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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 157, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/157\hfil Asymptotic behavior]
{Asymptotic behavior for Dirichlet problems of nonlinear Schr\"odinger equations
 with Landau damping on a half line}

\author[L. Esquivel \hfil EJDE-2017/157\hfilneg]
{Liliana Esquivel}

\address{Liliana Esquivel \newline
Universidad de Pamplona, Santander, Colombia}
\email{lesquivel@unipamplona.edu.co}


\thanks{Submitted  March 6, 2017. Published June 28, 2017.}
\subjclass[2010]{35Q55, 35B40}
\keywords{Fractional Schr\"odinger equation; boundary value; Landau damping}

\begin{abstract}
 This article is a continuation of the study in \cite{liliana},
 where we proved the existence of solutions, global in time,
 for the initial-boundary value problem
  \begin{gather*}
 u_{t}+iu_{xx}+i|u|^{2}u+|\partial _x|^{1/2}u=0,\quad t\geq 0,\;
 x\geq 0; \\
 u(x,0)=u_{0}(x),\quad x>0 \\
 u_x(0,t)=h(t),\quad t>0,
 \end{gather*}
 where $|\partial _x|^{1/2}$ is the module-fractional derivative
 operator defined by the modified Riesz Potential
 \begin{equation*}
 |\partial _x|^{1/2}=\frac{1}{\sqrt{2\pi }}\int_{0}^{\infty }
 \frac{\operatorname{sign}(x-y) }{\sqrt{|x-y|}}u_y(y)dy.
 \end{equation*}
 Here, we study the asymptotic behavior of the solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

Consider the initial-boundary value problem for a modified Schr\"odinger
equation with Landau damping on a half-line
\begin{equation} \label{nolineal}
\begin{gathered}
 u_{t}+\mathbb Ku+i|u|^{2}u=0,\quad t\geq0,\; x>0;\\
 u(x,0)=u_0(x), \quad x>0\\
 u(0,t)=h(t), \quad t>0,
\end{gathered}
\end{equation}
where the operator $\mathbb K$ is defined as
\begin{equation}
\mathbb K=\alpha u_{xx}+\lambda|\partial_x|^{\gamma}u, \label{operador K}
\end{equation}
with $\alpha, \lambda \in \mathbb C$, $\gamma \in \mathbb R$ and
$|\partial_x|^{\gamma}$ is the module-fractional derivative operator given by
$|\partial _x|^{\gamma}u=R^{\gamma}\partial _xu$.
Here $R^{\gamma}$ is the modified Riesz Potential
\begin{equation*}
R^{\gamma}u=\frac{1}{2\Gamma(\gamma) \sin ( \frac{
\pi }{2}\gamma) }\int_{0}^{\infty }
\frac{\operatorname{sign}(x-y) }{|x-y|^{ 1-\gamma}}u(y)dy.
\end{equation*}
In \cite{liliana}  we prove the existence solutions, global in time, to this
initial-boundary value problem (IBV problem),
as a continuation of this study in the present paper, we show the asymptotic
expansion for the solutions to
\eqref{nolineal}.  More precisely, the principal result in \cite{liliana}
is the following.

\begin{theorem}  \label{teorema de existencia}
 Suppose that $u\in  L^{1,\mu }(\mathbb{R}^{+})
\cap  L^{\infty }( \mathbb{R}^{+}) $ and
 $h\in  Y_{\beta }= H^{1,\beta}_{\infty}$ with
$\| u_{0}\| _{ Z}+\|h\| _{ Y_{\beta }}\leq \epsilon $,
where $\epsilon >0$ is sufficiently small and $\beta>1$. Then there exist
a unique global solution
\begin{equation*}
u\in  C( t[ 0,\infty ) ; L^{2}( \mathbb{R}
^{+}) ) \cap  C\Big( ( 0,\infty ) ; L^{^{2,
\frac{1}{2}( \frac{1}{2}+\mu ) }}( \mathbb{R}^{+})
\cap  L^{\infty }( \mathbb{R}^{+}) \Big) ,
\end{equation*}
with $\mu \in ( 0,1/2) $ to the initial-boundary value
problem $( \ref{nolineal}) $.
\end{theorem}

 Here $ L^{p,\mu}$ denote the function space
$ L^{p,\mu}:=\{\phi \in \mathbf{S}':\| \phi \| _{ L^{p,\mu}}<\infty \}$,
with the norm
$$
\| \phi \| _{ L^{p,\mu}}
= \Big(\int_{\mathbb{R^+}}(1+|x|)^{\mu p}|\phi (x)|^{p}dx\Big) ^{1/p}
$$
for $1\leq p<\infty $ and $\|\phi \| _{ L^{\infty, \mu }}
=\sup_{x\in \mathbb{R}^{+}}|(1+|x|)^{\mu}\phi (x)|$ for $p=\infty $.
We also use the notation $ L^{p}= L^{p,0}$.

The theory of asymptotic methods for nonlinear evolution equations is
relatively young and traditional questions of general theory
are far from being answered. A description of the large time asymptotic behavior
of solutions of nonlinear evolution equations requires principally new approaches
and the reorientation of points of view in the asymptotic methods.

The difficulty of the asymptotic methods is explained by the fact that they
need not only a global existence of solutions,
but also a number of additional a priori estimates of the difference between
the solution an the approximate solution (usually in the weighted norms).
Some key developments can be found in the book \cite{libroNHKS}, which is the
first attempt to give a systematic approach for
obtaining the large asymptotic representation of solutions to the nonlinear
 evolution equation with dissipation.


Some previous results concerning the nonlinear Schr\"odinger equation (NLS)
$\alpha _{2}=0$, which is the most closely related to our problem,
include \cite{potencial1},  \cite{potencial2} and  \cite{ozawa}.
In \cite{Ionescu} it was shown that \eqref{nolineal}
with $\alpha _{1}=0,\alpha _{2}=i$ admits global solutions whose long-time
behavior is not linear. For IBV-problems for the nonlinear Schrodinger equation,
there are  fewer amount of literature, in papers \cite{Bu} and
\cite{Holmer} with inhomogeneous Dirichlet boundary conditions there were
certain results.
Local existence in some Sobolev spaces. Weder \cite{Weder} proved that
the Dirichlet IBV-problem for the forced nonlinear Schr\"odinger equation
with a potential on the half-line, is locally and (under stronger conditions)
globally well posed.
Bu and Strauss \cite{BuStrauss} proved the existence of
global-in-time solution in the energy space for initial data in $ H^{1}$ and
the boundary data from $\mathbf{C}^3$ with a compact support.

Fokas\cite{Fokas2002}, assuming that a solution of the nonlinear Schr\"odinger
equation on the half-line exists,
showed that the solution can be represented in terms of the
solution of a matrix Riemann Hilbert, and in \cite{Fokas2005} the authors
prove that given appropriate initial and boundary conditions, the solution
of the nonlinear Schr\"odinger equation exists globally.
However, in spite of the importance, few works have considered the
IBV-problems for partial differential evolution equations with a fractional
derivative. Some key developments include the book \cite{bookhk}. This book
is the first attempt to develop systematically a general theory of
IBV-problems for evolution equations with pseudo-differential operators on a
half-line.The results of this book can be applied directly to study the
initial-boundary value problem for differential equations with fractional
Riemann-Liouville and Caputo derivatives.

A method for solving IBV-problems for linear partial differential evolution
equations with a general fractional derivative operator,
based on the Riemann-Hilbert theory, was
introduced in \cite{Ekaikina} and further developed in \cite{Fractal1}. It
was proved in  \cite{liliana,LK} that the above approach can be used
to the establish global existence in time of the solutions of \eqref{nolineal}
with Neumann and Dirichlet boundary data.


In this article, we use the factorization technique from paper \cite{LK}
for the free Schr\"odinger evolution group
\begin{equation}\label{e}
 \mathcal{G}(t)=\mathcal{B}_{s}\{e^{K(z)t}B_{s}\}, \quad K(z)=iz^{2}-\sqrt{z}.
\end{equation}
   Formula \eqref{e} is useful for studying the large time asymptotic behavior
of solutions of Fractional Schr\"odinger equations.
 The distorted operators
$\mathcal {B}_{s}^{\ast}$ and $\mathcal{B}_{s}$ will be defined in the following
section. Formula  \eqref{e} is obtained by
using the Hilbert transform with respect to the space variable and by the use of
techniques of complex analysis.
Our main goal is to evaluate the influence of the boundary data
on the asymptotic behavior of solutions. Theorem \ref{teorema de existencia}
shows that \eqref{nolineal} admits global
solutions and Theorem \ref{teoremaprincipal} shows that its long-time behavior
 essentially depends on the scattering properties
of the boundary data.

We believe that the results of this paper could be applied to study a wide class
 of dissipative nonlinear equations with a fractional derivative on a half-line.

\section{Preliminaries}

\subsection{Notation and main results}

 To state our results precisely, we introduce notation and function
spaces. We denote the usual Fourier transform  and inverse Fourier transform
by $\mathcal{F}$ and $\mathcal{F}^{-1}$ respectively.
The  Fourier sine transform $\mathcal{F}_{s}$ and the
Fourier cosine transform $\mathcal{F}_{c}$ are defined by
\begin{equation*}
\mathcal{F}_{s}\phi =\sqrt{\frac{2}{\pi }}\int_{R^{+}}\phi (x)\sin px\,dx,\quad
\mathcal{F}_{c}\phi =\sqrt{\frac{2}{\pi }}\int_{R^{+}}\phi (x)\cos px\,dx.
\end{equation*}
The usual direct and inverse Laplace transformation we denote by $\mathcal{L}$
and $\mathcal{L}^{-1}$
\begin{equation*}
\mathcal{L}\phi =\widehat{\phi }( \xi )
=\int_{\mathbf{0}}^{\infty }e^{-xp}\phi ( x) dx, \mathcal{L}^{-}\phi
=\frac{1}{2\pi i}\int_{i\mathbb{R}}e^{ix\xi }\widehat{\phi }( \xi ) d\xi .
\end{equation*}

For a complex value function $\phi $, which satisfies the H\"older
condition on the imaginary axis, we define sectionally analytic function
$\Phi (z)$ via the Cauchy type integral
\begin{equation*}
\Phi (z)=\frac{1}{2\pi i}\int_{i\mathbb{R}}\frac{\phi (q)}{q-z}dq,\quad
\operatorname{Re }z\neq 0.
\end{equation*}
We note that $\Phi (z)$ constitutes a function analytic in the left and
right semi-planes. Here and below these functions will be denoted
$\Phi^{+}(z)$ and $\Phi ^{-}(z)$ respectively. These functions can be defined for
all points of the imaginary axis $\operatorname{Re}p=0$ via their limiting
values $\Phi ^{+}(p)$ and $\Phi ^{-}(p)$, which are obtained on approaching
to contour from the left and from the right, respectively. First, we define
the sectionally analytic function
\begin{equation}
\mathcal{E}_{w}(x)=\frac{1}{2\pi i}\int_{i\mathbb{R}}\frac{
e^{-qx}e^{-\Gamma (q,K(z))}}{q-w}dq,
 \label{funcion e}
\end{equation}
for $\operatorname{Re}w\neq 0$, $K(z)=iz^{2}-\sqrt{z}$, $z\geq 0$,
where
\begin{equation}  \label{Gamma}
\begin{gathered}
\Gamma (w,\xi ) =\frac{1}{2\pi i}
\int_{0}^{\infty }\ln (q-w)\Big( \frac{K^{+\prime }(q)}{K^{+}(q)+\xi }-
\frac{K^{-\prime }(q)}{K^{-}(q)+\xi }\Big) dq,   \\
K^{\pm }(q) =iq^{2}+\sqrt{\mp iq}.
\end{gathered}
\end{equation}
We make a cut along to negative axis $w<0$. Denote by
\begin{gather*}
\Gamma ^{+}(s,\xi ) =\lim_{w\to s,
\operatorname{Im}w>0}\Gamma (w,\xi ),s>0 \\
\Gamma ^{-}(s,\xi )  =\lim_{w\to s,\operatorname{Im}w<0}\Gamma (w,\xi ),s>0.
\end{gather*}
Define the ``distorted" Fourier sine transform $\mathcal{B}_{s}$ and the
inverse ``distorted" Fourier sine transform $\mathcal{B}_{s}^{\ast }$  as
follows
\begin{equation}
\widehat{\phi }( p) =\mathcal{B}_{s}\phi =\int_{0}^{\infty }\psi_{s}( z,x)
\phi ( x) dx,\quad
\phi (x) =\mathcal{B}_{s}^{\ast }\widehat{\phi }=\frac{1}{2\pi}
\int_{0}^{\infty }\psi_{s}^{\ast }( z,x) \widehat{\phi }(z) dz,  \label{costranform}
\end{equation}
where
\begin{gather}
\psi _{s}(z,x)=\mathcal{E}_{iz}^{-}(x)-\mathcal{E}_{-iz}^{-}(x), \label{seno} \\
\psi _{s}^{\ast }(z,x)=e^{izx}e^{\Gamma (iz,K(z))}-e^{-izx}e^{\Gamma
(-iz,K(z))}+K'(z)\Theta(x,z),\label{senoestrella} \\
\Theta(x,z)= \frac{1}{2\pi }\int_{0}^{\infty }e^{-px}\psi(p,z)dp,
\label{auxiliar} \\
\psi(p,z)=\frac{\sqrt{2}}{2}\frac{e^{\Gamma (-p,K(z))}}
{(ip^{2}+(ip)^{1/2}+K(z))(ip^{2}+(-ip)^{1/2}+K(z))}.\label{lambda}
\end{gather}
For a detailed study of properties of $\mathcal{B}_{s}\phi $ and
$\mathcal{B}_{s}^{\ast }\widehat{\phi }$ see below in Lemmas
\ref{Estimativo en L2}. We introduce the Green operator on a half-line as
\begin{equation}
\mathcal{G}( t) =\mathcal{B}_{s}^{\ast }\{e^{tK(z)}\mathcal{B}_{s}\},
\label{Green}
\end{equation}
Moreover, denoting
\begin{equation}
 \mathring \psi_{s}(z,x)=e^{izx}e^{\Gamma (iz,K(z))}
-e^{-izx}e^{\Gamma(-iz,K(z))}+\frac{z}{K(z)}(\frac{5}{2}\sqrt{|z|}-2z^2)\Theta(x,z),
\end{equation}
we introduce the operator
\begin{equation}
\mathring{\mathcal{B}_{s}}\phi
=2i\int_{0}^{\infty }\mathring \psi_{s}( x,p) \phi ( x) dx,  \label{puntotransform}
\end{equation}
and the Boundary operator on a half- line
\begin{equation}
 \mathcal{H}(t)\phi =\mathring{\mathcal{B}_{s}}
\Big\{\frac{K(z)}{z}\int_0^{t}e^{K(z)(t-\tau)}h(\tau)d\tau\Big\}.
\label{GreenH}
\end{equation}
For a H\"older continuous function $\phi$ on the imaginary axis,
we define the operator
\begin{equation}\label{def:W}
\mathbb{J}\{\phi\}(z) =-\frac{1}{\pi }\int_{0}^{\infty }
\frac{\phi(p)}{p^{2}+z^{2}}( e^{-\Gamma^{+}(p,K(z))}
-e^{-\Gamma ^{-}(p,K(z))}) dp,
\end{equation}
To state the results of the present paper we give some
notations. We denote $\langle t\rangle =1+t$, $\{ t\} =\frac{t}{\langle t\rangle }$.
Moreover, we introduce  the functional $\mathcal{S}$ on $ L^{1,1}(\mathbb R)$ as
$$
\mathcal{S} u_0=\int_0^{\infty}f(y)u_0(y)dy,
$$
with $f(y)=y+\mathbb{J}\{e^{-py}-1\}|_{z=0}$.
The weighted Sobolev space is
\begin{equation*}
 H_{p}^{k,s}=\big\{ \phi \in S:\| \phi \| _{H_{p}^{k,s}}
=\| \langle x\rangle ^{s}\langle i\partial _x\rangle ^{k}\phi \| _{ L^{p}}\big\} ,
\end{equation*}
$k,s\in \mathbb{R}$, $1\leq p\leq \infty $.
We also use the notation $ H^{k,s}= H_{2}^{k,s}$ and
$ H^{k}= H_{2}^{k,0}$.

Different constants might be denoted by the same letter $C$.
For simplicity we put $\alpha _{1}=1,\alpha_{2}=1,\alpha _{3}=1$. Denote by
\begin{equation*}
\theta (s)=\begin{cases}
1,& s>1, \\
0,& s\leq 1
\end{cases}
\end{equation*}

Our main results read as follows.

\begin{theorem} \label{teoremaprincipal}
Let $u_{0}\in Z=H^{1}(\mathbb{R}^{+})\cap  H_{1}^{0,1}(\mathbb{R}^{+})$,
$h\in  Y=H_{\infty }^{1,\beta }$, $\beta >1/2$, be such that
$\|u_{0}\| _{ Z}+\| h\| _{ Y}\leq \epsilon $, where $\epsilon >0$ is sufficiently
small, and the compatibility condition $u_0(0)=h(0)$ is fulfilled. Then there
exists a unique global solution
\begin{equation*}
u\in C([ 0,\infty ) ; Z) .
\end{equation*}
Moreover the following asymptotic statement is valid,
\begin{equation}
u(x,t)=
h(t)\mathring{\mathcal{B}_{s}}\{z^{-1}\}+\theta(\beta)t^{-1}
\hat{h}(0) \Psi(xt^{-2}) +t^{-3}A \Lambda (x)+R.
\label{nl5}
\end{equation}
uniformly with respect $t\to \infty $, where
$\Psi, \Lambda \in  L^{\infty }(\mathbb R^{+})$
\begin{gather*}
\Psi(s)=-4\mathcal{F}_{s}\{e^{-\sqrt{z}}z^{-1/2}\}(s)\\
  \Lambda(s)=  \frac{\sqrt{2}}{8\pi}\Big[\int _{0}^{\infty}e^{-\sqrt{z}}
\sqrt{z}dz\Big]  \Big[\int _{0}^{\infty}e^{-ps}
\frac{\sqrt{p}}{(ip^2+(ip)^{1/2})(ip^2+(-ip)^{1/2})}dp \Big] \\
A=\mathcal{S}\Big(u_0+\int_{0}^{\infty}|u|^{2}u(\tau)d\tau \Big),\\
R=O(t^{-(3+\delta )})(\|u_0\|_{Z}+\| u\| _{\mathbf{X}}^3
+t^{-(1+\beta)}\|h\|_Y).
\end{gather*}
\end{theorem}

From this Theorem we conclude that the solution possesses the following
modified scattering behavior:
\begin{itemize}
 \item   If $\beta <1$, then there exist a function $\Psi \in  L
^{\infty }$ such that
\begin{equation*}
\sup_{t>0}\langle t\rangle ^{\beta+\gamma }\|
u-h(t)\mathring{\mathcal{B}_{s}}\{z^{-1}\}\| _{ L^{\infty }}\leq C\varepsilon.
\end{equation*}

 \item  If $\beta\geq 1$ then there exist  a constant $B$ and a function $\widetilde{\Lambda}
(\xi )\in  L^{\infty }$ such that
\begin{equation*}
\sup_{t>0}\langle t\rangle ^{1+\gamma }\|
u-t^{-1}B\widetilde \Lambda (xt^{-1/2})\| _{ L^{\infty }}\leq
C\varepsilon ,
\end{equation*}
\end{itemize}

\subsection{Linear problem}
Consider the linear fractional NLS equation posed on a half-line
\begin{equation}
\begin{gathered}
u_{t}+\mathbb{K}u=0,\quad t>0,\; x>0; \\
u(x,0)=u_{0}(x),\; x>0, \quad u(0,t)=h(t),\; t>0,
\end{gathered}
 \label{1.3}
\end{equation}
In the next lemma we prove that $\mathcal{G}(t)$ and $\mathcal{H}(t)$
given by \eqref{Green} and \eqref{GreenH} are the Green
and boundary operators of the problem \eqref{1.3}.

\begin{lemma}\label{factorizacion}
 Let the initial data $u_{0}\in Z=H^{1}(\mathbb{R}
^{+})\cap  H_{1}^{0,1}(\mathbb{R}^{+})$, and boundary data
$h\in  Y=H_{\infty }^{1,\beta }$, $\beta >\frac{1}{2}$. Then the
solution $u(x,t)$ of the initial-boundary value problem \eqref{1.3} has the
following integral representation
\begin{equation*}
u(x,t)=\mathcal{G}(t)u_{0}+\mathcal{H}(t)h,
\end{equation*}
where the operators $\mathcal{G}(t)$ and $\mathcal{H}(t)$ are given by
\eqref{Green} and \eqref{GreenH}.
\end{lemma}

\begin{proof}
 In \cite{liliana} we proved that the unique solution $u(x,t)$ to
\eqref{1.3} has the  integral representation
 $$
u(x,t)=\int _{0}^{\infty}G(x,y,t)u_0(y)dy+\int _{0}^{t}H(x,t-\tau)h(\tau)d\tau,
$$
 where
 \begin{equation}
  \begin{gathered}
   G( x,y,t) =\big(\frac{1}{2\pi i}\big)^2\int_{i\mathbb{R}}e^{\xi t}
\int_{i\mathbb{R}}e^{px} \frac{Y( p,\xi ) }{K( p) +\xi }
\Big(\mathcal{J}^{-}_{\varphi(\xi)}(y,\xi)-
\mathcal{J}^{-}_{p}(y,\xi) \Big) \,dy \, dp \, d\xi ,  \label{G1}\\
H(x,t)=-\big(\frac{1}{2\pi i}\big)^2\int_{i\mathbb{R}}e^{\xi t}
\int_{i\mathbb{R}}e^{px}\frac{Y(p,\xi)}{K(p)+\xi}
\Big(\mathcal{I}^{-}(\varphi(\xi),\xi)-\mathcal{I}^{-}(p,\xi)\Big) \, dp \, d\xi,
  \end{gathered}
 \end{equation}
with
\begin{equation} \label{funciones}
\begin{gathered}
\mathcal{J}_{z}(y,\xi)
= \frac{1}{2\pi i}\int_{i\mathbb{R}}\frac{e^{-qy}}{ q-z }\frac{1}{Y( q,\xi ) }dq,
\\
\mathcal{I}(z,\xi)=  iz-\frac{1}{2\pi}\int_{i\mathbb{R}}
\frac{\sqrt{|q|} }{q(q-z)}\frac{1}{Y( q,\xi ) }dq,
\end{gathered}
\end{equation}
and the ``analyticity switching''� function $Y(w,\xi)=e^{\Gamma(w,\xi)}$,
 $\operatorname{Re}\xi>0$, where $\Gamma$ is defined in \eqref{Gamma} and
$\varphi(\xi)$ is the only one root of the equation
$K(p)+\xi=0$ in the right-half complex plane, with the analytic extension of
the function $K(p)$ is given by
\begin{equation*}
K(p)=\begin{cases}
K^+(p)=ip^{2}+\sqrt{-ip} & \text{ if Im }p>0 \\
K^-(p)=ip^{2}+\sqrt{ip} & \text{ if Im }p<0.
\end{cases}
\end{equation*}

 Now we simplify the representation of $G(x,y,t)$. Via the Sokhotski-Plemelj
formula we obtain
\begin{align*}
G(x,y,t) &=   \big(\frac{1}{2\pi i}\big)^2\int_{i\mathbb{R}}e^{\xi t}
\int_{i\mathbb{R}}e^{px} \frac{Y( p,\xi ) }
 {K( p) +\xi }(\mathcal{J}^{-}_{\varphi(\xi)}(y,\xi)-
\mathcal{J}^{+}_{p}(y,\xi) ) dy \ dp \ d\xi\\
&\quad +\frac{1}{2\pi i}\int_{i\mathbb{R}}e^{px-K(p)t}dp.
\end{align*}
Remember that $\varphi (\xi )$ is the only root of the equation $K(p)+\xi=0$
on the right half plane, using this, we
change of variables $\xi =-K(z)$ and we obtain
\begin{align*}
G(x,y,t)
&=  - \big(\frac{1}{2\pi i}\big)^2
\int_{-i\infty e^{-i(\frac{\pi }{2})+\epsilon
}}^{-i\infty e^{i(\frac{\pi }{2})-\epsilon }}e^{-K(z) t} K'(z)
\int_{i\mathbb{R}}e^{px} \frac{Y( p,-K(z) ) }{K( p)-K(z) }\\
&\quad \times (\mathcal{E}^{-}_{z}(y,-K(z))-\mathcal{E}^{+}_{p}(y,-K(z)) )
dy \, dp \, dz \\
&\quad +\frac{1}{2\pi i}\int_{i\mathbb{R}}e^{px-K(p)t}dp,
\end{align*}
with
\begin{equation*}
\mathcal{E}_{w}(y)=\frac{1}{2\pi i}\int_{i\mathbb{R}}
\frac{e^{-qy}}{q-w}\frac{1}{Y(p,-K(z))}dq,\quad \text{for }\operatorname{Re}w\neq 0.
\end{equation*}
To change the contour of integration with respect to $p$ variable we apply
Cauchy Theorem. Taking residue in the point $p=-z$ we obtain
\begin{equation}
\begin{aligned}
G(x,y,t)  &=  -\big( \frac{1}{2\pi i}\big) ^{2}\int_{i\mathbb{R}
}e^{-K(z)t}e^{-zx+\Gamma(-z,-K(z))}( \mathcal{E}_{z}^{-}( y) -
\mathcal{E}_{-z}^{-}( y) ) dz \\
&\quad+ \big(\frac{1}{2\pi i}\big) ^{2}
\int_{i\mathbb{R}}e^{-K(z)t}zK'(z)\mathcal{E}_{z}^{-}(
y) \int_{0}^{\infty }e^{-px}\psi(p,z)dp\, dz \\
&\quad +\big( \frac{1}{2\pi i}\big) ^{2}
\int_{i\mathbb{R}}e^{-K(z)t}K'(z)
\int_{0}^{\infty }e^{-px}\mathcal{E}_{p}^{+}(y) \psi(p,z)dp\ dz,
\end{aligned}
\label{help1}
\end{equation}
where
\begin{equation}\label{psi}
\psi(p,z)=\frac{\sqrt{2}}{2}\frac{e^{\Gamma (-p,K(z))}}{(K^+(p)-K(z))(K^-(p)-K(z))}.
\end{equation}
Note that since integrand function is even with respect to $z$ variable
\begin{equation*}
\int_{i\mathbb{R}}e^{-K(z)t}K'(z)
\int_{0}^{\infty }e^{-px}\mathcal{E}_{p}^{+}(y) \psi(p,z)dp\ dz=0.
\end{equation*}
Consequently changing $z\mapsto iz$ into \eqref{help1} we obtain
\begin{equation}
G(x,y,t)  =  \big( \frac{1}{2\pi }\big) ^{2}
\int_{0}^{\infty }e^{-K(iz)t}\psi _{s}(z,y)\psi_{s }^{\ast }(x,z)dz,
\label{funcion Green}
\end{equation}
where the functions $\psi _{s}$ and $\psi _{s}^{\ast }$ was defined in
\eqref{seno}, \eqref{senoestrella}. Therefore, we obtain
\begin{equation}\label{Green01}
\mathcal{G}(t)\phi =\mathcal{B}_{s}^{\ast }
\{e^{K(p)t}\mathcal{B}_{s}\phi\},\quad K(z)=ip^{2}-\sqrt{p}.
\end{equation}
For the operator $\mathcal{H}(t)$ we  note that
$$
\mathcal{I}(\varphi(\xi),\xi)-\mathcal{I}(p,\xi)
=i(\varphi(\xi)-p)\Big[1-\frac{1}{2\pi i}\int _{i\mathbb R}\frac{\sqrt{|q|}}{q( q-p) ( q-\varphi
( \xi ) ) }\frac{1}{Y( q,\xi ) }dq \Big].
$$
 Denoting $K_1(q)=iq^{2}+\sqrt{|q|}$ we have
\begin{align*}
&\int_{i\mathbb{R}}\frac{\sqrt{|q|}}{q( q-p) ( q-\varphi
( \xi ) ) }\frac{1}{Y( q,\xi ) }dq \\
&=\int_{i\mathbb{R}}\frac{K_1( q) +\xi }{q( q-p) ( q-\varphi
( \xi ) ) }\frac{1}{Y( q,\xi ) }dq
-\int_{i\mathbb{R}}\frac{iq^{2}+\xi }{q( q-p) ( q-\varphi ( \xi
) ) }\frac{1}{Y( q,\xi ) }.
\end{align*}
Recalling that function $ \frac{K( \cdot) +\xi }{Y( \cdot,\xi )}$ is
analytic on the right half-plane,
via the Cauchy theorem we have
\begin{gather*}
 \int_{i\mathbb{R}}\frac{K( q) +\xi }{q( q-p) (
q-\varphi ( \xi ) ) }\frac{1}{Y( q,\xi ) }dq
=\frac{K( p) +\xi }{p( p-\varphi ( \xi )
) }\frac{1}{Y( p,\xi ) }+\frac{1}{2}\frac{\xi }{
p\varphi ( \xi ) }\frac{1}{Y( 0,\xi ) }-\frac{1}{2}, \\
 \int_{i\mathbb{R}}\frac{iq^{2}+\xi }{q( q-p) ( q-\varphi
( \xi ) ) }\frac{1}{Y( q,\xi ) }dq
=\frac{1}{2}-\frac{1}{2}\frac{\xi }{p\varphi ( \xi ) }\frac{1}{
Y( 0,\xi ) }.
\end{gather*}
Therefore,
\begin{equation*}
\int_{i\mathbb{R}}\frac{\sqrt{|q|}}{q( q-p) ( q-\varphi
( \xi ) ) }\frac{1}{Y( q,\xi ) }dq
=\frac{K( p) +\xi }{p( p-\varphi ( \xi ) ) }\frac{1
}{Y( p,\xi ) }+\frac{\xi }{p\varphi ( \xi ) }\frac{1}{Y( 0,\xi ) }-1,
\end{equation*}
and as consequence
\begin{equation}
 \mathcal{I}(\varphi(\xi),\xi)-\mathcal{I}(p,\xi)
=i(\varphi(\xi)-p)\Big[\frac{K( p) +\xi }{p( p-\varphi ( \xi ) ) }\frac{1
}{Y( p,\xi ) }+\frac{\xi }{p\varphi ( \xi ) }\frac{1}{Y( 0,\xi ) } \Big].
\end{equation}
Thus using the Cauchy Theorem we obtain
\begin{equation}
H(x,t)=\widetilde H_1(x,t)+\widetilde H_2(x,t),
\label{funcion H}
\end{equation}
where
\begin{equation} \label{H1}
\begin{gathered}
  \widetilde H_1(x,t)=\frac{1}{2\pi i}\int_{i\mathbb{R}}e^{\xi t}\,d\xi
 - \big( \frac{1}{2\pi i}\big)^{2}\int_{i\mathbb{R}}e^{\xi t}\xi
\hbox{--}\hskip-9pt\int _{i\mathbb{R}}e^{px}\frac{Y(p,\xi )}{Y(0,\xi )}
\frac{1 }{p(K( p) +\xi )}\,dp\,d\xi, \\
 \widetilde H_2(x,t)=\big(\frac{1}{2\pi i}\big)^{2}\int_{i\mathbb{R}}e^{\xi t}
\frac{\xi }{\varphi (\xi ) }\hbox{--}\hskip-9pt\int _{i\mathbb{R}}e^{px}\frac{Y(p,\xi )}{Y(0,\xi )}
\frac{1}{K( p) +\xi}\,dp\,d\xi.
\end{gathered}
\end{equation}
Using analytic properties of the integrand function via Jordan Lemma we have
\begin{equation} \label{H2}
\begin{aligned}
  \widetilde H_1(x,t)
&=-\frac{1}{2\pi i}\int_{i\mathbb{R}}e^{\xi t}
e^{-\varphi(\xi)x}\frac{Y(-\varphi(\xi),\xi)}{Y(0,\xi)}
\frac{\xi}{\varphi(\xi)}\frac{1}{K'(\varphi(\xi))}d\xi\\
&\quad - \big(\frac{1}{2\pi i}\big)^{2}
\int_{i\mathbb{R}}e^{\xi t}\xi \int_{-\infty}^{0}e^{px}\frac{Y(p,\xi )}{Y(0,\xi )}
\frac{\sqrt{(-ip)}-\sqrt{(ip)} }{p(K^+( p) +\xi )(K^-( p) +\xi )}\,dp\,d\xi .
\end{aligned}
\end{equation}
Changing of variable $\xi=-K(z)$ and remembering $Y(0,-K(z))=1$
(see Lemma \ref{cero}) we rewrite $\widetilde H_1$ as
\begin{equation}
\begin{aligned}
  \widetilde H_1(x,t)
&=-\frac{1}{2\pi i}\int_{i\mathbb{R}}e^{-K(z) t}
 e^{-zx+\Gamma(-z,-K(z))}\frac{K(z)}{z}dz\\
&\quad - \big(\frac{1}{2\pi i}\big)^{2}\int_{i\mathbb{R}}e^{-K(z) t}K(z)K'(z)
\int_{0}^{\infty}e^{-px}\psi(p,z) dp \, dz,
\end{aligned}
\end{equation}
since the integrand in the second integral expression is an odd function with
respect to $z$ variables we conclude
$$
\big(\frac{1}{2\pi i}\big)^{2}\int_{i\mathbb{R}}e^{-K(z) t}K(z)K'(z)
\int_{0}^{\infty}e^{-px}\psi(p,z) dp \ dz=0,
$$
and as a consequence
\begin{equation}\label{H3}
  \widetilde H_1(x,t)=\frac{1}{2\pi i}\int_{0}^{i\infty}e^{-K(z) t}
\frac{K(z)}{z}\left[e^{zx+\Gamma(z,-K(z))} -e^{-zx+\Gamma(-z,-K(z))} \right]dz.
\end{equation}
In a similar form we obtain
\begin{equation}\label{H4}
\begin{aligned}
  \widetilde H_2(x,t)
&=\frac{1}{2\pi i}\int_{0}^{i\infty}
e^{-K(z) t}\frac{K(z)}{z}\big[e^{-zt+\Gamma(-z,-K(z))}
-e^{zt+\Gamma(z,-K(z))} \big]dz\\
&\quad +\frac{1}{2\pi i}\hbox{--}\hskip-9pt\int _{-i\infty}^{i\infty}e^{-K(z) t}
\frac{K(z)}{z}K'(z)\Theta(x,z)dz.
\end{aligned}
\end{equation}
Since $K(z)K'(z)=-2z^3+\frac{5}{2}z\sqrt{|z|}+\frac{1}{2}$ and
$$
\frac{1}{4\pi i}\hbox{--}\hskip-9pt\int _{-i\infty}^{i\infty}e^{-K(z) t}\frac{1}{z}\Theta(x,z)dz=0,
$$
we reduce the function $\widetilde H_2$ as
\begin{equation}\label{H5}
\begin{aligned}
  \widetilde H_2(x,t)&=\frac{1}{2\pi i}\int_{0}^{i\infty}e^{-K(z) t}
\frac{K(z)}{z}[e^{-zt+\Gamma(-z,-K(z))} -e^{zt+\Gamma(z,-K(z))}]dz\\
 &\quad  +\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}e^{-K(z) t}
(\frac{5}{2}\sqrt{|z|}-2z^2)\Theta(x,z) dz.
\end{aligned}
\end{equation}
Applying \eqref{H3}--\eqref{H5} into \eqref{funcion H} we obtain
$$
H(x,t)=2i\int_{0}^{\infty}e^{-K(z) t}\frac{K(z)}{z}\mathring{\psi_{s}}(x,z)dz.
$$
From this we conclude
\[
\mathcal{H}(t)h =\mathring{\mathcal{B}_{s}}
\big\{\frac{K(p)}{p}\int_0^{t}e^{K(p)(t-\tau)}h(\tau)d\tau\big\},
\]
where $K(p)=ip^{2}-\sqrt{p}$ and the operator $\mathring{\mathcal{B}_{s}}$
was defined in \eqref{puntotransform}.
\end{proof}


\subsection{Large time asymptotic behavior for the evolution group
 and the boundary operator}

\begin{lemma}\label{psis}
 For $K(z)=-\sqrt{z}+iz^{2}$, and $\psi_{s}$ given by \eqref{seno}  we have
 $$
\psi _{s}(z,x)= e^{-izx}e^{-\Gamma
(-iz,K(z))}-e^{izx}e^{-\Gamma (iz,K(z))}+z\mathbb{J}\{e^{-px}\}(z),
$$
where $\mathbb J$ was given by \eqref{def:W}.
\end{lemma}

The proof of the above lemma is obtained
using analytic properties of the integrand function via Jordan Lemma,
 and the Cauchy Theorem.

\begin{lemma}  \label{Estimativo en L2}
 For $u_0 \in \mathbf{Z=H}^{1}(\mathbb{R}^{+})\cap  H_{1}^{0,1}(\mathbb{R}^{+})$,
the Green operator $\mathcal G(t)$ satisfies the asymptotic expansion
 $$
\mathcal G(t)u_0=t^{-3} \Lambda(x)\mathcal{S}u_0+O(t^{-(3+\gamma)})\| u_0\|_{ Z},
$$
 where $\Lambda \in  L^{\infty}(\mathbb{R}^+)$,
 \begin{equation}\label{258}
  \Lambda(x)=   \frac{\sqrt{2}}{8\pi}
\Big[\int _{0}^{\infty}e^{-\sqrt{z}} \sqrt{z}dz\Big]
\Big[\int _{0}^{\infty}e^{-px}\frac{\sqrt{p}}{(ip^2+(ip)^{1/2})
 (ip^2+(-ip)^{1/2})}dp \Big],
\end{equation}
and
$$
\mathcal{S}u_0=\int_0^{\infty}f(y)u_0(y)dy,
$$
with $f(y)=y+\mathbb{J}\{e^{-py}-1\}|_{z=0}$.
\end{lemma}

\begin{proof}
 From the equality  obtained in Lemma \ref{psis} for the function $\psi_{s}$,
and Cauchy Theorem we obtain
\begin{align*}
  \psi_{s}(z,y)
&= \sin zy+(e^{-\Gamma(iz,K(z))}-1) (e^{-izy}-1)\\
&\quad -(e^{-\Gamma(-iz,K(z))}-1)(e^{izy}-1)  +zW(z,y),
\end{align*}
where $W(z,y) =\mathbb{J}\{e^{-py}-1\}(z)$.
Via Taylor theorem we have
$$
|W(0,y)|\leq Cy\int_{0}^{\infty }\frac{1}{p}
( \{p\}^{\gamma}\langle p\rangle ^{-3/2}) dp<C,
$$
moreover
\begin{equation}
 |W(z,y)-W(0,y)|\leq Cyz \label{W}.
\end{equation}
Via Lemma \eqref{lema de gamma} we have
$$
e^{-\Gamma(\pm iz,K(z))}-1=O(z^{\gamma}), \quad \delta >0.
$$
Combining this with \eqref{W} we conclude
\begin{equation}
 \psi_{s}(z,y)=zf(y)+O(yz^{1+\gamma}), \quad f(y)=y+W(0,y),\label{q1}
\end{equation}
and as a consequence
\begin{equation}
 \mathcal{B}_{s}u_0= z\mathcal{S} u_0 +O(z^{1+\gamma})\|u_0\|_{ L^{1,1}}, \label{q3}
\end{equation}
where the functional $\mathcal{S}$ is  given by
$$
\mathcal{S} u_0=\int _0^{\infty}f(y)u_0(y) dy.
$$
On the other hand, from the definition of the function $\psi_{s}^{*}$,
given by \eqref{seno}, we note
\begin{equation}
\begin{aligned}
\psi_{s}^{*}(x,z)
&=\frac{\sqrt{2}}{8\pi}\frac{1}{\sqrt{z}}\int _{0}^{\infty}e^{-px}
\frac{\sqrt{p}}{(ip^2+(ip) ^{1/2})(ip^2+(-ip)^{1/2})}dp \\
&\quad +R_1(x,z)+R_2(x,z),
\end{aligned} \label{i2}
\end{equation}
where
\begin{gather*}
  R_1(x,z)= e^{izx+\Gamma^+ (z)}-e^{-izx+\Gamma^+ (-z)}+z\Theta(x,z),\\
  R_2(x,z)= \frac{1}{4\pi} \frac{1}{\sqrt{z}}
\int _{0}^{\infty}e^{-px}p\left[e^{\Gamma^+(p,z)}\lambda(p,z)
-e^{\Gamma^+(p,0)}\lambda(p,0)\right]dp.
\end{gather*}
Using
$$
|\Theta(x,z)|\leq C\int _{0}^{\infty }\frac{\sqrt{p}}{(ip^{2}
+(-ip)^{1/2} )(ip^{2}+(ip)^{1/2}  )}\leq C,
$$
and via Lemma \ref{lema de gamma}
$|e^{izx+\Gamma^+ (z)}-e^{-izx+\Gamma^+ (-z)}|\leq C$, we conclude
$$
R_1(x,z)=O(\langle z \rangle).
$$
Now, we estimate $R_2$. We have
\begin{equation}
 \begin{aligned}
& e^{\Gamma^+(p,z)}\lambda(p,z)-e^{\Gamma^+(p,0)}\lambda(p,0)\\
&= [ e^{\Gamma^{+}(p,z)}-e^{\Gamma^{+}(p,0)}]\lambda(p,z)
+e^{\Gamma^{+}(p,0)}\left[\lambda(p,z)-\lambda(p,0) \right],
 \end{aligned}
\end{equation}
Using
$$
|\lambda(p,z)-\lambda(p,0)|
\leq C \frac{|K(z)|^{1-\gamma}|p|}{|ip^{2}+(-ip)^{1/2}|^{2}
|ip^{2}+(ip)^{1/2}|^{2-\gamma}},
$$
and via Lemma \ref{lema de gamma}
$$
e^{\Gamma^{+}(p,z)}-e^{\Gamma^{+}(p,0)}=O(z^{\gamma}),
$$
therefore
\begin{align*}
&R_2(x,z) \\
&= \frac{1}{4\pi}\frac{1}{\sqrt{z}}\int _{0}^{\infty}e^{-px}
 \Big([ e^{\Gamma^{+}(p,z)}-e^{\Gamma^{+}(p,0)}]\psi(p,z)
+e^{\Gamma^{+}(p,0)}[\psi(p,z)-\psi(p,0)]\Big) dp\\
&=O(z^{\gamma-\frac{1}{2}}).
\end{align*}
Thus via \ref{i2}
\begin{equation}
 \psi_{s}^{*}(x,z)
=\frac{\sqrt{2}}{8\pi}\frac{1}{\sqrt{z}}\int _{0}^{\infty}
e^{-px}\frac{\sqrt{p}}{(ip^2+(ip)^{1/2})
 (ip^2+(-ip)^{1/2})}dp +O(z^{\gamma-\frac{1}{2}}).\label{q4}
\end{equation}
Combining \eqref{q3} and \eqref{q4} we conclude
\begin{equation}
  \mathcal G(t)u_0=t^{-3} \Lambda(x)\mathcal{S}u_0
+O(t^{-(3+\gamma)})\| u_0\|_{ L^{1,1}}\label{i11},
\end{equation}
where $\Lambda  \in  L^{\infty}$,
\begin{equation}
\begin{gathered}
  \Lambda(x)=   \frac{\sqrt{2}}{8\pi}
\Big[\int _{0}^{\infty}e^{-\sqrt{z}} \sqrt{z}dz\Big]
  \Big[\int _{0}^{\infty}e^{-px}\frac{\sqrt{p}}{(ip^2+(ip)^{1/2})(ip^2
+(-ip)^{1/2})}dp \Big].    \\
  \mathcal{S}u_0= \int _{0}^{\infty}f(y)u_0(y) \ dy.
\end{gathered}
\end{equation}
Thus, Lemma \eqref{Estimativo en L2} is proved.
\end{proof}


\begin{lemma}\label{lema de h}
 Let $h\in  Y= H_{\infty }^{1,\beta }(\mathbb{R}^{+})$, $\beta >1/2$,
 then the following asymptotic expansion for large time $t$ holds
\begin{equation}
 \mathcal{H}(t)h =h(t)\mathring{\mathcal{B}_{s}}\{z^{-1}\}
+\theta(\beta)t^{-1} \hat{h}(0) \Psi(xt^{-2})
+O(t^{-1-\beta})\|h\|_Y,
\end{equation}
where $\theta$ is the characteristic function of the interval $[1,\infty)$ and $\Psi\in  L^{\infty}(\mathbb R^{+})$ is given by
\begin{equation}\label{Psi}
 \Psi(s)=-4\mathcal{F}_{s}\{e^{-\sqrt{z}}z^{-1/2}\}(s).
\end{equation}
\end{lemma}

\begin{proof}
First, we recall the definition of the operator $\mathcal H$ given in \eqref{GreenH}:
 \begin{equation}\label{bo1}
  \mathcal{H}(t)h=\mathring{\mathcal{B}_{s}}
\big\{\frac{K(z)}{z}[h_1(z,t)+h_{2}(z,t)]\big\},
 \end{equation}
 where
 $$
h_1(z,t)=\int_{t/2}^{t}e^{K(z)(t-\tau)}h(\tau)d\tau, \quad
h_2(z,t)=\int_{0}^{t/2}e^{K(z)(t-\tau)}h(\tau)d\tau.
$$
Integrating by parts,
\begin{equation}\label{bo2}
  K(z)h_1(z,t)= h(t)-e^{K(z)\frac{t}{2}}h(\frac{t}{2} )
-\int_{t/2}^{t}e^{K(z)(t-\tau)}h'(\tau)d\tau.
\end{equation}
Recalling that $h\in  H^{1,\beta}_{\infty}$,
\begin{equation}\label{bo4}
\begin{gathered}
\mathring{\mathcal{B}_{s}}\big\{\frac{1}{z}e^{K(z)t}h(\frac{t}{2})\big\}
=O(\langle t\rangle^{-(1+\beta)})\|h\|_{ Y},\\
  \int_{t/2}^{t}e^{K(z)(t-\tau)}h'(\tau)d\tau=O(h'(t)),
\end{gathered}
\end{equation}
and therefore
\begin{equation}\label{bo5}
  \mathring{\mathcal{B}_{s}}
\Big\{\frac{1}{z}\int_{t/2}^{t}e^{K(z)(t-\tau)}h'(\tau)d\tau\Big\}
 =O(h'(t)).
\end{equation}
Thus, \eqref{bo2}--\eqref{bo5} imply
\begin{equation}\label{bo51}
 \mathring{\mathcal{B}_{s}}\big\{\frac{K(z)h_{1}(z,t)}{z}\big\}
=h(t)\mathring{\mathcal{B}_{s}}\{z^{-1}\}
+O(\langle t\rangle^{-(1+\beta)})\|h\|_{ Y}.
\end{equation}
On the other hand
\begin{equation}
h_2(z,t)=e^{K(z)t}\int_{0}^{t/2}h(\tau )d\tau
+\int_{0}^{t/2}e^{K(z)(t-\tau )}( 1-e^{K(z)\tau }) h(\tau )d\tau ,
\label{bo6}
\end{equation}
now, from the definition of the function $\mathring{\psi}_s(z,x)$
given by \eqref{puntotransform} we have
\begin{equation*}
 \frac{K(z)}{z}\mathring{\psi_s}(z,x)=\frac{2i\sin xz}{\sqrt{z}}
+R_1(z,x)+R_2(z,x)+R_3(z,x),
\end{equation*}
with
\begin{gather*}
 R_1(z,x)=z(e^{ixz+\Gamma(iz,-K(z))}-e^{-ixz+\Gamma(-iz,-K(z))} ),\\
 R_2(z,x)=\frac{e^{ixz}(e^{\Gamma(iz,-K(z))}-1 )
-e^{-ixz}(e^{\Gamma(-iz,-K(z))}-1 )}{\sqrt{z}},\\
 R_3(z,x)=\big(\frac{5}{2}\sqrt{|z|}-2z^2\big)\Theta(x,z)
\end{gather*}
From Lemma \ref{lema de gamma} $e^{\Gamma(\pm iz,-K(z))}=O(1)$ and
$e^{\Gamma(\pm iz,-K(z))}-1=O(z^{\gamma})$, $\gamma\in (0,1)$ and as
consequence $R_1(z,x)+R_2(z,x)=O(z^{\gamma-\frac{1}{2}})$. Using
$$
|\Theta(x,z)|\leq C\int _{0}^{\infty }\frac{\sqrt{p}}{(ip^{2}+(-ip)^{1/2} )
(ip^{2}+(ip)^{1/2}  )}\leq C,
$$
we conclude $R_3(z,x)=O(z^\gamma)$. Therefore
\begin{equation}
 \frac{K(z)}{z}\mathring{\psi_s}(z,x)
=\frac{2i\sin xz}{\sqrt{z}}+O(z^{\gamma-\frac{1}{2}})\,. \label{bo7}
\end{equation}
From \eqref{bo6}, \eqref{bo7} and
$| 1-e^{K(z)\tau }| \leq Cz^{\frac{\gamma }{2}}\tau ^{\gamma }$ we conclude
\begin{equation}\label{bo8}
\begin{aligned}
 &\mathring{\mathcal{B}_{s}}\big\{\frac{K(z)h_{2}(z,t)}{z}\big\}\\
 &=t^{-1} \hat{h}(0) \Psi(xt^{-2}) +O(t^{-(1+\gamma)})\|h\|_Y
 +\int_{0}^{t/2}O(( t-\tau ) ^{-2-\gamma })
 \tau ^{\gamma }h(\tau )d\tau\\
&=t^{-1} \hat{h}(0) \Psi(xt^{-2}) +O(t^{-(1+\gamma)})\|h\|_Y
 +\max (t^{-2-\gamma },t^{-1-\beta -\gamma })\| h\| _{ Y},
\end{aligned}
\end{equation}
where $\Psi(s)=-4\mathcal F_{s}\{e^{-\sqrt{z}}z^{-1/2}\}(s)$.
Finally,  from \eqref{bo1} along to \eqref{bo2}-\eqref{bo8} we have
\begin{equation*}
 \mathcal{H}(t)h =h(t)\mathring{\mathcal{B}_{s}}\{z^{-1}\}
+\theta(\beta)t^{-1} \hat{h}(0) \Psi(xt^{-2})+O(t^{-1-\beta})\|h\|_Y,
\end{equation*}
where $\theta$ is the characteristic function of the interval $[1,\infty)$.
The proof is complete.
\end{proof}

In this Lemma we exhibit several properties of the ``analyticity switching"
function $Y(w,K(z))=e^{\Gamma (w,K(z))}$, where
\begin{equation*}
\Gamma (w,\xi )
=\frac{1}{2\pi i}\int_{0}^{\infty }\ln (q-w)( \frac{
K^{+\prime }(q)}{K^{+}(q)+\xi }
-\frac{K^{-\prime }(q)}{K^{-}(q)+\xi })
dq.
\end{equation*}
We make a cut along to negative axis $w<0$. Denote by
\begin{align*}
\Gamma ^{+}(s,\xi )  =\lim_{w\to s, \operatorname{Im}w>0}\Gamma (w,\xi ),\; s>0 \\
\Gamma ^{-}(s,\xi )  =\lim_{w\to s, \operatorname{Im}w<0}\Gamma (w,\xi ),\; s>0.
\end{align*}

\begin{lemma}\label{lema de gamma}
We have for $s>0$, $\arg \xi \in ( -\frac{\pi }{2}-\frac{\pi
}{4},\frac{\pi }{2}+\frac{\pi }{4}) $
\begin{equation*}
\frac{e^{\Gamma ^{+}(s,\xi )}}{e^{\Gamma ^{-}(s,\xi )}}
=\frac{K^{+}(p)+\xi }{K^{-}(p)+\xi }.
\end{equation*}
Moreover the following formula is valid for
 $z\in \mathbb{R}$ and $w\in \mathbb{C}/w>0$,
\begin{gather*}
| Y(w,K(z))| \leq C,\quad | e^{-\Gamma^{+}(iz,K(z))}| \leq C, \\
\Gamma (w,-K(z))  =O(\{ w\}^{\gamma }+\{ z\} ^{\gamma }),\\
\partial_{z}\Gamma(iz,K(z)) =O(\{z\}^{-1}\langle z \rangle^{-2}).
\end{gather*}
\end{lemma}

The proof of the above Lemma can be found in \cite{LK}.

\begin{lemma}\label{cero}
 For $w\in \mathbb{C}/w>0$ we have
 $$
\Gamma(-w,-K(z))=\Gamma(w,-K(z)),
$$ a
s consequence
 $Y(0,-K(z))=1$.
\end{lemma}

\begin{proof}
Integrating by parts and via Cauchy Theorem we rewrite $\Gamma$ as
$$
\Gamma(w,-K(z))=-\frac{1}{2\pi}\int _0^{\infty}\frac{1}{q-w}
\ln \Big( \frac{K(q)-K(z)}{q^2-z^2} \Big)dq.
$$
As consequence,  via the change of variables $v=-q$, we obtain
\begin{equation}
 \begin{aligned}
  \Gamma(-w,-K(z)) &=
   -\frac{1}{2\pi }\int _0^{\infty}\frac{1}{q+w}\ln ( \frac{K(q)-K(z)}{q^2-z^2} )dq\\
 &=   \frac{1}{2\pi }\int _0^{\infty}\frac{1}{v-w}\ln ( \frac{K(v)-K(z)}{v^2-z^2} )dq\\
  &=-\Gamma(w,-K(z)).
 \end{aligned}
\end{equation}
Therefore $\Gamma(0,-K(z))=0$, this guarantees that $Y(0,-K(z))=1$.
\end{proof}

\section{Proof of Theorem \ref{teoremaprincipal}}

 It follows from Lemma \ref{factorizacion} and Duhamel principle that
the solution of \eqref{nolineal} is given by
\begin{equation}\label{duhamel}
 u(x,t)=\mathcal{G}(t)u_0+\mathcal{H}(t)h
+\int_0^{t}\mathcal{G}(t-\tau)\mathcal{N}(u)(\tau)d\tau,
\end{equation}
Let us define the function spaces $ Z= H^{1}(\mathbb R^+)\cap  H^{0,1}(\mathbb R^+)$
$$
 X=\{\phi \in  C([0,\infty); Z ): \|\phi\|_{ X}<\infty\},
$$
where
$$
\|\phi\|_{ X}=\sup_{0\leq t}\{\langle t\rangle^{1/2}\|\phi(t)\|_{ H^{1}}
+\|\phi(t)\|_{ H^{0,1}}+\langle t\rangle^{\rho}\|\phi(t)\|_{ L^{\infty}}\},
$$
with $\rho=\min \{1,\beta\}$.
 By the contraction mapping principle we can prove that there exist an unique
solution $u$ to \eqref{nolineal} in $ X$, since
$$
X\subset  C([ 0,\infty ) ; L^{2}( \mathbb{R}
^{+}) ) \cap  C\Big( ( 0,\infty ) ; L^{^{2,
\frac{1}{2}( \frac{1}{2}+\mu ) }}( \mathbb{R}^{+})
\cap  L^{\infty }( \mathbb{R}^{+}) \Big)
$$
the uniqueness guarantee that the solution given by  Theorem
\ref{teorema de existencia} is the same solution $u\in  X$.

Now we prove the asymptotic formula for the solution.
From Lemmas \ref{Estimativo en L2} and \ref{lema de h} we obtain
\begin{equation}
\mathcal{G}(t)u_{0}+\mathcal{H}(t)h
= h(t)\mathring{\mathcal{B}_{s}}\{z^{-1}\}
+\theta(\beta)t^{-1} \hat{h}(0) \Psi(xt^{-2})
+t^{-3}\Lambda (x)\mathcal{S} {u}_{0} +R,  \label{las dos}
\end{equation}
with $\theta (\beta )=1$ for $\beta >1$ and $\theta (\beta )=0$ for
$\beta \leq 1$, $\widehat{h}(p)=\mathcal{L}h$,
$\Lambda, \Psi \in  L^{\infty }(\mathbb{R}^+)$  defined by \eqref{258}, \eqref{Psi}
 respectively and
\begin{equation*}
R=t^{-(3+\gamma) }\| u_{0}\| _{ Z}+t^{-(1 +\beta) }\| h\| _{ Y}.
\end{equation*}
Thus we observe for $t>1$,
\begin{equation}
\begin{aligned}
&\int_{0}^{t}\mathcal{G}( t-\tau ) \mathcal{N}(u)(\tau )\,d\tau\\
&=\int_{0}^{t/2}\mathcal{G}( t) \mathcal{N}(u)(\tau )\,d\tau
 +\int_{0}^{t/2}[ \mathcal{G}( t-\tau ) -
\mathcal{G}( t)] \mathcal{N}(u)(\tau )\,d\tau \\
&\quad +\int_{t/2}^{t}
\mathcal{G}( t-\tau ) \mathcal{N}(u)(\tau )\,d\tau .
\end{aligned} \label{nl1}
\end{equation}
Via \ref{las dos} we note that
\begin{align*}
&\int_{0}^{t/2}\mathcal{G}( t) \mathcal{N} (u)(\tau )\,d\tau\\
&= t^{-3}\Lambda (x)\int_{0}^{\infty }\mathcal{S}\mathcal{N}(u)(\tau)d\tau
 -t^{-3}\Lambda (x)\int_{t/2}^{\infty }
\mathcal{S} \mathcal{N}(u)(\tau )d\tau \\
&\quad  +O(t^{-(3+\delta )})\int_{0}^{t/2}
 ( \| \mathcal{N}(u)(\tau )\| _{ H_{1}^{0,1}}
 +\| \mathcal{N}(u)(\tau )\| _{ H^{1}}) d\tau,
\end{align*}
since $|\mathcal{S}\phi|\leq C \|\phi\|_{ L^{1,1}}$ we observe
$$
|\mathcal{S}\mathcal{N}u|
\leq C \|\mathcal{N}u\|_{ L^{1,1}}\leq C \|u\|_{ L^{\infty}}\|u\|_{ H^1}
\leq C\langle \tau \rangle^{-(1+\rho)}\|u\|_{\mathbf{X}}^3,
$$
and as consequence
\begin{equation}
 \int_{0}^{t/2}\mathcal{G}( t) \mathcal{N} (u)(\tau )\,d\tau
=t^{-3}\Lambda (x)\int_{0}^{\infty }\mathcal{S}\mathcal{N}(u)(\tau)d\tau
+O(t^{-(3+\gamma)})\|u\|_{\mathbf{X}}^3, \quad \gamma >0
\end{equation}

By Lemma \ref{Estimativo en L2}, we have
$$
\mathcal G(t)u_0=t^{-3} \Lambda(x)\mathcal{S}u_0+O(t^{-(3+\gamma)})\| u_0\|_{ Z},
$$
by properties of asymptotic representation
we obtain  $\| \partial _{t}\mathcal{G}(t)\phi \| _{\mathbf{\infty }}\leq
Ct^{-4}(\| \phi \| _{ H^{0,1}}+\| \phi \| _{ H^{1}})$ we obtain
\begin{equation}
\int_{0}^{t/2}\left[ \mathcal{G}( t-\tau ) -\mathcal{G}
( t) \right] \mathcal{N}(u)(\tau )\,d\tau =O(t^{-4})\| u\| _X^3.  \label{nl3}
\end{equation}
By Lemma \ref{Estimativo en L2} we have
\begin{equation}
\begin{aligned}
\int_{t/2}^{t}\mathcal{G}( t-\tau ) \mathcal{N}(u)(\tau)\,d\tau
&=\| u\| _{\mathbf{X}}^3\int_{t/2}^{t}O(\langle t
 -\tau \rangle ^{-3}\langle \tau \rangle ^{-(1+\gamma )})d\tau \\
&=O(t^{-(3+\delta )})\| u\| _{X}^3
\end{aligned}\label{nl4}
\end{equation}
From \eqref{duhamel}-\eqref{nl4} we obtain
\begin{equation}
u(x,t)=
h(t)\mathring{\mathcal{B}_{s}}\{z^{-1}\}+\theta(\beta)t^{-1}
\hat{h}(0) \Psi(xt^{-2})+t^{-3}A \Lambda (x)+R.
\end{equation}
where
$$
A=\mathcal{S}(u_0+\int_{0}^{\infty}|u|^{2}u(\tau)d\tau ),
$$
and
$$
R=O(t^{-(3+\delta )})(\|u_0\|_{Z}+\| u\| _{\mathbf{X}}^3).
$$
Hence, Theorem \ref{teoremaprincipal} is proved.

\begin{thebibliography}{00}

\bibitem{BiondiniBui} G. Biondini, A. Bui;
\emph{On the nonlinear Schr\"odinger
equation on half line with homogeneous Robin boundary conditions},
Stud.Appl.Math.129(3)(2012)249--271.

\bibitem{Bu} Q. Bu;
\emph{On well-posedness of the forced nonlinear Schr\"odinger equation},
Appl.Anal.46 (3--4) (1992), 219--239.

\bibitem{BuStrauss} C. Q. Bu, W. Strauss;
\emph{An inhomogeneous boundary value problem for nonlinear Schr\"odinger
equations}, J. Differential equations 173(2001)79--91.

\bibitem{potencial1} J. Dong, M. Xu;
\emph{Space-time fractional Schr\"odinger equation with time-independent potentials}.
 J. Math. Anal. Appl., \textbf{344} (2008), no. 2, 1005--1017.

\bibitem{liliana} L. Esquivel;
\emph{Nonlinear Schr\"odinger equation with Landau damping on a half-line}.
Differ. Equ. Appl. 7 (2015), no. 2, 221--244.

\bibitem{LK} L. Esquivel, E. Kaikina;
\emph{Neumann problem for nonlinear Schrodinger equation with the Riezs fractional
derivative operator}, J. Differential Equations. (2016).

\bibitem{LK2} L. Esquivel, E. Kaikina;
\emph{A forced fractional Schr\"odinger equation with a Neumann boundary condition},
Nonlinearity, Volume 29, Number 7.

\bibitem{Fokas2002} A. S. Fokas;
\emph{Integrable nonlinear evolution equations on the half-line}.
 Commun. Math. Phys. 230 1-39. 2002

\bibitem{Fokas2005} A. S. Fokas, A. R. Its, L-Y Sung;
\emph{The nonlinear Schr\"odinger equation on the half-line}.
 Nonlinearity 18(4), 1771--1822. 2005.

\bibitem{Gak} F. D. Gakhov;
\emph{Boundary Value Problems}. Dover Publications, INC. New York. 1966.

\bibitem{aplication} X. Guo, M. Xu;
\emph{Some physical applications of fractional Schr\"odinger equation},
Journal of Mathematical Physics 47. 2006. 082104.

\bibitem{Gua} B. Guo, Z. Huo;
\emph{Global Well-Posedness for the Fractional Nonlinear Schr\"odinger Equation,}
Communications in Partial Differential Equations. 2010

\bibitem{Ionescu} A. D. Ionescu, F. Pusateri;
\emph{Nonlinear fractional Schr\"odinger equations in one dimension}.
Journal of Functional Analysis 266. 2014.

\bibitem{bookhk} N. Hayashi, E. Kaikina;
\emph{Nonlinear theory of pseudodifferential equations on a half-line}.
North-Holland Mathematics Studies, 194. Elsevier Science B. V., Amsterdam,
 2004, 319 pp.

\bibitem{Naumkin} Nako Hayashi, Pavle I. Naumkin;
\emph{Asymptotics of small solutions to nonlinear Schr\"odinger equations with cubic
nonlinearities}. Int. J. Pure Appl. Math 3(2002), no. 3, 255-273.

\bibitem{libroNHKS} N. Hayashi, E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev;
\emph{Asymptotics for Dissipative Nonlinear Equations}, Lecture Notes in
Math., vol. 1884,Springer-Verlag, Berlin (2006) 557 pp.

\bibitem{Holmer} J. Holmer;
\emph{The initial boundary value problem for the 1D nonlinear Schr\"odinger
equation on the half-line}, Differential Integral Equations 18 (6) (2005), 647--668.

\bibitem{Ekaikina} E. Kaikina;
\emph{Fractional derivative of Abel type on
a Half Line}. Transactions of the American Mathematical Society. Vol. 364,
No. 10, October 2012, Pages 5149-5172.

\bibitem{Fractal1} E. I. Kaikina;
\emph{A new unified approach to study fractional PDE equations on a half-line}.
Complex Var. Elliptic Equ. 58 (2013), no. 1, 55--77.

\bibitem{Kaikina} E. I. Kaikina;
\emph{Asymptotics for inhomogeneous Dirichlet initial-boundary value problem for
the nonlinear Schr\"odinger equation.} J. Math. Phys. 54 (2013), no. 11 111504,
15 pp.

\bibitem{Laskin} N. Laskin;
\emph{Fractional Schr\"odinger equation.}
Phys. Rev. E (3) 66 (2002), no. 5, 056108, 7 pp.

\bibitem{potencial2} E. K. Lenzi, H. V. Ribeiro, M. A. F. dos Santos, R.
Rossato, R. S. Mendes;
\emph{Time dependent solutions for a fractional Schr\"odinger equation with
delta potentials}. J. Math. Phys. \textbf{54}
(2013), no. 8, 082107, 8 pp.

\bibitem{Kotlyarov} A. Monvel, V. Kotlyarov, D. Shepelsky;
\emph{Decaying long-time asymptotics for the focusing NLS equation with periodic
boundary condition}, Stud. Appl. Math. 129 (3) (2012), 249--271.

\bibitem{ozawa} T. Ozawa;
\emph{Long range scattering for nonlinear Schr\"odinger equations in one space
dimension}, Commun. Math. Phys., \textbf{139}(1991), pp. 479-493. 28 (10) (2005),
1237--1255.

\bibitem{SA} J. Sabatier, O. P. Agrawal, J. A. T. Machado;
\emph{Advances in Fractional Calculus: Theoretical Developments and Applications
in Physics and Engineering}, Springer, Netherlands, 2007.

\bibitem{Samko} S. G. Samko, A. A. Kilbas, O. I. Marichev;
\emph{Fractional Integrals and Derivatives}. Theory and Applications,
Gordon and Breach, Yverdon (1993).

\bibitem{SG} E. Scalas, R. Gorenflo, F. Mainardi;
\emph{Fractional calculus and continuous-time finance},
Physica A: Statistical Mechanics and its
Applications 284 (1--4) (2000), 376--384.

\bibitem{Weder} R. Weder;
\emph{The forced nonlinear Schr\"odinger equation with
potential on the half-line}, Math. Methods Appl. Sci.

\bibitem{Dan} D. Wu;
\emph{Existence and stability of standing waves for
nonlinear fractional Schr\"odinger equations with Hartree type nonlinearity},
 J. Math. Anal. Appl., 411. 2014.

\bibitem{S} W. R. Schneider, W. Wyss;
\emph{Fractional diffusion and wave equations},
Journal of Mathematical Physics 30 (1) (1989), 134--144.

\bibitem{SW} W. Wyss;
\emph{The fractional diffusion equation}, Journal of
Mathematical Physics 27 (11) (1986), 2782--2785.

\end{thebibliography}

\end{document}