\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 59, pp. 1--12.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/59\hfil Unique continuation]
{Unique continuation property for the Kadomtsev-Petviashvili (KP-II) equation}
\author[M. Panthee\hfil EJDE-2005/59\hfilneg]
{Mahendra Panthee}  

\address{Mahendra Panthee \hfill\break
Central Department of Mathematics,
Tribhuvan University, Kirtipur, Kathmandu, Nepal; and \hfill\break
Department of Mathematics, CMAGDS, IST, 1049--001,  Av. Rovisco Pais,
Lisbon, Portugal}
 \email{mpanthee@math.ist.utl.pt}

\date{}
\thanks{Submitted April 9, 2005. Published June 10, 2005.}
\thanks{Partially supported by FAPESP-Brazil and CMAGDS through FCT/POCTI/FEDER, IST,
\hfill\break\indent
 Portugal}

\subjclass[2000]{35Q35, 35Q53}
\keywords{Dispersive equations; KP equation; unique continuation property;
\hfill\break\indent
 smooth solution; compact support}

\begin{abstract}
 We generalize a method introduced by Bourgain in \cite{Borg} based on
 complex analysis to address two spatial dimensional models and  prove
 that if a sufficiently smooth solution to the initial value problem
 associated with the Kadomtsev-Petviashvili (KP-II) equation
\begin{equation*}
(u_t+u_{xxx}+uu_{x})_{x} +u_{yy}=0, \quad
(x, y) \in \mathbb{R}^2, \;t\in\mathbb{R},
\end{equation*}
 is supported compactly in a nontrivial time interval then
 it vanishes identically.
\end{abstract}

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

\section{Introduction}

 Let us consider the  following initial value problem (IVP) associated
 with the Kadomtsev-Petviashvili (KP) equation,
\begin{equation}
\begin{gathered}
(u_t+u_{xxx}+uu_{x})_{x} = \alpha u_{yy}, \quad(x, y) \in \mathbb{R}^2, \,\,t\in\mathbb{R}\\
u(x,y,0)=\phi(x,y),
\end{gathered}
\label{ivpkp}
\end{equation}
where $u = u(x, y, t)$ is a real valued function and $\alpha = \pm 1$.
 This model was derived by Kadomtsev and Petviashvili
\cite{KP.1} to describe the propagation of weakly nonlinear long waves 
on the surface of fluid, when the wave motion is essentially one-directional 
with weak transverse effects along y-axis. Equation \eqref{ivpkp} is known as 
KP-I or KP-II equation according as $\alpha = 1$ or $\alpha = -1$ and is considered 
 a two dimensional generalization of the Korteweg-de Vries (KdV) equation
\begin{equation}\label{i.kdv}
u_t +u_{xxx} +uu_x =0, \quad x,\,t\in \mathbb{R},
\end{equation}
  which arises in modeling the evolution of one dimensional surface gravity waves
   with small amplitude  in a shallow channel of water. The KdV model is a widely studied
    model and arises in various physical contexts. It has very rich mathematical structure 
    and can also be solved by using inverse scattering technique.


The next generalization of the KdV model in two space dimension is the Zakharov-Kuznetsov (ZK) equation
\begin{equation}\label{i.zk}
u_t +(u_{xx}+u_{yy})_x +uu_x =0.
\end{equation}
The equation \eqref{i.zk} derived by
Zakharov and Kuznetsov in \cite{ZK} models the propagation of nonlinear
ion-acoustic waves in magnetized plasma. Much effort has been devoted
to study several properties including well-posedness issues and existence and stability of solitary 
wave solutions for the ZK model, see for example \cite{BL}, \cite{FA}, \cite{MP1} and references therein. 
In particular, the author in \cite{MP1} considered the following question:  If a
sufficiently smooth real valued solution $u=u(x, y, t)$ to the IVP
associated to (\ref{i.zk}) is supported compactly on a certain time
interval, is it true that $u\equiv 0$?  In some sense, it is a weak
version of the unique continuation property (UCP) which is defined as
follows (cf. \cite{SS}):

\begin{definition}\label{def-ucp} \rm
 Let $L$ be an evolution operator acting on functions defined on some connected open set
  $\Omega$ of $\mathbb{R}^n\times \mathbb{R}_t$. The operator $L$ is said to have unique continuation
  property (UCP) if every solution $u$ of $Lu = 0$ that vanishes on some nonempty open
  set $\Omega_1\subset\Omega$ vanishes in the horizontal component of $\Omega_1$ in $\Omega$.
\end{definition}


After  Carleman \cite{CA} initiated studies of UCP based on the
weighted  estimates for the associated solutions, many authors
improved and extended Carleman's method to address parabolic and
hyperbolic operators (see \cite{LH} and \cite{SM}). As far as we
know the first work dealing with the UCP for a general class of
dispersive equations in one space dimension is due to Saut and
Scheurer \cite{SS} that also  includes the KdV equation and uses
Carleman type estimates. Also, D. Tataru in \cite {DT} derived
some Carleman type estimates to  prove the UCP for Schr\"odinger
equation. Further,  Isakov in \cite{VI} considered  UCP for a
large class of evolution equations with nonhomogeneous principal
part.  Later, Zhang in \cite{ZHA}  obtained a slightly stronger
 UCP for the KdV equation than that in \cite{SS} using inverse
scattering theory. Using Miura's transformation Zhang in
\cite{ZHA}  obtained UCP for the modified KDV (mKdV) equation as
well.   Recently, Kenig, Ponce and Vega in \cite{KPV12} considered
the generalized KdV equation and proved UCP for it by deriving new
Carleman's type estimate. Further, Kenig, Ponce and Vega in
\cite{KPV1.5} obtained much stronger type of UCP for generalized
KdV equation by proving that, if $u_1, u_2\in C(\mathbb{R}, H^s(\mathbb{R}))$ are
two solutions of the generalized KdV equation with $s>0$ large
enough and if there exist $t_1 \neq t_2$ and $\alpha \in \mathbb{R}$ such
that $u_1(x, t_j) = u_2(x, t_j)$ for any $x\in (\alpha, \infty)$
or $(-\infty, \alpha)$ for $j = 1,2$ then $u_1 \equiv u_2$.
Bourgain in \cite{Borg} introduced a new approach to address a
wider class of evolution equations using complex variables
techniques. The method introduced in \cite{Borg} is more general
and can also be applied to models in higher spatial dimensions.
Quite recently, Carvajal and Panthee \cite{CP, CP-2} extended the
argument introduced in \cite{Borg} and \cite{KPV12} to prove the
UCP for the nonlinear Schr\"odinger-Airy equation. Also, it is
worth to mention the works of  I\'orio in \cite{IR-1, IR-2} and
Kenig, Ponce, Vega in \cite{KPV1.3} dealing with the UCP for
Benjamin-Ono type and  nonlinear Schr\"odinger equation.

The author in \cite{MP1} generalized the method introduced in
\cite{Borg}  to address a bi-dimensional (spatial) model and
provided with an affirmative answer to the question posed above
for the ZK model. Although, employing this method, one can deduce
UCP for the linear problem almost immediately, the same is not so
simple for the nonlinear problem and is quite involved.  The
symbol associated with the linear operator and the appropriate
choice of the parameter play important role in the approach we
used. The positive result obtained for the ZK model motivated us
to think for the similar result for the KP equation. Unlike ZK
model, there is singularity in the associated symbol of the linear
KP operator. So, one needs to handle the analysis with utmost
care. The structure of the associated symbol has also influenced a
lot in the well-posedness results for the Cauchy problem for the
KP equation. In this sense, the KP-II equation is much better than
the KP-I equation, see for example  \cite{Borg.1},
\cite{IN-1}--\cite{IM.3}, \cite{MST.1}, \cite{T.1}, \cite{TTz.1}
and \cite{Tz.1}. The structure of the associated symbol has also
affected our result on UCP for the KP equation. Here, we are able
to handle only the KP-II equation by choosing appropriate
parameters, see Remark \ref{rem.f} below. Therefore, from here
onwards, we concentrate our work on KP-II equation  (i.e., the IVP
\eqref{ivpkp} with $\alpha = -1$) and obtain UCP for it. More
precisely, using the scheme employed for the Zakharov-Kuznetsov
equation we prove the following theorem.

\begin{theorem}
\label{ucpkp}
Let $u\in C(\mathbb{R}; H^s(\mathbb{R}^2))$ be a  solution to the IVP 
associated with the KP-II equation with $s>0$ large enough. If there exists
 a non trivial time interval $I = [-T, T]$ such that for some $B>0$
$$
\mathop{\rm supp} u(t) \subseteq [-B, B] \times [-B, B],
\quad \forall \,t\in I,
$$ then $u \equiv 0$.
\end{theorem}

\begin{remark} \rm
In the context of the KdV equation \eqref{i.kdv}, several types
unique continuation properties  exist in the recent literature,
see for example \cite{KPV12}, \cite{KPV1.5}, \cite{SS} and
\cite{ZHA}. Since KP equation is known to be a two-dimensional
version of the KdV equation, we believe that the other forms of
the UCP as mentioned in the above references could be proved for
the KP equation too, but this needs to be done. As far as we know,
our result in this article is the first UCP for the KP type
models.
\end{remark}

  To prove this theorem, using the principle of Duhamel,
we write the IVP associated with the KP-II equation
    in the equivalent integral equation form,
\begin{equation}
\label{inteqkp1}
u(t) = S(t)\phi -\int_{0}^{t} S(t-t')(uu_x)(t')\,dt',
\end{equation}
where $S(t)$ given by,
\begin{equation}
\label{unigrkp1}
S(t)f(x, y) = \frac1{2\pi}\int_{\mathbb{R}^2} e^{i(t(\xi^3 -
\frac{\eta^2}{\xi})+x\xi+y\eta)}\hat{f}(\xi, \eta)\,d\xi d\eta,
\end{equation}
is the unitary group which describes the solution to the linear
 problem
\begin{equation}
\begin{gathered}
(u_t+u_{xxx})_{x}+u_{yy}=0,\\
u(x,y,0)=f(x,y).
\end{gathered}
\label{ivpkpl}
\end{equation}

Significant amount of work has been devoted to address the Cauchy problem 
associated with the KP equation, see for example \cite{Borg.1}, \cite{IN-1}--\cite{IM.3}, 
\cite{MST.1}, \cite{T.1}, \cite{TTz.1}, \cite{Tz.1} and references therein. Here we are 
not going to deal with this problem. For our purpose $H^1$-well-posedness of the associated 
Cauchy problem is enough. Finally,
let us record that the  quantities
\begin{gather}\label{con1}
\int_{\mathbb{R}^2}u^2\,dx\,dy,\\
\label{con2}
\frac12\int_{\mathbb{R}^2}\big[u_x^2 - (\partial_x^{-1}u_y)^2 - \frac13u^3\big]\,dx\,dy,
\end{gather}
are  conserved by the KP-II flow which are useful to get global
solution to the associated Cauchy problem in certain Sobolev spaces.

We organize this article as follows. In Section \ref{sec-2} we
establish some basic estimates needed in the proof of the main
result. The proof of the Theorem \ref{ucpkp} will be presented in
Section \ref{sec3}.

Before leaving this section let us introduce some notations that will
be used throughout this article. We use $\hat{f}$ to denote the
Fourier transform of $f$ and is defined as,
$$\hat{f}(\lambda) =
\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^n}e^{-i\mathbf{x}\cdot
\lambda}f(\mathbf x)\,d\mathbf{x}.
$$
We denote the $L^2$-based Sobolev space of order $s$ by $H^s$. The
various constants whose exact values are immaterial will be denoted by
$c$. We use the notation $A\lesssim B$ if there exists a constant $c>0$
such that $A\leq cB$.

\section{Basic Estimates}\label{sec-2}

 This section is devoted to establish some basic estimates that
will play fundamental role in our analysis. These estimates are not new and can be found in 
\cite {Borg} and the author's previous work in \cite{MP1}. We will not give the details of 
the proofs rather we just sketch the idea of the proof. Let us begin with the following result.

\begin{lemma} \label{lmm1}
Let $u \in C([-T, T]; H^s(\mathbb{R}^2))$ be a sufficiently smooth
solution to the IVP (\ref{ivpkp}). If for some $B>0$,
$\mathop{\rm supp}u(t) \subseteq \mathcal{B}:= [-B, B] \times [-B, B]$,
 then for all $\lambda =(\xi, \eta), \sigma =(\theta, \delta) \in \mathbb{R}^2$,
we have
\begin{equation}
|\widehat{u(t)}(\lambda + i \sigma)| \lesssim e^{c|\sigma | B} .
\end{equation}
Where we have used $|(x, y)| = \max \{|x|, |y|\}$.
\end{lemma}

\begin{proof} The proof follows  using the Cauchy-Schwarz inequality and
the conservation law (\ref{con1}) with the argument similar to the one
given in the proof of Lemma 2.1 in \cite{MP1}.
\end{proof}

For $\lambda =(\xi, \eta)$  and $\lambda' = (\xi', \eta')$  define
\begin{gather}
\label{def-u*}
u^{*}(\lambda) = \sup_{t\in I} |\widehat{u(t)}(\lambda)|,\\
\label{def-a}
a(\lambda) = \sup_{|\xi'|\geq |\xi|\atop |\eta'|\geq |\eta|}
 |u^{*}(\lambda')|.
\end{gather}

Considering $\phi$ sufficiently smooth and taking in to account  the
well-posedness theory for the IVP (\ref{ivpkp})
(see for example, \cite{Borg.1}), we have the following result.

\begin{lemma} \label{lem111}
Let $u \in C([-T, T] ; H^s(\mathbb{R}^2))$ be a
sufficiently smooth solution to the IVP (\ref{ivpkp}) with
$\mathop{\rm supp} u(t)\subseteq{\mathcal{B}}$, $t\in I$, then
for some constant $B_1$, we have,
\begin{equation}\label{lex.1}
a(\lambda )\lesssim \frac{B_1}{1+|\lambda |^4}.
\end{equation}
\end{lemma}

\begin{proof}
 The Cauchy-Schwarz inequality and the conservation law (\ref{con1}) yield
\begin{equation}
\label{CS11}
\int_{\mathbb{R}^2}|u(t)(\lambda)|\,d\lambda \leq |\mathcal{B}|^{1/2}
\|u(t)\|_{L^2}\lesssim 1.
\end{equation}
Now, using properties of the Fourier transform and (\ref{CS11}) we get
\begin{equation}
\label{CS12}
|\widehat{u(t)}(\lambda)|\leq \|\widehat{u(t)}\|_{L^{\infty}} \leq c \|u(t)\|_{L^1} \lesssim
 1.
\end{equation}
Therefore,
\begin{equation}
\label{CS123}
\sup_{t} |\widehat{u(t)}(\lambda)|\leq c.
\end{equation}

From the local well-posedness result (see \cite{Borg.1}), we have
\begin{equation}
\label{loc123}
\|D^s u(t)\|_{L_{T}^{\infty}L_{xy}^{2}} \leq c.
\end{equation}
Since,\; $\mathop{\rm supp} u(t)\subseteq{\mathcal{B}}$ and
\begin{equation*}|
\lambda |^s \widehat{u(t)}(\lambda) = \widehat{D^su(t)}(\lambda) =
\frac{1}{2\pi}\int_{\mathbb{R}^2}D^su(t)(x, y) e^{-i(x\xi + y\eta)}\,dxdy,
\end{equation*}
 using the Cauchy-Schwarz inequality and the estimate (\ref{loc123}) we obtain
\begin{equation}
\label{loc111}
|\lambda|^s|\widehat{u(t)}(\lambda)| \leq c \int_{\mathbb{R}^2} |D^su(t)(x,
 y)|\,dx dy\leq c \Big(\int_{\mathbb{R}^2}|D^su(t)(x,
y)|^2 \,dxdy\Big)^{1/2}\leq c_1.
\end{equation}
Therefore,
\begin{equation}
|\widehat{u(t)}(\lambda)|\leq \frac{c_1}{|\lambda|^s}.
\label{loc122}
\end{equation}

If we consider $s = 4$ (which is possible, since we have local
well-posedness for the  IVP (\ref{ivpkp}), for eg., in $H^1$) and combine
(\ref{CS123}) and (\ref{loc122}) we get
\begin{equation}
\sup_{t} |\widehat{u(t)}(\lambda)|\leq \frac{B_1}{1+|\lambda|^4}.
\end{equation}

If $\lambda'$ is such that $|\xi'|\geq |\xi|$ and $|\eta'| \geq
|\eta|$, then $\frac{1}{1+|\lambda|^4}\geq\frac{1}{1+|\lambda'|^4}$.
Hence
\begin{equation*}
a(\lambda) = \sup_{|\xi'|\geq |\xi|\atop |\eta'| \geq
|\eta|}\sup_{t\in I} |\widehat{u(t)}(\lambda')|
\leq \sup_{|\xi'|\geq |\xi|\atop |\eta'| \geq
|\eta|}\frac{B_1}{1+|\lambda'|^4}
\leq \frac{B_1}{1+|\lambda|^4},
\end{equation*}
as required.
\end{proof}

\begin{proposition} \label{prop111}
 Let $u(t)$ be compactly supported and suppose that there exists $t\in
 I$ with $u(t)\neq 0$. Then there exists a number $c>0$ such that
 for any large number $Q>0$ there are arbitrary large
 $|\lambda|$-values such that
\begin{gather}\label{ineq1}
a(\lambda) > c(a*a) (\lambda), \\
\label{ineq2}
a(\lambda) > e^{-\frac{|\lambda|}{Q}}.
\end{gather}
\end{proposition}

\begin{proof}
The main ingredient in the proof is the estimate \eqref{lex.1} in
Lemma \ref{lem111}. The argument is similar to the one given in the proof
of lemma in page 440 in \cite{Borg}, so we omit it.
\end{proof}

Using the definition of $a(\lambda)$ and Proposition \ref{prop111} we
choose  $|\lambda|$ large (with $|\xi|,\,|\eta|$ large) and
$t_1 \in I$ such that
\begin{equation}
\label{ineq333}
|\widehat{u(t_1)}(\lambda)| = u^{*}(\lambda) = a(\lambda) >
 c(a*a)(\lambda) + e^{-\frac{|\lambda|}{Q}}.
\end{equation}

Now we prove some estimates for derivative of an entire
function. Let us begin with the following lemma whose proof is given
in  \cite{Borg}.

\begin{lemma} \label{Bolema}
Let $\phi : \mathbb{C} \to \mathbb{C}$ be an entire function which is bounded and
integrable on the real axis and satisfies
\begin{equation*}
|\phi(\xi + i\theta)| \lesssim e^{|\theta|B}, \quad
 \xi,\,\theta\in \mathbb{R}.
\end{equation*}
Then, for $\lambda_1\in \mathbb{R}^{+}$ we have
\begin{equation}
|\phi'(\lambda_1)|\lesssim B \Big(\sup_{\xi'\geq
 \lambda_1}|\phi(\xi')|\Big)\Big[1+ \big|\log\Big(\sup_{\xi'\geq
 \lambda_1}|\phi(\xi')|\Big)\big|\Big].
\end{equation}

\end{lemma}

In the sequel we use this lemma to obtain some more estimates which
are crucial in the proof of our main result. The details of the proof of
these estimates can be found in \cite{MP1}. For the sake of clarity,
we mention main ingredients and sketch of the proof.

\begin{lemma} \label{lm111}
Let $\Phi : \mathbb{C}^2 \to\mathbb{C} $ be an entire function
satisfying
\begin{equation*}
|\Phi(\lambda +i\sigma)|\lesssim e^{c|\sigma|B}\,\,\,\,\,\,\,\,
 \lambda,\,\sigma \in \mathbb{R}^2,
\end{equation*}
such that for $z_2$ fixed, $\Phi_1(z_1):= \Phi(z_1, z_2)$ and for
$z_1$ fixed, $\Phi_2(z_2):= \Phi(z_1, z_2)$ are bounded and integrable
on the real axis. Then for $\lambda_1,\, \lambda_2 \in \mathbb{R}^{+}$ we have
\begin{equation}
\label{lm123}
|\nabla \Phi(\lambda_1, \lambda_2)| \lesssim B\Big(\sup_{\xi'\geq
 \lambda_1\atop\eta'\geq \lambda_2}|\Phi(\xi', \eta')|\Big)\Big[1+
 \big| \log\Big(\sup_{\xi'\geq \lambda_1\atop\eta'\geq
 \lambda_2}|\Phi(\xi', \eta')|\Big)\big|\Big].
\end{equation}
\end{lemma}

\begin{proof}
 Let us take $\lambda' = (\xi', \eta')$. First we apply  Lemma \ref{Bolema}
 to  $\Phi_1$ for fixed $z_2$ to get
\begin{equation}
\label{llm223}
|\Phi'_{1}(\lambda_1)|\lesssim B\Big(\sup_{\xi'\geq
 \lambda_1}|\Phi_1(\xi')|\Big)\Big[ 1+ \big|\log
 \Big(\sup_{\xi'\geq \lambda_1}|\Phi_1(\xi')|\Big)\big|\Big].
\end{equation}
Next, we apply Lemma \ref{Bolema} to $\Phi_2$ for fixed $z_1$ to obtain
\begin{equation}
\label{llm2233}
|\Phi'_{2}(\lambda_2)|\lesssim B\Big(\sup_{\eta'\geq
 \lambda_2}|\Phi_1(\eta')|\Big)\Big[ 1+ \big|\log
 \Big(\sup_{\eta'\geq \lambda_2}|\Phi_2(\eta')|\Big)\big|\Big].
\end{equation}
Now using \eqref{llm223}, \eqref{llm2233} and the definition
\begin{equation*}
\nabla \Phi(\lambda_1, \lambda_2) :=
(\Phi_1'(\lambda_1),\Phi_2'(\lambda_2)),
\end{equation*}
we obtain the required result.
\end{proof}

\begin{corollary} \label{cor11}
Let $\sigma\in \mathbb{R}^2$ be such that
\begin{equation}
\label{coreq11}
|\sigma|\leq B^{-1}\Big[ 1+ \big|\log \Big(\sup_{\xi'\geq
 \lambda_1>0\atop\eta'\geq\lambda_2>0}|\Phi(\xi',
 \eta')|\Big)\big|\Big]^{-1}.
\end{equation}
Then
\begin{equation}
\label{coreq12}
\sup_{\xi'\geq
\lambda_1\atop\eta'\geq\lambda_2}|\Phi(\lambda'+i\sigma)|\leq 4
\,\sup_{\xi'\geq \lambda_1\atop\eta'\geq\lambda_2}|\Phi(\lambda')|
\end{equation}
and
\begin{equation}
\label{coreq123}
\sup_{\xi'\geq
\lambda_1\atop\eta'\geq\lambda_2}|\nabla\Phi(\lambda'+i\sigma)|\lesssim
B\,\Big(\sup_{\xi'\geq
\lambda_1\atop\eta'\geq\lambda_2}|\Phi(\lambda')|\Big)\Big[1+
\big|\log \Big(\sup_{\xi'\geq
\lambda_1\atop\eta'\geq\lambda_2}|\Phi(\lambda')|\Big)\big|\Big].
\end{equation}
\end{corollary}

\begin{proof}
To prove (\ref{coreq12}), first fix $\eta'\geq \lambda_2$ and use Corollary 2.9 in
\cite{Borg} and then fix $\xi'\geq \lambda_1$ and use the same corollary to get
\begin{align*}
\sup_{\xi'\geq \lambda_1\atop\eta'\geq\lambda_2}|\Phi(\xi'+i\theta,
\eta'+i\delta)| & \leq  \sup_{\eta'\geq\lambda_2}\Big(2
\,\sup_{\xi'\geq\lambda_1}|\Phi(\xi', \eta'+i\delta)|\Big) \\
& \leq  4\,\sup_{\xi'\geq
\lambda_1\atop\eta'\geq\lambda_2}|\Phi(\lambda')|.
\end{align*}
To prove (\ref{coreq123}), we  define $\tilde{\Phi}(z) =
\Phi(z+i\sigma)$. Then $\tilde{\Phi}$ is an entire function and
 satisfies the hypothesis of Lemma \ref{lm111}. So, by the same Lemma we get
\begin{equation*}
|\nabla\tilde{\Phi}(\lambda_1, \lambda_2)|\lesssim B\,
 \Big(\sup_{\xi'\geq \lambda_1\atop\eta'\geq
 \lambda_2}|\tilde{\Phi}(\xi', \eta')|\Big)\Big[1+ \big|
 \log\Big(\sup_{\xi'\geq \lambda_1\atop\eta'\geq
 \lambda_2}|\tilde{\Phi}(\xi', \eta')|\Big)\big|\Big].
\end{equation*}
 for any $\lambda_1,\lambda_2 \in \mathbb{R}^{+}$.

Now    the definition of $\tilde{\Phi}$ and use of  (\ref{coreq12}) imply
\begin{equation}
\begin{split}
&|\nabla\Phi(\lambda_1 + i\theta, \lambda_2 + i\delta)|\\
 & \lesssim
B\, \Big(\sup_{\xi'\geq \lambda_1\atop\eta'\geq
\lambda_2}|\Phi(\xi'+i\theta, \eta'+i\delta)|\Big)\Big[1+ \big|
\log\Big(\sup_{\xi'\geq \lambda_1\atop\eta'\geq
\lambda_2}|\Phi(\xi'+i\theta, \eta'+i\delta)|\Big)\big|\Big]\\
& \lesssim  4B\, \Big(\sup_{\xi'\geq
\lambda_1\atop\eta'\geq \lambda_2}|\Phi(\xi', \eta')|\Big)\Big[1+
\big| \log\Big(4 \sup_{\xi'\geq \lambda_1\atop\eta'\geq
\lambda_2}|\Phi(\xi', \eta')|\Big)\big|\Big]\\
& \lesssim
 B\,\Big(\sup_{\xi'\geq \lambda_1\atop\eta'\geq
\lambda_2}|\Phi(\lambda')|\Big)\Big[(1+ \log 4)+(1 + \log 4) \big|
\log\Big(\sup_{\xi'\geq \lambda_1\atop\eta'\geq
\lambda_2}|\Phi(\lambda')|\Big)\big|\Big]\\
& \lesssim
 B\,\Big(\sup_{\xi'\geq \lambda_1\atop\eta'\geq
\lambda_2}|\Phi(\lambda')|\Big)\Big[1 + \big| \log\Big(\sup_{\xi'\geq
\lambda_1\atop\eta'\geq \lambda_2}|\Phi(\lambda')|\Big)\big|\Big].
\end{split}
\end{equation}
Which yields the desired estimate (\ref{coreq123}).
\end{proof}

\begin{corollary} \label{cor22}
Let $t\in I$, $\Phi(z) = \widehat{u(t)}(z)$, $\sigma$ be as in Corollary
\ref{cor11} and $a(\lambda)$ be  as in (\ref{def-a}). Then, for
$|\sigma'|\leq |\sigma|$ fixed, we have
\begin{equation}
\label{coreq221}
|\nabla\Phi(\lambda-\lambda'+i\sigma')|\lesssim B\, \big[a(\lambda) +
 a(\lambda-\lambda')\big]\big[1 + | \log a(\lambda)|\big].
\end{equation}
\end{corollary}

\begin{proof}
Define  $\tilde{\Phi}(z) := \Phi(z + i \sigma')$,
$z=(z_1, z_2) =(\xi+i\theta, \eta+i\delta)$.
First, we use  (\ref{coreq123}) with
$\sigma = 0$ and then use (\ref{coreq12}) to get the required result.
The details of the proof can be found in \cite{MP1}.
\end{proof}

\section{Proof of the main result}\label{sec3}

This section is devoted to supply proof of Theorem \ref{ucpkp}.
Although the scheme of the proof is analogous to the one we
employed to get similar result for the ZK equation in \cite{MP1},
one needs to overcome some additional technical difficulties
arising from the structure of the Fourier symbol associated to the
linear KP-II equation.

\begin{proof}[Proof of Theorem \ref{ucpkp}]
If possible, suppose $u(t) \neq 0$ for some $t \in I$. Our aim is
to  get a contradiction by using the estimates derived in the
previous section.

Let $t_1, t_2\in I$, with $t_1$ as in \eqref{ineq333}.  Using
Duhamel's principle, we have
\begin{equation}
u(t_2) = S(t_2-t_1) u(t_1) - \frac{1}{2}\int_{t_1}^{t_2}
S(t_2-t')(u^2)_x(t')\,dt'.
\end{equation}

Taking Fourier transform in the space variables with $\lambda = (\xi,
\eta)$ we get
\begin{equation}
\widehat{u(t_2)}(\lambda) =
e^{i(t_2-t_1)(\xi^3-\frac{\eta^2}{\xi})}\widehat{u(t_1)}(\lambda)
-\frac{i\xi}{2}\int_{t_1}^{t_2}e^{i(t_2-t')(\xi^3-\frac{\eta^2}{\xi})}
\widehat{u^2(t')}(\lambda)\,dt'.
\end{equation}
Let $t_2-t_1 = \Delta t$ and then make a change of variables $s =
t'-t_1$ to get
\begin{equation}
\widehat{u(t_2)}(\lambda) = e^{i\Delta t
(\xi^3-\frac{\eta^2}{\xi})}\Big[\widehat{u(t_1)}(\lambda)-\frac{i\xi}{2}\int_{0}^{\Delta
t}e^{-is(\xi^3-\frac{\eta^2}{\xi})}\widehat{u^2(s+t_1)}(\lambda)\,ds\Big].
\end{equation}
Since $u(t), t\in I$ has compact support, by Paley-Wiener theorem,
$\widehat{u(t)}(\lambda)$ has analytic continuation in
$\mathbb{C}^2$, and we have for $\sigma = (\theta, \delta)$
\begin{equation}\label{e.s.1}
\begin{split}
\widehat{u(t_2)}(\lambda + i\sigma)
& = e^{i\Delta
t\{(\xi+i\theta)^3-\frac{(\eta+i\delta)^2}{\xi+i\theta}\}}\Big[\widehat{u(t_1)}(\lambda+i\sigma)\\
&\quad - \frac{i(\xi+i\theta)}{2}\int_{0}^{\Delta
t}e^{-is\{(\xi+i\theta)^3-\frac{(\eta+i\delta)^2}{\xi+i\theta}\}}\widehat{u^2(s+t_1)}(\lambda+i\sigma)\,ds\Big].
\end{split}
\end{equation}
Since
\begin{align*}
(\xi+i\theta)^3-\frac{(\eta+i\delta)^2}{\xi+i\theta} & = \xi^3
-3\xi\theta^2
-\frac{1}{\xi^2+\theta^2}(\xi\eta^2-\xi\delta^2+2\eta\theta\delta)\\
&\quad + i\{3\xi^2\theta -\theta^3 -\frac{1}{\xi^2
+\theta^2}(2\xi\eta\delta-\theta\eta^2+\theta\delta^2)\},
\end{align*}
using Lemma \ref{lmm1} we get from \eqref{e.s.1}
\begin{equation}\label{eneq1kp1}
\begin{split}
&c e^{\Delta t\{3\xi^2\theta -\theta^3-\frac{2\xi\eta\delta
 -\theta(\eta^2 -\delta^2)}{\xi^2 +\theta^2}\}} \\
& \geq
 \big|\widehat{u(t_1)}(\lambda+i\sigma)\big|
 -\frac{|\xi+i\theta|}{2}\int_{0}^{\Delta t} e^{s\{3\xi^2\theta
 -\theta^3-\frac{2\xi\eta\delta -\theta(\eta^2 -\delta^2)}{\xi^2
 +\theta^2}\}}\big|\widehat{u^2(s+t_1)}(\lambda+i\sigma)\big|\,ds.
 \end{split}
\end{equation}
Let us take $|\lambda| = \max \{|\xi|, |\eta|\}$ very large such that
\begin{equation}\label{sup112}
\xi\eta > 0\quad\quad {\rm{and}}\quad\quad |\xi |\sim |\eta |.
\end{equation}
Also, let us choose $\sigma = \sigma(\lambda)$ with
$|\sigma| = \max \{|\theta|, |\delta|\}\approx 0$ with
\begin{equation}\label{sup122}
\theta \Delta t < 0 \quad\quad{\rm{and}}\quad\quad \delta \Delta t >0.
\end{equation}
Moreover, let us suppose the following conditions are satisfied
\begin{equation}
\label{sup1232}
\frac{1}{|\xi|}\ll |\theta|,\,|\delta| \quad{\rm and}\quad\frac{1}{|\eta|}\ll
|\theta|,\,|\delta|.
\end{equation}
With these choices, (\ref{eneq1kp1}) can be written as
\begin{equation}
\label{eneq2akp2}
\begin{split}
 e^{\Delta t\{3\xi^2\theta-\frac{2\xi\eta\delta -\theta\eta^2}{\xi^2
 +\theta^2}\}} & \gtrsim \big|\widehat{u(t_1)}(\lambda+i\sigma)\big|\\
 &  {} - |\xi|\int_{0}^{\Delta t} e^{s\{3\xi^2\theta
 -\frac{2\xi\eta\delta -\theta\eta^2}{\xi^2 +\theta^2}
 \}}\big|\widehat{u^2(s+t_1)}(\lambda+i\sigma)\big|\,ds.
 \end{split}
\end{equation}
Now, taking into consideration of (\ref{sup112}) and
(\ref{sup122}), the estimate (\ref{eneq2akp2}) yields
\begin{equation}\label{eq1akpa1}
\begin{split}
&e^{-|\Delta t|\{3\xi^2|\theta|+\frac{2|\xi\eta\delta
 |+|\theta|\eta^2}{\xi^2 +\theta^2}\}} \\
 &\gtrsim
 \big|\widehat{u(t_1)}(\lambda+i\sigma)\big|
 - |\xi|\int_{0}^{|\Delta t|} e^{-s\{3\xi^2|\theta |+\frac{2|\xi\eta\delta| +|\theta|\eta^2}{\xi^2 +\theta^2}
 \}}\big|\widehat{u^2(t_1\pm s)}(\lambda+i\sigma)\big|\,ds.
 \end{split}
\end{equation}
Where ``$+$'' sign corresponds to $\Delta t > 0$ and $'-'$ sign
corresponds to $\Delta t < 0$. In what follows we consider the case
$\Delta t >0$ ( the case $\Delta t < 0$ is similar). Since $e^{-x}<1$
for $x>0$, (\ref{eq1akpa1}) can be written as
% \label{eq1akpb1}
\begin{align*}
 &e^{-\{3\xi^2+\frac{\eta^2}{\xi^2+\theta^2}\}|\theta\Delta t|}\\
 & \gtrsim
  \big|\widehat{u(t_1)}(\lambda+i\sigma)\big| - |\xi|\int_{0}^{|\Delta
  t|} e^{-s\{3\xi^2|\theta |+\frac{2|\xi\eta\delta|
  +|\theta|\eta^2}{\xi^2 +\theta^2} \}}\big|\widehat{u^2(t_1+
  s)}(\lambda+i\sigma)\big|\,ds.
\end{align*}
Finally, this last estimate can be written as
\begin{equation}\label{eqkpI123}
\begin{split}
&e^{-\{3\xi^2+\frac{\eta^2}{\xi^2+\theta^2}\}|\theta\Delta t|}\\
 & \gtrsim
  \big|\widehat{u(t_1)}(\lambda)\big| - |\xi|\int_{0}^{|\Delta t|}
 e^{-s\{3\xi^2|\theta |+\frac{2|\xi\eta\delta| +|\theta|\eta^2}{\xi^2
 +\theta^2} \}}\big|\widehat{u^2(t_1+ s)}(\lambda)\big|\,ds\\
 &  {}\, -\big|\widehat{u(t_1)}(\lambda+i\sigma)
 -\widehat{u(t_1)}(\lambda)\big| \\
 &  {}\,-|\xi|\int_{0}^{|\Delta t|} e^{-s\{3\xi^2|\theta
 |+\frac{2|\xi\eta\delta| +|\theta|\eta^2}{\xi^2 +\theta^2}
 \}}\big|\widehat{u^2(t_1+ s)}(\lambda+i\sigma) - \widehat{u^2(t_1+
 s)}(\lambda)\big|\,ds\\
 & :=  I_1 - I_2 - I_3.
 \end{split}
\end{equation}
Now, we proceed to obtain appropriate estimates for $I_1$, $I_2$ and
$I_3$ to arrive at  a  contradiction in \eqref{eqkpI123}.
To obtain estimate for  $I_1$, we use definition of $u^{*}(\lambda)$, i.e.,
(\ref{def-u*}) and the estimate (\ref{ineq1}) to get
\begin{align*}
& |\xi|\int_{0}^{|\Delta t|} e^{-s\{3\xi^2|\theta
|+\frac{2|\xi\eta\delta| +|\theta|\eta^2}{\xi^2+\theta^2}\}}\big|\widehat{u(t_1+
s)}\big|*\big|\widehat{u(t_1+ s)}\big|(\lambda)\,ds \\
&\leq
 |\xi|(u^{*}*u^{*})(\lambda)\int_{0}^{|\Delta t|} e^{-s\{3\xi^2|\theta
|+\frac{2|\xi\eta\delta| +|\theta|\eta^2}{\xi^2+\theta^2}\}}ds \\
 & \leq
|\xi|(a*a)(\lambda)\frac{1- e^{-|\Delta t|\{3\xi^2|\theta
|+\frac{2|\xi\eta\delta| +|\theta|\eta^2}{\xi^2+\theta^2} \}}}{3\xi^2|\theta
|+\frac{2|\xi\eta\delta| +|\theta|\eta^2}{\xi^2+\theta^2} } \\
& \leq  \frac{|\xi |
(a*a)(\lambda)}{3|\xi||\xi\theta|}
\lesssim
\frac{a(\lambda)}{3|\xi\theta|}.
\end{align*}
Therefore,
\begin{equation}
\label{estI1}
I_1 \gtrsim a(\lambda) - \frac{a(\lambda)}{3|\xi\theta|} \geq
 \frac{a(\lambda)}{3}.
\end{equation}
To get estimate for $I_2$, let us define $\Phi(z) = \widehat{u(t_1)}(z)$,
  $z = ( z_1, z_2)\in \mathbb{C}^2$. Now, using (\ref{ineq333}) one can obtain
\begin{equation}
\label{I21}
|\Phi(\lambda)| = |\widehat{u(t_1)}(\lambda)| = \sup_{|\xi'|\geq
 |\xi|\atop |\eta'|\geq|\eta|}|\Phi(\lambda')|= a(\lambda).
\end{equation}
Let us choose $|\sigma|$ satisfying \begin{equation}
\label{sigma1}
|\sigma| \lesssim B^{-1} \big[1+ |\log a(\lambda)|\big]^{-1},
\end{equation}
 and use Corollary \ref{cor11} to obtain
\begin{equation}\label{estI2}
I_2  \lesssim  |\sigma |
\sup_{|\xi'|\geq |\xi|\atop
|\eta'|\geq|\eta|}|\nabla\widehat{u(t_1)}(\lambda'+i\sigma)|
\lesssim  |\sigma | B\,a(\lambda) \big[1 + |\log a(\lambda)|\big]
\lesssim  a(\lambda)
\lesssim  \frac{1}{15} a(\lambda).
\end{equation}
To obtain  estimate for $I_3$, we
use Proposition \ref{prop111}, Corollary \ref{cor22} and
$|\sigma|$ as in (\ref{sigma1}) to get
\begin{equation*}
\begin{split}
&\big|\widehat{u^2(t_1+s)}(\lambda+i\sigma)-\widehat{u^2(t_1+s)}(\lambda)\big|\\
&\leq
\int_{\mathbb{R}^2}\big|\widehat{u(t_1+s)}(\lambda-\lambda'+i\sigma)
-\widehat{u(t_1+s)}(\lambda-\lambda')\big|
\big|\widehat{u(t_1+s)}(\lambda')\big|\,d\lambda'\\
&\leq
|\sigma|\int_{\mathbb{R}^2}\sup_{|\sigma'|\leq|\sigma|}|\nabla\widehat{u(t_1+s)}
(\lambda-\lambda'+i\sigma')\big|\,a(\lambda')\,d\lambda'\\
& \leq  \int_{\mathbb{R}^2}[a(\lambda)
+ a(\lambda-\lambda')]\, a(\lambda')\,d\lambda'\\
&\leq
a(\lambda)c_2 + (a*a)(\lambda) \\
& \leq
a(\lambda)(c_2+c^{-1})\lesssim a(\lambda).
\end{split}
\end{equation*}
Therefore,
\begin{equation} \label{estI3}
\begin{aligned}
I_3 & \lesssim  |\xi| a(\lambda)\int_{0}^{|\Delta t|}
e^{-s\{3\xi^2|\theta |+\frac{2|\xi\eta\delta| +|\theta|\eta^2}{\xi^2+\theta^2}\}}ds
\\
& =  |\xi| a(\lambda)\; \frac{1-e^{-|\Delta t
|\{3\xi^2|\theta |+\frac{2|\xi\eta\delta| +|\theta|\eta^2}{\xi^2+\theta^2} \}}}{3\xi^2|\theta
|+\frac{2|\xi\eta\delta| +|\theta|\eta^2}{\xi^2+\theta^2} }\\
& \leq
\frac{|\xi|a(\lambda)}{3|\xi^2\theta|}\\
&\lesssim
\frac{a(\lambda)}{|\xi\theta|}   < \frac{a(\lambda)}{15}.
\end{aligned}
\end{equation}
Now inserting (\ref{estI1}), (\ref{estI2}) and (\ref{estI3}) in
(\ref{eqkpI123}) and using the estimate \eqref{ineq2} we get
\begin{equation}
\label{estbf}
e^{-\{3\xi^2 + \frac{\eta^2}{\xi^2+\theta^2}\}|\theta\Delta t|} \gtrsim
\frac{a(\lambda)}{3} -\frac{a(\lambda)}{15}-\frac{a(\lambda)}{15}
 =  \frac{1}{3} a(\lambda)   \gtrsim
e^{-\frac{|\lambda|}{Q}}.
\end{equation}
On the other hand, using (\ref{sup112}) and (\ref{sup1232}) one can
easily deduce
\begin{equation}
\label{estfkp1}
e^{-\{3\xi^2+\frac{\eta^2}{\xi^2+\theta^2}\}|\theta||\Delta t|}\leq
e^{-|\lambda||\Delta t|}.
\end{equation}
Hence, using (\ref{estfkp1}) in (\ref{estbf}), we arrive at
$$ e^{-|\lambda||\Delta t|} \gtrsim e^{-\frac{|\lambda|}{Q}},
$$
which is a contradiction for $|\lambda|$ large, if we choose $Q $
large enough such that $\frac{1}{Q}<|\Delta t|$.
\end{proof}

\begin{remark}\label{rem.f} \rm
Note that the Fourier symbol associated with the linear KP-I
operator is $\xi^3+\frac{\eta^2}{\xi}$. In this case we cannot make
choice as in \eqref{sup112} and \eqref{sup122} to obtain estimate like
in \eqref{eq1akpa1} with term of the form $e^{-|\Delta t| \gamma}$,
for some $\gamma > 0$. As seen in the proof of Theorem \ref{ucpkp},
existence of such term in the RHS of  \eqref{eq1akpa1} is very essential
in the argument we employed. It would be interesting to obtain UCP for
the KP-I equation employing some other argument or modification.
\end{remark}

\subsection*{Acknowledgements}
The author is grateful to F. Linares, M. Scialom and X. Carvajal
for their valuable conversations and support  during his
stay at UNICAMP, Campinas, where the first version of this article
was prepared.


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