begin{lemma}\label{L4.5}
Let the hypothesis of Lemma~\ref{L4.2} be satisfied for $n=3$,
$I=(0,1)$, $k=0$ and, in addition, $u_1=u_2\equiv 0$. Consider a
solution to the problem \eqref{3.1}, \eqref{1.2}, \eqref{1.5} in
the class $X_0(Q_T)$. Then for any $t\in (0,T]$
\begin{equation} \label{4.15}
\begin{aligned}
&\iint_{\mathbb{R}_+^2} u^2(t,x,y)\rho(x)\,dx dy + \iiint_{Q_t}
(3u_x^2+u_y^2) \rho'(x)\,dx\,dy\,d\tau+
\iint_{B_t} u_x^2\big|_{x=0}\,dy d\tau  \\
&= \iint_{\mathbb{R}_+^2} u_0^2\rho\,dx dy + \rho(1)\iint_{B_t}
u_3^2\,dy d\tau+ 2\iiint_{Q_t} fu\rho\,dx dy\,d\tau,
\end{aligned}
\end{equation}
where either $\rho\equiv 1$ or $\rho\equiv 1+x$.
\end{lemma}

\begin{proof}
Similarly to Lemma~\ref{L4.3} for smooth solutions (\ref{4.15}) is obtained
via multiplication of \eqref{3.1} by $2u(t,x,y)\rho(x)$ and
consequent integration
and then for weak ones via closure.
\end{proof}

\section{Proof of the main result}\label{S5}

This section contains the proof of Theorem~\ref{T2.1} consisting of several
lemmas. The first one is devoted to local well-posedness.

\begin{lemma}\label{L5.1}
Let the hypothesis of Theorem~\ref{T2.1} be satisfied. Then there
exists $t_0\in (0,T]$ such that any of the considered
initial-boundary value problems for the equation \eqref{1.1} is
well posed in $X_k((0,t_0)\times I\times\mathbb{R})$.
\end{lemma}

\begin{proof}
For $t_0\in (0,T]$ introduce a set of functions
$$
Y_k((0,t_0)\times I\times\mathbb{R})=\bigl\{v\in X_k((0,t_0)\times
I\times\mathbb{R}):
\partial_t^m v\bigl|_{t=0}=\Phi_m\ \text{for } m\leq m_0-1\bigr\}
$$
and define on this set a map $\Lambda$ in such a way: $u=\Lambda
v$ is a solution in $Y_k((0,t_0)\times I\times\mathbb{R})$ to a
corresponding initial-boundary value linear problem for an
equation
\begin{equation}\label{5.1}
u_t+u_{xxx}+u_{xyy}=f-vv_x
\end{equation}
with the initial profile \eqref{1.2} and one of the three boundary
conditions \eqref{1.3}, \eqref{1.4} or \eqref{1.5}. Note that the
functions $\widetilde\Phi_m$, written for these problems, coincide
for $m< k/3$ with the functions $\Phi_m$ written for the original
problems. Therefore the compatibility conditions of the order $k$
are satisfied. Moreover, by virtue of Lemma~\ref{L2.1} $vv_x \in
M_k((0,t_0)\times I\times\mathbb{R})$, so Lemma~\ref{L4.2}
provides existence of the map $\Lambda$ and according to
(\ref{4.3}) and (\ref{2.7})
\begin{equation}\label{5.2}
\|u\|_{X_k((0,t_0)\times I\times\mathbb{R})}\leq
c(T,k)\Bigl(\widetilde c+ t^{1/6}_0\|v\|^2_{X_k((0,t_0)\times
I\times\mathbb{R})}\Bigr),
\end{equation}
where the constant $\widetilde c$ depends on the norms of $u_0$,
$\EuScript{U}^n$ and $f$ in the corresponding spaces. It follows
from (\ref{5.2}) that for considerably large $R>0$ and
considerably small $t_0^*\in (0,T]$ the map $\Lambda$ transforms
for any $t_0\in (0,t_0^*]$ a ball $Y_{k,R}((0,t_0)\times
I\times\mathbb{R})=\{v\in Y_k((0,t_0)\times I\times\mathbb{R}):
\|v\|_{X_k((0,t_0)\times I\times\mathbb{R})}\leq R\}$ into itself.

Next, consider two functions $v$ and $\widetilde v$ from the set
$Y_{k,R}((0,t_0)\times I\times\mathbb{R})$. Similarly to
(\ref{5.2})
$$
\|\Lambda v-\Lambda\widetilde v\|_{X_k((0,t_0)\times
I\times\mathbb{R})}\leq c(T,k)t^{1/6}_0R \|v-\widetilde
v\|_{X_k((0,t_0)\times I\times\mathbb{R})}
$$
and therefore $\Lambda$ is a contraction in $Y_{k,R}((0,t_0)\times
I\times\mathbb{R})$ for considerably small $t_0$.

Continuous dependence is established in a similar way.
\end{proof}

The next lemma is devoted to one global conditional a priori estimate
valid for all three considered problems.

\begin{lemma}\label{L5.2}
Let the hypothesis of Theorem~\ref{T2.1} be satisfied for $k\geq
2$. Let $u(t,x,y)$ be a solution to any of the three
initial-boundary value problems for the equation \eqref{1.1} in
the class $X_k((0,T')\times I\times\mathbb{R})$ for some $T'\in
(0,T]$. Then uniformly with respect to $T'$,
\begin{equation}\label{5.3}
\begin{aligned}
&\|u\|_{X_k((0,T')\times I\times\mathbb{R})} \\
&\leq c\bigl(T,k, \|u_0\|_{H^k(I\times\mathbb{R})},
  \|\EuScript{U}^n\|_{\EuScript{B}_n^k(T)},
  \|f\|_{M_k((0,T)\times I\times\mathbb{R})},
  \|u\|_{X_{k-1}((0,T')\times I\times\mathbb{R})}\bigr).
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Consider $u$ as a solution to the corresponding problem for the equation
(\ref{5.1}), where $v\equiv u$. Then the inequalities (\ref{2.8}) and
(\ref{4.3}) yield that similarly to (\ref{5.2})
$$
\|u\|_{X_k((0,t_0)\times I\times\mathbb{R})}\leq
c(T,k)\Bigl(\widetilde c+ t^{1/6}_0\|u\|_{X_{k-1}((0,T')\times
I\times\mathbb{R})} \|u\|_{X_k((0,t_0)\times
I\times\mathbb{R})}\Bigr),
$$
whence (\ref{5.3}) follows by the standard argument.
\end{proof}

The next lemma provides a global a priori estimate for the problem in
$\Pi_T^+$ in the class $X_1(\Pi_T^+)$ and thus completes the proof of
Theorem~\ref{T2.1} in this case.

\begin{lemma}\label{L5.3}
Let the hypothesis of Theorem~\ref{T2.1} be satisfied for $n=1$,
$I=\mathbb{R}_+$, $k=1$. Let $u(t,x,y)$ be a solution to the
problem \eqref{1.1}--\eqref{1.3} in the class $X_1(\Pi_{T'}^+)$
for some $T'\in (0,T]$. Then uniformly with respect to $T'$
\begin{equation}\label{5.4}
\|u\|_{C([0,T'];H^1(\mathbb{R}_+^2))} \leq
c(T,\|u_0\|_{H^1(\mathbb{R}_+^2)},\|u_1\|_{H^{2/3,2}(B_T)},
\|f\|_{L_2(0,T;H^1(\mathbb{R}_+^2))}).
\end{equation}
\end{lemma}

\begin{proof}
We reproduce here in brief the proof from \cite{faminskii06}.
Extend the function $u_1$ in the class $H^{2/3,2}$ to the whole
plane $\mathbb{R}^2$ such that $u_1(t,y)=0$ for $t\leq -1$. Let
\begin{equation}\label{5.5}
U(t,x,y)\equiv u(t,x,y)-J_+(t,x,y;u_1).
\end{equation}
Write down for the function $U$ the equality (\ref{4.12}), then for $t\in [0,T']$
\begin{equation}\label{5.6}
\iint_{\mathbb{R}_+^2} U^2(t,x,y)\, dx\,dy + \iint_{B_t}
U_x^2\Bigl|_{x=0}dy\,d\tau \leq c+ 2 \iiint_{\Pi_t^+}
(f-uu_x)U\,dx\,dy\,d\tau.
\end{equation}
Since
\begin{equation}\label{5.7}
uu_xU=\Bigl(\frac {U^3}3+J_+\frac{U^2}2\Bigr)_x+
\partial_xJ_+\Bigl(\frac {U^2}2+J_+U\Bigr)
\end{equation}
and $U\bigl|_{x=0}=0$, it follows from (\ref{5.6}) and (\ref{3.16}) that
\begin{equation}\label{5.8}
\|u\|_{C([0,T'];L_2(\mathbb{R}_+^2))}+\|u_x\bigl|_{x=0}\|_{L_2(B_{T'})}\leq
c.
\end{equation}

Next, write down for the function $U$ the inequality (\ref{4.13}), then
by virtue of the already established estimate (\ref{5.8}) for any
$t\in [0,T']$
\begin{equation}\label{5.9}
\begin{aligned}
&\iint_{\mathbb{R}_+^2} (U_x^2+U_y^2-\frac 1 3 U^3)\rho\,dx\,dy +
\frac 1 2  \iiint_{\Pi_t^+}(U_{xx}^2+U_{xy}^2+U_{yy}^2)\rho'\,dx\,dy\,d\tau\\
&\leq c + c\iiint_{\Pi_t^+}(U_x^2+U_y^2)\rho\,dx\,dy\,d\tau+
\iiint_{\Pi_t^+}uu_xU^2\rho\,dx\,dy\,d\tau\\
&\quad + c\iint_{B_t} u_1^2 u_x^2\bigl|_{x=0}dy\,d\tau
+2 \iiint_{\Pi_t^+}uu_xU_x\rho'\,dx\,dy\,d\tau\\
&\quad +2 \iiint_{\Pi_t^+}(J_+U_x+u\partial_xJ_+)(U_{xx}+U_{yy})
 \rho\,dx\,dy\,d\tau.
\end{aligned}
\end{equation}
The inequality (\ref{2.1}) and the estimates (\ref{3.16}), (\ref{5.8})
 yield that
\begin{align*}
\iiint_{\Pi_t^+} uu_xU^2\rho\,dx\,dy\,d\tau
&= \iiint_{\Pi_t^+} \Bigl(\partial_xJ_+uU^2\rho-
\frac 1 3 (J_+\rho)_xU^3-\frac 1 4 U^4\rho'\Bigr)\,dx\,dy\,d\tau \\
&\leq c\iiint_{\Pi_t^+} (U_x^2+U_y^2)\rho\,dx\,dy\,d\tau+c,
\end{align*}
\begin{align*}
&2\iiint_{\Pi_t^+}uu_xU_x\rho'\,dx\,dy\,d\tau\\
&=-\iint_{B_t}u_1^2(U_x\rho')\bigl|_{x=0}dy\,d\tau-
\iiint_{\Pi_t^+}u^2(U_{xx}\rho'+U_x\rho'')\,dx\,dy\,d\tau \\
&\leq \frac 1 6  \iiint_{\Pi_t^+} U_{xx}^2\rho'\,dx\,dy\,d\tau +
c\iiint_{\Pi_t^+} (U_x^2+U_y^2)\rho\,dx\,dy\,d\tau+c,
\end{align*}
\begin{align*}
\iint_{B_t} u_1^2 u_x^2\bigl|_{x=0}dy\,d\tau
&\leq \varepsilon \iiint_{\Pi_t^+}U_{xx}^2\rho'\,dx\,dy\,d\tau \\
&\quad +c(\varepsilon)\int_0^t \Bigl(1+\sup_{y\in \mathbb{R}}
u_1^4\Bigr) \iint_{\mathbb{R}_+^2}U_x^2\rho\,dx\,dy\,d\tau+c,
\end{align*}
where $\|u_1\|_{L_4(0,T;C_b(\mathbb{R}))}\leq
c\|u_1\|_{H^{2/3,2}(\mathbb{R}^2)}$ (see, e.g. \cite{besov}) and
$\varepsilon>0$ can be chosen arbitrarily small. In the above
arguments we also use the obvious interpolational inequality
\begin{equation}\label{5.10}
|\varphi(0)|\leq c \Bigl(\int_{\mathbb{R}_+}
(\varphi')^2\rho'\,dx\Bigr)^{1/4} \Bigl(\int_{\mathbb{R}_+}
\varphi^2\rho\,dx\Bigr)^{1/4}+ c\Bigl(\int_{\mathbb{R}_+}
\varphi^2\rho\,dx\Bigr)^{1/2}.
\end{equation}
Finally,
\begin{align*}
&2 \iiint_{\Pi_t^+}(J_+U_x+u\partial_xJ_+)(U_{xx}+U_{yy})\rho\,dx\,dy\,d\tau \\
&\leq \frac 1 6  \iiint_{\Pi_t^+} (U_{xx}^2+U_{yy}^2)\rho'\,dx\,dy\,d\tau\\
&\quad +c\int_0^t \sup_{(x,y)\in \mathbb{R}_+^2}
\Bigl[\bigl((\partial_xJ_+)^2+J_+^2\bigr) \frac
{\rho^2}{\rho'}\Bigr]
\iint_{\mathbb{R}_+^2}(U_x^2\rho+u^2)\,dx\,dy\,d\tau \\
&\leq \frac 1 6  \iiint_{\Pi_t^+}
(U_{xx}^2+U_{yy}^2)\rho'\,dx\,dy\,d\tau+ \int_0^t \gamma(\tau)
\iint_{\mathbb{R}_+^2}U_x^2\rho\,dx\,dy\,d\tau+c,
\end{align*}
where $\|\gamma\|_{L_1(0,T)}\leq c$ since $\rho^2(\rho')^{-1}\leq c(1+x)^{3/2}$
and we can apply the inequalities (\ref{3.16}) and (\ref{3.22}).
Therefore the inequality (\ref{5.9}) provides (\ref{5.4}).
\end{proof}

The last lemma establishes a global a priori estimate for the problem in
$Q_T$ in the class $X_3(Q_T)$ and thus completes the proof of
Theorem~\ref{T2.1}.

\begin{lemma}\label{L5.4}
Let the hypothesis of Theorem~\ref{T2.1} be satisfied for $n=3$,
$I=(0,1)$, $k=3$. Let $u(t,x,y)$ be a solution to the problem
\eqref{1.1}, \eqref{1.2}, \eqref{1.5} in the class $X_3(Q_{T'})$
for some $T'\in (0,T]$. Then uniformly with respect to $T'$
\begin{equation}\label{5.11}
\|u\|_{C([0,T'];H^3(\Sigma))} \leq
c\bigl(T,\|u_0\|_{H^3(\Sigma)},\|\EuScript{U}^3\|_{\EuScript{B}_3^3(T)},
\|f\|_{M_3(Q_T)}\bigr).
\end{equation}
\end{lemma}

\begin{proof}
Let $v(t,x,y)\in X_3(Q_T)$ be a solution to the linear problem
\eqref{3.1}, \eqref{1.2}, \eqref{1.5}. Let
\begin{equation}\label{5.12}
U(t,x,y)\equiv u(t,x,y)-v(t,x,y),
\end{equation}
then the function $U\in X_3(Q_{T'})$ is a solution to a problem
\begin{gather*}
U_t+U_{xxx}+U_{xyy}=-uu_x,\\
U\big|_{t=0}=0,\quad U\big|_{x=0}=U\big|_{x=1}=U_x\big|_{x=1}=0.
\end{gather*}
First write down the equality (\ref{4.15}) for the function $U$
in the case $\rho\equiv 1$. Then using the equality (\ref{5.7}),
where $J_+$ is substituted by $v$, similarly to (\ref{5.8}) we derive
an estimate
\begin{equation}\label{5.13}
\|u\|_{C([0,T'];L_2(\Sigma))}+\|u_x\bigl|_{x=0}\|_{L_2(B_{T'})}\leq c.
\end{equation}
Next, again use the equality (\ref{4.15}) for the function $U$ but now
in the case $\rho\equiv 1+x$. Then by virtue of the already proved estimate
(\ref{5.13})
\begin{equation}\label{5.14}
\iiint_{Q_{T'}} (U_x^2+U_y^2)\,dx\,dydt\leq c
-2\iiint_{Q_{T'}} uu_xU\rho\,dx\,dydt.
\end{equation}
Again applying the corresponding analogue of (\ref{5.7}) we find that
$$
2\iint_{\Sigma} uu_xU\rho\,dx\,dy =
\iint_{\Sigma} \Bigl(-\frac 23 U^3+(v_x\rho-v)U^2+2vv_xU\rho\Bigr)dx\,dy
$$
and by virtue of the interpolation inequality (\ref{2.1}) derive
from (\ref{5.14}) an estimate
\begin{equation}\label{5.15}
\|u\|_{L_2(0,T';H^1(\Sigma))}\leq c.
\end{equation}

Next, by induction with respect to $j$ we prove that for $j\leq 3$
\begin{equation}\label{5.16}
\|\partial_y^ju\|_{C([0,T'];L_2(\Sigma))}+
\|\partial_y^ju\|_{L_2(0,T';H^1(\Sigma))}+
\|\partial_y^ju_x\bigl|_{x=0}\|_{L_2(B_{T'})}\leq c.
\end{equation}
For $j=0$ this estimate coincides with (\ref{5.13}), (\ref{5.15}).
For $j\geq 1$ write down for the function $\partial_y^jU$ the equality
(\ref{4.15}) in the case $\rho\equiv 1+x$:
\begin{equation} \label{5.17}
\begin{aligned}
&\iint_{\mathbb{R}_+^2} (\partial_y^jU)^2\rho\,dx dy + \iiint_{Q_t}
\bigl(3(\partial_y^jU_x)^2+(\partial_y^{j+1}U)^2\bigr)\,dx\,dy\,d\tau\\
&+ \iint_{B_t} (\partial_y^jU_x)^2\big|_{x=0}\,dy d\tau \\
&= -2\iiint_{Q_t} \partial_y^j(uu_x)\partial_y^jU\rho\,dx\,dy\,d\tau.
\end{aligned}
\end{equation}
Here
\begin{equation}\label{5.18}
\begin{aligned}
&\Bigl|2\iiint_{Q_t} (u\partial_y^jU_x+u_x\partial_y^jU)
 \partial_y^jU\rho\,dx\,dy\,d\tau\Bigr|\\
&= \Bigl|\iiint_{Q_t} (u_x\rho-u)(\partial_y^jU)^2\,dx\,dy\,d\tau\Bigr| \\
&\leq c\int_0^t\Bigl(\iint_{\Sigma}(u_x^2+u^2)\,dx\,dy\Bigr)^{1/2}
\Bigl(\iint_{\Sigma}(\partial_y^j U)^4\,dx\,dy\Bigr)^{1/2}\,d\tau \\
&\leq \varepsilon \iiint_{Q_t} \bigl((\partial_y^jU_x)^2+
(\partial_y^{j+1}U)^2\bigr)\,dx\,dy\,d\tau
+c(\varepsilon)\int_0^t \gamma(\tau) \iint_{\Sigma} (\partial_y^jU)^2\,
dx\,dy\,d\tau,
\end{aligned}
\end{equation}
where $\|\gamma\|_{L_1(0,T')}\leq c$ and $\varepsilon>0$ can be chosen
arbitrarily small. All other terms in the right part of (\ref{5.17}) are also
similarly estimated by the right part of (\ref{5.18}) (plus certain appropriate
constant) and so (\ref{5.17}) yields the estimate (\ref{5.16}).

Now let $v_1(t,x,y)\in X_0(Q_T)$ be a solution to a problem
\begin{gather*}
v_{1t}+v_{1xxx}+v_{1xyy}=f_t,\\
v_1\big|_{t=0}=\Phi_1,\quad v_1\big|_{x=0}=u_{1t},\quad
v_1\big|_{x=1}=u_{2t},\quad v_{1x}\big|_{x=1}=u_{3t}.
\end{gather*}
Let
$$
U_1(t,x,y)\equiv u_t(t,x,y)-v_1(t,x,y),
$$
then the function $U_1\in X_0(Q_{T'})$ is a solution to a problem
\begin{gather*}
U_{1t}+U_{1xxx}+U_{1xyy}=-(uu_x)_t,\\
U_1\big|_{t=0}=0,\quad
U_1\big|_{x=0}=U_1\big|_{x=1}=U_{1x}\big|_{x=1}=0.
\end{gather*}
Writing down for the function $U_1$ the equality (\ref{4.15}) in the
case $\rho\equiv 1+x$ we obtain the corresponding analogue of (\ref{5.17})
and estimating nonlinear terms similarly to (\ref{5.18})
derive an estimate
\begin{equation}\label{5.19}
\|u_t\|_{C([0,T'];L_2(\Sigma))}+
\|u_t\|_{L_2(0,T';H^1(\Sigma))}+
\|u_{tx}\bigl|_{x=0}\|_{L_2(B_{T'})}\leq c.
\end{equation}

Write down the equation \eqref{1.1} in a form
\begin{equation}\label{5.20}
u_{xxx}=f-u_{xyy}-uu_x-u_t.
\end{equation}
By virtue of (\ref{5.16}) and (\ref{5.19}) the right part of this equality
can be represented as a sum $g_0+g_{1x}$, where
$g_j\in C([0,T'];L_2(\Sigma))$ with appropriate estimates on the
corresponding norms. Therefore it follows from (\ref{5.20})
(together with (\ref{5.16}) for $j=2$) that
\begin{equation}\label{5.21}
\|u\|_{C([0,T'];H^2(\Sigma))}\leq c.
\end{equation}
Finally, we apply the inequality (see, e.g. \cite{lions})
\begin{equation}\label{5.22}
\|g\|_{H^2(\Sigma)}\leq c\bigl(\|\Delta g\|_{L_2(\Sigma)}+
\|g\big|_{\partial\Sigma}\|_{H^{3/2}(\mathbb{R})}+
\|g\|_{H^1(\Sigma)}\bigr)
\end{equation}
for the function $g\equiv u_x$. It follows from \eqref{1.1} that
$$
\Delta u_x=f-uu_x-u_t\in C([0,T'];L_2(\Sigma)).
$$
Moreover, by virtue of (\ref{5.16}), (\ref{5.19}) and embedding
$H^2(\Sigma)\subset H^{3/2}(\partial\Sigma)$ (see \cite{lions})
\begin{align*}
&\|u_x\big|_{x=0}\|_{C([0,T'];H^{3/2}(\mathbb{R}))} \\
&\leq \|u_{0x}\big|_{x=0}\|_{H^{3/2}(\mathbb{R})}
+2\|u_{tx}\big|_{x=0}\|_{L_2(B_{T'})}^{1/2}
\|u_x\big|_{x=0}\|_{L_2(0,T';H^3(\mathbb{R}))}^{1/2} \leq c,
\end{align*}
\begin{align*}
\|u_x\big|_{x=1}\|_{C([0,T'];H^{3/2}(\mathbb{R}))}
&= \|u_3\|_{C([0,T'];H^{3/2}(\mathbb{R}))} \\
&\leq \|u_3(0,\cdot)\|_{H^{3/2}(\mathbb{R})}
+c\|u_3\|_{H^{1,3}(B_T)} \leq c_1.
\end{align*}
Therefore, (\ref{5.22}) together with (\ref{5.21}) and (\ref{5.16})
for $j=3$ provide (\ref{5.11}).
\end{proof}

\section{Weak solutions}\label{S6}

Solutions considered in Theorem~\ref{T2.1} are at least in the space
$X_1$ because of the use of the inequality (\ref{2.7}), which is
valid only if $k\geq 1$. Of course, the contraction principle is not the
unique method to construct solutions. For example, they can be obtained
as limits of certain sequences of solutions to regularized problems,
if appropriate uniform estimates on these regularized solutions are
established (uniqueness in this situation requires its own special approaches).
In order to describe classes of weak solutions, which are obtained on this way,
we need some additional notation.

By $C_w(\overline{I};\EuScript{B})$, where $\EuScript{B}$ is a certain Banach space,
we denote a space of weakly continuous mappings from $\overline{I}$ to
$\EuScript{B}$. Note that $C_w(\overline{I};\EuScript{B})\subset
L_\infty(I;\EuScript{B})$ for bounded intervals $I$ (see, e.g. \cite{gajewski})
and this space becomes the Banach space supplied with a norm
$$
\|f\|_{C_w(\overline{I};\EuScript{B})}=\sup_{t\in \overline{I}}
\|f(t)\|_{\EuScript{B}}.
$$
By analogy with (\ref{1.16}) let
$$
\lambda^+(u;T)=\sup_{m\geq 0}
\int_0^T\!\!\int_m^{m+1}\!\!\int_\mathbb{R} (u_x^2+u_y^2)\,dy\,dx\,dt.
$$

\begin{theorem}\label{T6.1}
Let $(1+x)^\beta u_0\in L_2(\mathbb{R}_+^2)$ for certain
$\beta\geq 0$, $u_1\in H^{s/3,s}(B_T)$ for certain $s>3/2$ and
$T>0$, $(1+x)^\beta f\in L_1(0,T;L_2(\mathbb{R}_+^2))$. Then there
exists a solution $u(t,x,y)$ to the problem
\eqref{1.1}--\eqref{1.3} such that
$$
(1+x)^\beta u\in C_w([0,T];L_2(\mathbb{R}_+^2)),\quad
\lambda^+(u;T)<\infty.
$$
If, in addition, $\beta>0$ then
$$
(1+x)^{\beta-1/2}u\in L_2(0,T;H^1(\mathbb{R}_+^2)).
$$
The problem \eqref{1.1}--\eqref{1.3} is well-posed in this class
if $\beta\geq 1$.
\end{theorem}

\begin{proof}
This result was established in the paper \cite{faminskii02} under
other (more complicated) hypothesis on the boundary data $u_1$.
This hypothesis in \cite{faminskii02} ensured that the boundary
potential $J_+(t,x,y;u_1)$, which was constructed in the form
(\ref{3.23}), possessed the following properties:
\begin{equation}\label{6.1}
J_+\in C([0,T];L_2(\mathbb{R}_+^2))\cap
L_2(0,T;H^1(\mathbb{R}_+^2)), \quad
\|\partial_{x,y}J_+(t,\cdot,\cdot;u_1)\|_{C_b(\overline{\mathbb{R}}_+^2)}
\in L_1(0,T),
\end{equation}
which were crucial for global a priori estimates in the space
$C([0,T];L_2(\mathbb{R}_+^2))$ (see (\ref{5.8})) and on the value
$\lambda^+(u,T)$. But these properties are also provided by the
inequalities (\ref{3.18}) and (\ref{3.20}) under the hypothesis of
the present theorem.
\end{proof}

\begin{theorem}\label{T6.2}
Let $u_0\in L_2(\Sigma)$, $u_1,u_2\in H^{s/3,s}(B_T)$ for certain
$s>3/2$ and $T>0$, $u_3\in L_2(B_T)$, $f\in L_1(0,T;L_2(\Sigma))$.
Then the problem \eqref{1.1}, \eqref{1.2}, \eqref{1.5} is
well-posed in a class $C_w([0,T];L_2(\Sigma))\cap
L_2(0,T;H^1(\Sigma))$.
\end{theorem}

\begin{proof}
The main global a priori estimate in this class is obtained similarly
to (\ref{5.13}), (\ref{5.15}), where in the formula (\ref{5.12}) the
function $v$ is taken in a form
\begin{equation}\label{6.2}
v(t,x,y)\equiv J_+(t,x,y;u_1)\sigma(1-x)+
J_+(-t,1-x,y;\widetilde u_2)\sigma(x),
\end{equation}
where $\widetilde u_2(t,y)\equiv u_2(-t,y)$, $u_1$ and $u_2$ are
extended in the class $H^{s/3,s}$ to the whole plane
$\mathbb{R}^2$ such that $u_1(t,y)\equiv 0$ for $t\leq -1$,
$u_2(t,y)\equiv 0$ for $t\geq T+1$, and $\sigma(x)$ is a certain
smooth "cut-off" function, namely, $\sigma(x)=0$ for $x\leq 1/4$,
$\sigma(x)=1$ for $x\geq 3/4$, $\sigma'(x)>0$ for $x\in
(1/4,3/4)$. Then the estimates (\ref{3.18}), (\ref{3.20}) and
(\ref{3.22}) provide the required properties of the function $v$
similar to (\ref{6.1}), where $\mathbb{R}^2_+$ is substituted by
$\Sigma$, and $v_x|_{x=1}$ is evaluated in $L_2(B_T)$ by virtue of
the estimate on $\partial_xJ_+$ in
$C_b(\overline{\mathbb{R}}_+;L_2(B_T))$.

Uniqueness and continuous dependence are established on the base
of the equality (\ref{4.15}) in the case $\rho\equiv 1+x$
similarly to Theorem~\ref{T6.1} (here, of course, additional
decay of solutions at infinity is not required), where the boundary
data are made zero similarly to (\ref{5.12}), (\ref{6.2}).
\end{proof}

In the paper \cite{baykova} existence of global solutions to the
problem \eqref{1.1}, \eqref{1.2}, \eqref{1.4} was established in a
class of functions $u\in C_w([0,T];L_2(\mathbb{R}_-^2))$ such that
$$
\lambda^-(u;T)=\sup_{m\geq 0}
\int_0^T\!\!\int_{-m-1}^{-m}\!\!\int_\mathbb{R}
(u_x^2+u_y^2)\,dy\,dx\,dt<\infty
$$
under the hypothesis $u_0\in L_2(\mathbb{R}_-^2)$, $u_2\in
H^{s/3,s}(B_T)$ for certain $s>3/2$, $u_3\in L_2(B_T)$,
$f\in L_1(0,T;L_2(\mathbb{R}_-^2))$.

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