\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 13, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/13\hfil Fourth-order Schr\"odinger equation]
{On the Schr\"odinger equations with isotropic and anisotropic fourth-order 
dispersion}

\author[E. J. Villamizar-Roa, C. Banquet \hfil EJDE-2016/13\hfilneg]
{Elder J. Villamizar-Roa, Carlos Banquet}

\address{Elder J. Villamizar-Roa \newline
Universidad Industrial de Santander,
Escuela de Matem\'aticas,
A.A. 678, Bucaramanga, Colombia}
\email{jvillami@uis.edu.co}

\address{Carlos Banquet \newline
 Universidad de C\'ordoba, Departamento de Matem\'aticas y  Estad\'istica,
 A.A. 354, Monter\'ia, Colombia}
\email{cbanquet@correo.unicordoba.edu.co}

\thanks{Submitted August 22, 2015. Published January 7, 2016.}
\subjclass[2010]{35Q55, 35A01, 35A02, 35C06}
\keywords{Fourth-order Schr\"{o}dinger equation; biharmonic equation;
\hfill\break\indent local and global solutions}

\begin{abstract}
 This article concerns the Cauchy problem associated with the nonlinear
 fourth-order Schr\"odinger equation with isotropic and anisotropic
 mixed dispersion. This model is given by the equation
 $$
 i\partial_tu+\epsilon \Delta u+\delta A u+\lambda|u|^\alpha u=0,\quad
 x\in\mathbb{R}^{n},\; t\in \mathbb{R},
 $$ 
 where $A$ is either the operator $\Delta^2$ (isotropic dispersion)
 or $\sum_{i=1}^d\partial_{x_ix_ix_ix_i}$, $1\leq d<n$
 (anisotropic dispersion), and $\alpha, \epsilon, \lambda$ are  real
 parameters. We obtain local and global well-posedness results in
 spaces of initial data with low regularity, based on weak-$L^p$ spaces.
 Our analysis also includes the biharmonic and anisotropic biharmonic
 equation $(\epsilon=0)$; in this case, we obtain the existence of
 self-similar solutions because of their scaling invariance property.
 In a second part, we analyze the convergence of solutions for the 
 nonlinear fourth-order Schr\"odinger equation
 $$
 i\partial_tu+\epsilon \Delta u+\delta \Delta^2 u+\lambda|u|^\alpha u=0,
 \quad x\in\mathbb{R}^{n},\; t\in \mathbb{R},
 $$
 as $\epsilon$ approaches zero, in the $H^2$-norm, to the solutions of
 the corresponding biharmonic equation
 $i\partial_tu+\delta \Delta^2 u+\lambda|u|^\alpha u=0$.
\end{abstract}

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

\section{Introduction}

This article is devoted to the study of the Cauchy problem associated with 
the  fourth-order Schr\"{o}dinger equation
in $\mathbb{R}^n\times \mathbb{R}$,
\begin{equation}\label{FoSch}
\begin{gathered}
i\partial_tu+\epsilon \Delta u+\delta A u+f(|u|)u=0, \quad
 x\in \mathbb{R}^{n},\; t\in \mathbb{R}, \\
u(x,0)=u_0(x), \quad x\in \mathbb{R}^{n}, \\
\end{gathered}
\end{equation}
where the unknown $u(x,t)$ is a complex-valued function in space-time 
$\mathbb{R}^n\times \mathbb{R}$, $n\geq 1$, $u_0$ denotes the initial
data and  $\epsilon$, $\delta$, are real parameters. 
The operator $A$ is defined by
\begin{equation}\label{DefOpe}
A u=\begin{cases}
\Delta^2u=\Delta\Delta u, &\text{(isotropic dispersion)}, \\
\sum_{i=1}^du_{x_ix_ix_ix_i},\; 1\leq d<n, &\text{(anisotropic dispersion)}. 
\end{cases}
\end{equation}
The nonlinear term is given by  $f(|u|)u$, where
 $f:\mathbb{R}\to \mathbb{R}$ satisfies
 \begin{equation}\label{fprop}
|f(x)-f(y)|\leq C_f|x-y|(|x|^{\alpha-1}+|y|^{\alpha-1}),
\end{equation}
for some $1\leq \alpha<\infty$, $f(0)=0$, and the constant $C_f >0$
is independent of $x,y \in \mathbb{R}$. A typical case of a function
$f$ is  $f(x)=| x|^\alpha$.

The class of fourth-order Schr\"{o}dinger equations has been widely
used in many branches of  applied science such as nonlinear
optics, deep water wave dynamics, plasma physics, superconductivity,
quantum mechanics and so on \cite{AceAngTur, Dysthe, Hirota, Ivano,
Kar, KarSha, WenFan}. If we consider $\epsilon=0$ in \eqref{FoSch},
the resulting equation is the fourth-order nonlinear Schr\"{o}dinger
equation
\begin{equation}\label{FoNLS}
i\partial_tu+\delta A u+f(|u|)u=0.
\end{equation}
In particular, if we take $A=\Delta^2$ in \eqref{FoNLS} we obtain the well-known
biharmonic equation 
\begin{equation}\label{BNLS}
i\partial_tu+\delta \Delta^2 u+f(|u|)u=0,
\end{equation}
introduced by Karpman \cite{Kar}, and Karpman and Shagalov
\cite{KarSha} to take into account the role played by the higher
fourth-order dispersion
 terms in formation and propagation of intense laser beams in a bulk medium 
with Kerr nonlinearity  \cite{Ivano}.
 Historically, \eqref{BNLS} has been extensively studied in Sobolev spaces, 
see for instance
\cite{Fibich,Miao,Miao1,Pau2,Pau,Pau1,Segata,Wang,ZhuYanZha} and references
therein. Fibich et al \cite{Fibich} established sufficient conditions for 
the  existence of global solutions to \eqref{BNLS}, for $\delta<0$  and $\delta>0$,
with initial data in $H^2(\Omega)$ being $\Omega$ a smooth bounded
domain of $ \mathbb{R}^n$. Global existence and scattering theory
for the defocusing biharmonic equation, in $H^2(\mathbb{R}^n)$, was established in
Pausander \cite{Pau2, Pau}.
 Wang in \cite{Wang} showed the global existence of
solutions  and a scattering result for biharmonic equation (with a
nonlinearity of the form $|u|^pu$) with
 small initial radial data in the homogeneous Sobolev space
 ${\dot{H}^{s_c}(\mathbb{R}^n)}$ and
 dimensions $n\geq 2$. Here $s_c=\frac{n}{2}-\frac{4}{p}$ and 
$s_c>-\frac{3n-2}{2n+1}$.
 The main ingredient of \cite{Wang} is the improvement of 
 the Strichartz estimatives associated with \eqref{BNLS} for radial initial data; 
see also Zhu,  Yang and Zhang \cite{ZhuYanZha}, where some results on blow-up 
solitons for biharmonic  equation are established. 
More recently, Guo in \cite{Guo6} analyzed the 
existence of global solutions in Sobolev spaces and the asymptotic behavior
for the Cauchy problem associated with \eqref{BNLS}  with combined
power-type nonlinearities. Finally, we recall a recent result of
Miao et al \cite{Miao} about the defocusing
energy-critical nonlinear biharmonic equation  $iu_t
+ \Delta^2u = -|u|^{\frac{8}{d-4}}u$, which establishes that any
finite energy solution is global and scatters both forward and
backward in time for dimensions $d\geq 9$.

When  $\epsilon\neq 0$ and $A$ is the biharmonic operator, equation 
\eqref{FoSch} corresponds to the
following nonlinear Schr\"{o}dinger equation with isotropic
mixed-dispersion:
\begin{equation}\label{INLS}
i\partial_tu+\epsilon \Delta u+\delta\Delta^2 u+f(|u|)u=0.
\end{equation}
This equation was also introduced by Karpman \cite{Kar}, and Karpman
and Shagalov \cite{KarSha}, and it has been used as a model to
investigate the role played by the higher-order dispersion terms, in
formation and propagation of solitary waves in magnetic materials
where the effective quasi-particle mass becomes infinite. From the
mathematical point of view, equation \eqref{INLS}  has been studied
extensively in Sobolev and Besov spaces,
 see for instance \cite{GuoCui4,GuoCui3,Guo,GuoCui5,Fibich} and some 
references therein.
Fibich  et al \cite{Fibich} investigated the  existence of
global solutions to \eqref{INLS} in the class $C(\mathbb{R};
H^2(\mathbb{R}^n))$ by using the conservation laws. Moreover, the dynamic 
of the solutions and numerical simulations were also analyzed. 
These results were improved by Guo and Cui in
\cite{GuoCui4}. Local well-posedness of the Cauchy problem associated with
\eqref{INLS} in Sobolev spaces $H^s(\mathbb{R}^n)$, with 
$f(u)=|u |^\alpha$, $\frac{\alpha}{2}\geq\frac{4}{n}$,
$s>s_0:=\frac{n}{2}-\frac{4}{\alpha}$, was obtained by Cui and
Guo in \cite{GuoCui5}. Additionally,  by using the
local existence and the conservation laws, a global well-posedness results
in $H^2(\mathbb{R}^n)$ was also established. 
In \cite{GuoCui3} the authors proved some
results of local and global well-posedness on Besov spaces for
 dimensions $1\leq n\leq 4$; more exactly, the authors proved that
 the Cauchy problem associated with \eqref{INLS}, with $f(u)=| u |^\alpha$, 
is local well possed in
 $C([-T,T];\dot{B}^{s_\alpha}_{2,q}(\mathbb{R}^n))$ and
 $C([-T,T];B_{2,q}^s(\mathbb{R}^n))$ for some $T>0$, where
 $s_\alpha=\frac{n}{2}-\frac{4}{\alpha}$, $s>s_\alpha$, $1\leq q \leq \infty$.
With respect to the global well-posedness in Sobolev space,  Guo in \cite{Guo}, 
considering $f(u)=| u |^{2m}$, and using the I-method, proved the existence 
of global solutions in $H^s(\mathbb{R}^n)$ for 
$s>1+\frac{mn-9+\sqrt{(4m-mn+7)^2+16}}{4m}$, $4<mn<4m+2$.

Another important model considered  in \eqref{FoSch} is given by the 
case of anisotropic dispersion, that is,
\begin{equation}\label{ANLS}
i\partial_tu+\epsilon \Delta u+\delta\sum_{i=1}^du_{x_ix_ix_ix_i}
+f(|u|)u=0.
\end{equation}
This model appears in the propagation of ultrashort laser pulses in a
 planar wave\-guide medium with
anomalous time-dispersion, and the propagation of solitons in fiber 
arrays (see Wen and Fan \cite{WenFan}
and Acevedes  {\it et al} \cite{AceAngTur}).  Results of local and global 
well-posedness for initial data in $H^s$-spaces were given in \cite{GuoCui5} 
and \cite{ZhaGuoSheWei}

In this article we are interested in the local and global well-posedness 
of the general fourth-order Schr\"{o}dinger equation outside the framework 
of finite energy $H^s$-spaces. More exactly, we analyze the existence of local
 and global solutions for the Cauchy problem \eqref{FoSch} in a new class 
of initial data based on
weak-$L^p$ spaces. Weak-$L^p$ spaces, also denoted by $L^{(p,\infty)}$, 
are natural extensions of Lebesgue
spaces $L^p$, in view of the Chebyshev inequality \cite{BL}. 
They contain singular functions with infinite $L^2$-mass such as 
homogeneous functions of degree $-\frac{n}{p}$. However, 
$L^{(p,\infty)}\subset L^2_{\rm loc}$ for $p>2$.
 Making a comparison between weak-$L^p$ spaces and $H^{s,l}$-spaces, 
it is known that the continuous inclusion 
$H^{s,l}(\mathbb{R}^{n})\subset L^{(p,\infty)}(\mathbb{R}^{n})$
holds for $s\geq0$ and $\frac{1}{p}\geq\frac{1}{l}-\frac{s}{n}$, and
$H^{s,l}$-spaces do not contain any weak-$L^{p}$ spaces if $s\in\mathbb{R}$,
$1\leq l\leq2$ and $l\leq p$. In particular, $L^{(p,\infty)}(\mathbb{R}%
^{n})\not \subset H^{s,2}(\mathbb{R}^{n})=H^{s}(\mathbb{R}^{n})$ for all
$s\in\mathbb{R}$, when $p\geq2$. On the other hand, comparing equations 
\eqref{FoNLS} with \eqref{INLS} and \eqref{ANLS},
we observe that equation \eqref{FoNLS}, with $f(| u |)=|
u|^\alpha$, unlike equations \eqref{INLS} and \eqref{ANLS}, is invariant 
under the group of transformations
$u(x,t)\to u_\lambda(x,t)$, where
$u_\lambda(x,t)=\lambda^{\frac{4}{\alpha}}u(\lambda x,\lambda^4t)$,
$\lambda>0$. Solutions which are invariant under the transformation
$u\to u_\lambda$ are called self-similar solutions.
As pointed out in Dudley  et al \cite{Dudley} (see also \cite{LucEld1}), 
self-similarity type properties appear in a wide range of physical situations and they reproduce the structure of a phenomena in different spatio-temporal scales. A universal law governing self-similar scale invariance reveals the existence of internal symmetry and structure in a system. Thus, self-similar solutions naturally provide such a law for system \eqref{FoNLS}. In ultrafast nonlinear optics, self-similar dynamics have attracted a lot of interest and constitute an increasing field of research (see  \cite{Dudley} and references therein). For instance, in Fermann {\it et al.} \cite{Fermann} was showed that a type of self-similar parabolic pulse is an asymptotic solution to a nonlinear Schr\"{o}dinger equation with gain. In
order to obtain self-similar solutions we need to consider a norm
$\| \cdot\|$ defined on a space of initial data $u_0$, which
is invariant with respect to the group of transformations
$u\to u_\lambda$, that is, 
$\| {u_0}_\lambda\|=\|{u_0}\|$ for all $\lambda>0;$ therefore $u_0$ must be a
homogeneous function of degree $-\frac{4}{\alpha}$. However, $H^s$-spaces 
are not well adapted for studying this kind of solutions.
This fact represents an additional motivation to study the existence of 
global solutions of the Cauchy problem associated with \eqref{FoNLS} with 
initial data outside  $H^s$-spaces, by using norms based on $L^{(p,\infty)}$. 
As consequence, the existence of forward
self-similar solutions for \eqref{FoNLS} is obtained by assuming
$u_0$ a sufficiently small homogeneous function of degree
$-\frac{4}{\lambda}$. Because equation appearing in \eqref{FoSch} does not
verify any scaling symmetries (in particular equations \eqref{INLS} 
and \eqref{ANLS}), it is not likely to possess self-similar solutions. 
However, by using time decay estimates for the
respective fourth-order Schr\"{o}dinger group in weak-$L^p$ spaces,
we are able to obtain a result of existence of global solutions for
the Cauchy problem \eqref{FoSch} in a class of function spaces
generated by the scaling of the biharmonic equation \eqref{BNLS}
with $f(| u |)=| u|^\alpha$. In relation to the
existence of local in time solutions for \eqref{FoSch} and in
particular, the Cauchy problem associated with the equation
\eqref{FoNLS}, we will prove a result of existence and uniqueness
for a large class of singular initial data, which includes
homogeneous functions of degree $-\frac n p$ for adequate values of $p$.
 The solutions obtained here can be physically interesting because, 
as was said, elements of $L^{(p,\infty)}$ have local finite $L^2$-mass 
(that is, they belong to $L^2_{\rm loc}$), for $p > 2$.  
In addition, for initial data in $H^s(\mathbb{R}^n)$, the corresponding 
solution belongs to $H^s(\mathbb{R}^n)$, which shows that the constructed 
data-solution map in $L^{(p,\infty)}$ recovers the $H^s$-regularity and 
it is compatible with the $H^s$-theory.

It is worthwhile to remark that the existence of local and global
solutions for dispersive equations with initial data outside the
context of finite $L^2$-mass, such as weak-$L^r$ spaces, has been
analyzed for the classical Schr\"{o}dinger equation, coupled
Schr\"{o}dinger equations, Davey-Stewartson system, which are
models characterized by having scaling relation 
(cf. \cite{WC, LucEld1, LucEldPab,  EldJean}). Existence of solutions in the
framework of weak-$L^r$ spaces for models which have no scaling
relation, have been explored in the case of Boussinesq and
Schor\"{o}dinger-Boussinesq system in \cite{CaLuEld, Fer-Bousq} 
and more recently, in the context of Klein-Gordon-Schr\"{o}dinger 
system \cite{CaLuEld1}.


To state our results, we  establish the definition of mild solution 
for the Cauchy problem \eqref{FoSch}. A mild solution for \eqref{FoSch} 
is a function  $u$ satisfying the  integral equation
\begin{equation}\label{IntEqu}
u(x,t)=G_{\epsilon,\delta}(t)u_0(x)
+i\int_0^tG_{\epsilon,\delta}(t-\tau)f(|u(x,\tau)|)u(x,\tau)d\tau,
\end{equation}
where  $G_{\epsilon,\delta}(t)$ is the free group associated with the
linear Fourth-order Schr\"{o}dinger equation, that is,
\begin{equation}\label{DefGe}
G_{\epsilon,\delta}(t)\varphi=\begin{cases}
J_{\epsilon,\delta}(\cdot,t)\ast\varphi, &\text{if } A=\Delta^2,\\
I_{\epsilon,\delta}(\cdot,t)\ast\varphi, &\text{if }
A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i},
\end{cases}
\end{equation}
 for all $\varphi\in \mathcal{S}'(\mathbb{R}^n)$, where
\begin{align*}
J_{\epsilon,\delta}(x,t)
&=(2\pi)^{-n}\int_{\mathbb{R}^n}e^{ix \xi 
 -it\left(\epsilon|\xi|^2-\delta|\xi|^4\right)}d\xi\\
I_{\epsilon,\delta}(x,t)
&=\Big((2\pi)^{-d}\prod_{j=1}^d
\int_{\mathbb{R}}e^{ix_j\xi_j-it(\epsilon\xi_j^2-\delta\xi_j^4)}d\xi_j\Big)\\
&\quad \times \Big((2\pi)^{-(n-d)}\prod_{j=d+1}^n\int_{\mathbb{R}}e^{ix_j\xi_j
 -it\epsilon\xi_j^2}d\xi_j\Big)\\
&\equiv I^1_{\epsilon,\delta}(x,t)I^2_{\epsilon,\delta}(x,t).
\end{align*}

Before to precise our results, briefly we recall some notation and
facts about Lorentz spaces, see Bergh and L\"{o}fstr\"{o}m \cite{BL}, 
which will be our scenario to
establish existence results. Lorentz spaces $L^{(p,d)}$ are
defined as the set of measurable function $g$ on $\mathbb{R}^{n}$
such that the quantity
\begin{equation*}
\| g\|_{(p,d)}=\begin{cases}
\Big( \frac{p}{d}\int_0^{\infty }[
t^{1/p}g^{\ast \ast }(t)] ^{d}\frac{dt}{t}\Big)^{1/d}, &
\text{if }  1<p<\infty, \; 1\leq d<\infty, \\
\sup_{t>0}t^{1/p}g^{\ast \ast }(t), & 
\text{if }  1<p\leq \infty ,\; d=\infty ,
\end{cases}
\end{equation*}
is finite. Here $g^{\ast \ast }(t)=\frac{1}{t}\int_0^{t}g^{\ast }(s)\text{ }ds$
and
\begin{equation*}
 g^{\ast}(t)=\inf \{s>0:\mu ( \{{x}\in \Omega :|g({x})|>s\}) \leq t\},\quad t>0,
\end{equation*}
with $\mu $ denoting the Lebesgue measure. In particular,
$L^{p}(\Omega)=L^{(p,p)}(\Omega )$ and, when 
$d=\infty$, $L^{(p,\infty )}(\Omega )$ are called weak-$L^{p}$ spaces.
Furthermore, $L^{(p,d_1)}\subset L^{p}\subset L^{(p,d_{2})}\subset
L^{(p,\infty )}$ for $1\leq d_1\leq p\leq d_{2}\leq \infty $. In
particular, weak-$L^{p}$ spaces contain singular functions with
infinite $L^{2}$-mass such as homogeneous functions of degree
$-\frac n p$.  Finally, a helpful fact about Lorentz
spaces is the validity of the H\"{o}lder inequality, which reads
\begin{equation*}
\| gh\|_{(r,s)}\leq C(r)\| g\|_{(p_1,d_1)}\|
h\|_{(p_{2},d_{2})},
\end{equation*}
for $1<p_1\leq \infty $, $1<p_{2}$, $r<\infty $,
$\frac{1}{p_1}+\frac{1}{p_{2}}<1$,
$\frac{1}{r}=\frac{1}{p_1}+\frac{1}{p_{2}}$, and
$s\geq 1$ satisfies $\frac{1}{d_1}+\frac{1}{d_{2}}\geq \frac{1}{s}$.

In this paper we obtain new results for the existence of local and global  solutions
to Schr\"{o}dinger equations with isotropic
and anisotropic fourth-order dispersion. First, we prove the existence 
of local-in-time solutions to the
integral equation \eqref{IntEqu} (see Theorem
\ref{LocalTheo}).
For the existence of local solutions, fixed $0<T<\infty$, we
consider the space $\mathcal{G}_{\beta}^T$ of Bochner measurable
functions $u:(-T,T)\to L^{(p(\alpha+1),\infty)}$ such that
\begin{equation*}
\|u\|_{\mathcal{G}_{\beta}^T}=\sup_{-T<t<T}|t|^{\beta}\|u(t)\|_{(p(\alpha+1),\infty)},
\end{equation*}
where
\begin{equation}\label{Defbeta}
\beta=\begin{cases}
\frac{n\alpha}{4p(\alpha+1)}, &\text{if } A=\Delta^2,\\[4pt]
\frac{(2n-d)\alpha}{4p(\alpha+1)}, &\text{if }A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i},
\end{cases}
\end{equation}
and $p$ is such that the pair $(\frac{1}{p},\frac{1}{p(\alpha+1)})$
belongs to  the set $\Xi_0\setminus\partial \Xi_0$ where $\Xi_0$ is
the quadrilateral $R_0P_0BQ_0$, with $ B=(1,0)$, $P_0=(2/3,0)$,
$Q_0=(1,1/3)$ and $R_0=( 1/2,1/2)$. The exponent
$\beta$ in  \eqref{Defbeta}, and the restriction of $p$, correspond
to the time decay of the group $G_{\epsilon, \delta}(t)$ on Lorentz spaces
(see Proposition \ref{LinEstLoc} below). The initial data is such
that $\|G_{\epsilon,\delta}(t)u_0\|_{\mathcal{G}_{\beta}^T}$ is
finite. As a consequence, some results of local existence in Sobolev 
spaces can be recovered (see Remark \ref{rem1}).

We also analyze the existence of
global-in-time solutions (see Theorem \ref{GlobalTheo}). For that we
define the space $\mathcal{G}^{\infty}_{\sigma}$ as the set of
Bochner measurable functions 
$u:(-\infty ,\infty )\to L^{(\alpha+2,\infty )}$ such that
\[
\| u\|_{\mathcal{G}^{\infty}_{\sigma}}=\sup_{-\infty <t<\infty}|t|^\sigma\|
u(t)\|_{(\alpha+2,\infty )}<\infty,
\]
where $\sigma$ is  given by
\begin{equation}\label{Defsigma}
\sigma=\begin{cases}
\frac{1}{\alpha}-\frac{n}{4(\alpha+2)}, &\text{if } A=\Delta^2,\\[4pt]
\frac{1}{\alpha}-\frac{2n-d}{4(\alpha+2)}, &\text{if }
A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}.
\end{cases}
\end{equation}
Observe that the value
$\sigma=\frac{1}{\alpha}-\frac{n}{4(\alpha+2)}$ in \eqref{Defsigma}
is the unique one such that the norm 
$\|u\|_{\mathcal{G}^{\infty}_{\sigma}}$ becomes invariant by the
scaling of  biharmonic equation with $f(u)=| u |^\alpha$. In order
to obtain existence of global solutions,  we consider the following
class of initial data
\begin{equation}\label{initial_data}
\mathcal{D}_\sigma\equiv\{\varphi\in \mathcal{S}'(\mathbb{R}^n):
\sup_{-\infty <t<\infty}t^\sigma\|
G_{\epsilon,\delta}(t)\varphi\|_{(\alpha+2,\infty)}<\infty\}.
\end{equation}
Consequently, if we consider the biharmonic or anisotropic 
biharmonic equation, that is, $\epsilon=0$ in \eqref{FoSch}, we obtain the
existence of self-similar solutions by assuming $u_0$ a sufficiently
small homogeneous function of degree $-\frac{4}{\alpha}$ 
(see Corollary \ref{self}).

As it was said, formally, when we drop the second order  dispersion
term in $i\partial_tu+\epsilon \Delta u+\delta \Delta^2
u+f(|u|)u=0$, that is,  taking $\epsilon=0$, we obtain the biharmonic
equation $i\partial_tu+\delta \Delta^2 u+f(|u|)u=0$.
 However, to the best of our
knowledge, the vanishing second order dispersion limit has not been
addressed. We observe that the analysis of vanishing dispersion
limits can be seen as an interesting issue in dispersive PDE theory,
because it permits to describe qualitative properties between
different models. We recall, for instance, that in fluid mechanics,
the vanishing viscosity limit of the incompressible Navier-Stokes
equations is a classical issue \cite{FPV,Iftimie}. This is the
motivation of the second aim of this paper. We study the convergence as $\epsilon$
goes to zero, in the $H^2$-norm, of the solution of Cauchy problem
\eqref{FoSch}, with $A=\Delta^2$, to the corresponding Cauchy
problem associated with the biharmonic equation \eqref{BNLS}. In the anisotropic case, that is,
$A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}$, the vanishing second order dispersion limit is not clear, because we are not able to bound $\| \nabla u_\epsilon\|_{L^2}$  or $ \| u_\epsilon\|^2_{H^1}+\sum_{i=1}^d\|  u_{\epsilon_{x_ix_i}}\|^2_{L^2}$ in terms of the conserved quantities associated  to \eqref{FoSch} and  independently of $\epsilon$ (see Remark \ref{AniDisp}). This is an interesting question to be considered as future research.

The rest of this article is organized as follows. 
In Section 2 we establish some linear and nonlinear estimates which 
are fundamental for obtaining our results of local and global mild solutions.
In Section 3 we state and prove our results of local and global
solutions. Finally, in Section 4, we give a result about vanishing second order
dispersion limit.

\section{Linear and nonlinear estimates}

In this section we establish some linear and nonlinear estimates which are 
fundamental for obtain our results of local and global mild solutions.
We start by rewriting Theorem 2, Section 3, of Cui \cite{Cui1} for the
case $n = 1$ and Theorem 2, Section 3, of Cui \cite{Cui2}
for the case $n \geq 2$ (see also Lemma 2.1 in Guo and Cui
\cite{GuoCui, GuoCui5}). For this purpose we denote $\Xi_0$ the quadrilateral
$R_0P_0BQ_0$ in the $(1/p,1/q)$ plane, where
\begin{equation*}
B=(1,0), \quad P_0=(2/3,0),\quad Q_0=(1,1/3), \quad  R_0=( 1/2,1/2).
\end{equation*}
$\Xi_0$ comprises the apices $B, R_0$ and all the edges $BP_0$,
$BQ_0$, $P_0R_0$ and $Q_0R_0$, but does not comprise the apices
$P_0$ and $Q_0$.

\begin{proposition}\label{LemmaCui} 
Given  $T>0$ and a pair of positive numbers $(p,q)$ satisfying 
$(1/p,1/q)\in\Xi_0$, there exists a  constant $C=C(T,p,q)>0$ such that for 
any $\varphi\in L^p(\mathbb{R}^n)$ and $-T\leq t\leq T$ it holds
\[
\| G_{\epsilon,\delta}(t)\varphi\|_{L^q}
\leq C|t|^{-b_l}\| \varphi \|_{L^{p}},
\]
where
\begin{equation}\label{e1}
b_l=\begin{cases}
\frac{n}{4}\big(\frac{1}{p}-\frac{1}{q}\big), &\text{if } A=\Delta^2,\\[4pt]
\frac{2n-d}{4}\big(\frac{1}{p}-\frac{1}{q}\big), &\text{if }
A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}.
\end{cases}
\end{equation}
Moreover, if $\epsilon=0$ the above estimate holds for all
$t\neq 0$.
\end{proposition}

The above inequality is not convenient to obtain a result of global
well-posedness  because the constant $C$ depends on $T$. To
overcome this problem we establish a different result which follows
from a standard scaling argument.

\begin{lemma}\label{LemmaNoT}
If $\frac 1 p+\frac 1{p'}=1$ with $p\in[1,2]$, then there exists a constant $C$
independent of $\epsilon,\delta$ and $t$ such that
\[
\|G_{\epsilon,\delta}(t)\varphi\|_{L^{p'}}
\leq C |t|^{-b_g}\|\varphi\|_{L^p},\quad \varphi\in L^p(\mathbb{R}^n),
\]
for all $t\neq 0$, where
\begin{equation}\label{e2}
b_g=\begin{cases}
\frac{n}{4}(\frac{2}{p}-1), & \text{if } A=\Delta^2,\\[4pt]
\frac{2n-d}{4}(\frac{2}{p}-1), & \text{if } 
A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}.
\end{cases}
\end{equation}
\end{lemma}

\begin{proof} 
It is clear that
$\|G_{\epsilon,\delta}(t)\varphi\|_{L^2}=\|\varphi\|_{L^2}$, in both
cases, the isotropic and anisotropic dispersion.  Now,  for the
isotropic case, we define $h(\xi):=\frac{z\xi}{t}-(\epsilon
\xi^2-\delta\xi^4)$ and since $|h^{(4)}(\xi)|=24$, we can use
\cite[Proposition VIII. 2]{SteinLibro1}   to obtain
\[
 \Big|\int_{-\infty}^{\infty}e^{ith(\xi)}d\xi\Big|
\leq C|t|^{-1/4}.
\]
Note that the constant $C$ given above does not depend on
$\epsilon$ and $\delta$. From Young inequality we have 
\[
\|G_{\epsilon,\delta}(t)\varphi\|_{L^\infty}\leq C
|t|^{-1/4}\|\varphi\|_{L^1}.
\]
 Then the result follows by real interpolation.

 The anisotropic case is obtained in a similar
way. Indeed, we only need to note that
\[
|I^1_{\epsilon,\delta}(x,t)|\leq C_1|t|^{-d/4}\quad\text{and}\quad
|I^1_{\epsilon,\delta}(x,t)|\leq C_2|t|^{-(n-d)/2},
\]
where $C_1$ and $C_2$ are independent of $t, \epsilon$ and $\delta$. 
Consequently
\[
|I_{\epsilon,\delta}(x,t)|=|I^1_{\epsilon,\delta}(x,t)I^2_{\epsilon,\delta}(x,t)|
\leq C|t|^{-\frac{2n-d}{4}}.
\]
The proof is finished.
\end{proof}

\begin{lemma}\label{LinEstLoc} 
Let $T>0$, $1\leq d\leq \infty $  and  $1\leq p,q\leq \infty$ 
satisfying $(1/p,1/q)\in\Xi_0\setminus \partial \Xi_0$. 
Then, there exists a positive
constant $C=C(T,p,q)>0$ such that
\begin{equation}
\| G_{\epsilon,\delta}(t)\varphi\|_{(q,d)}
\leq C| t|^{-b_l}\| \varphi\|_{(p,d)},
\end{equation}
for all  $-T\leq t\leq T$ and $\varphi\in L^{(p,d)}$. Here $b_l$ is
defined in \eqref{e1}. Moreover, if $\epsilon=0$ the above
estimate holds for all $t\neq 0$.
\end{lemma}

\begin{proof} 
We prove only the isotropic case; the anisotropic case 
can be proved in an analogous way.
Since $\Xi_0$ is convex we can chose $(1/{p_0},1/{q_0})$,
$(1/{p_1},1/{q_1})\in \Xi_0$  such that 
$\frac{1}{p}=\frac{\theta }{p_0}+\frac{1-\theta }{p_1}$ and
$\frac{1}{q}=\frac{\theta }{q_0}+\frac{1-\theta }{q_1}$, with $0<\theta <1$.
From Proposition \ref{LemmaCui} we have 
$G_{\epsilon,\delta}(t):L^{p_0}\to L^{q_0}$ and
$G_{\epsilon,\delta}(t):L^{p_1}\to L^{q_1}$, with
norms bounded by
\begin{gather*}
\| G_{\epsilon,\delta}(t)\|_{p_0\to q_0}\leq C| t| ^{-n/4(1/{p_0}-1/{q_0})},\\
\| G_{\epsilon,\delta}(t)\|_{p_1\to q_1}\leq C| t| ^{-n/4(1/{p_1}-1/{q_1})}.
\end{gather*}
Since $L^{p}=L^{(p,p)}$, using real interpolation we obtain
\begin{align*}
\| G_{\epsilon,\delta}(t)\|_{(p,d)\to (q,d)}
&\leq C| t|^{-n/4(1/{p_0}-1/{q_0})\theta}| t| ^{-n/4(1/{p_1}-1/{q_1})(1-\theta)}\\
&=C| t| ^{-n/4(1/p-1/q)},
\end{align*}
which completes the proof.
\end{proof}

In the same spirit of Lemma \ref{LinEstLoc} one can obtain the next result,
 which gives a linear estimate in Lorentz spaces. The proof follows 
from Lemma \ref{LemmaNoT} and real interpolation. We omit it.

\begin{lemma}\label{LinEstGlo} 
Let $1\leq d\leq \infty $, $1< p< 2$ and $p^{\prime }$ such that 
$\frac{1}{p}+\frac{1}{p^{\prime }}=1$. Then, there exists a positive
constant $C$ such that
\begin{equation}
\| G_{\epsilon,\delta}(t)\varphi\|_{(p^{\prime },d)}\leq
C| t|^{-b_g}\| \varphi\|_{(p,d)},
\end{equation}
for all $t\neq 0$ and $\varphi\in L^{(p,d)}$. Here $b_g$ is defined
in \eqref{e2}.
\end{lemma}

For the rest of this article, we denote the nonlinear part of the integral 
equation \eqref{IntEqu} by
\begin{equation*}
\mathcal{F}(u)=i\int_0^t
G_{\epsilon,\delta}(t-\tau)f(|u(x,\tau)|)u(x,\tau)d\tau.
\end{equation*}
In the next lemma we estimate the nonlinear term $\mathcal{F}(u)$ in
the norm $\|\cdot \|_{\mathcal{G}_{\sigma}^{\infty }}$, which
is crucial in order to obtain existence of global mild solutions.

\begin{lemma}\label{EstNonGlo}
Let $1\leq \alpha<\infty $ and  assume that $(\alpha +1)\sigma<1$.
Then

(1) If $\frac{n\alpha}{4(\alpha+2)}<1$ and $A=\Delta^2$, then there exists 
a constant $C_1>0$ such that
\begin{equation} \label{Global-estim}
\begin{aligned}
& \| \mathcal{F}(u)-\mathcal{F}(v)\|_{\mathcal{G}_{\sigma}^{\infty }}\\
&\leq  C_1\sup_{-\infty<t<\infty}|t|^{\sigma}\| u-v\|_{(\alpha+2,\infty)}
\sup_{-\infty <t<\infty}|t|^{\alpha\sigma }
\big[ \| u\|_{(\alpha+2,\infty)}^{\alpha}+\| v \|_{(\alpha+2,\infty )}^{\alpha}\big],
\end{aligned}
\end{equation}
for all $u,v$ such that the right hand side of \eqref{Global-estim}
is finite.

(2) If $\frac{(2n-d)\alpha}{4(\alpha+2)}<1$ and 
$A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}$, then there exists a constant $C_2>0$ 
such that
\begin{equation} \label{Global-estim2}
\begin{aligned}
& \| \mathcal{F}(u)-\mathcal{F}(v)\|_{\mathcal{G}_{\sigma}^{\infty }} \\
&\leq C_2\sup_{-\infty<t<\infty}|t|^{\sigma}\| u-v\|_{(\alpha+2,\infty)}
\sup_{-\infty <t<\infty}|t|^{\alpha\sigma }
\big[ \| u\|_{(\alpha+2,\infty)}^{\alpha}+\| v \|_{(\alpha+2,\infty )}^{\alpha}\big],
\end{aligned}
\end{equation}
for all $u,v$ such that the right hand side of \eqref{Global-estim2}
is finite.
\end{lemma}

\begin{proof}  
Without loss of generality we  consider only the case $t>0$. 
Using Lemma  \ref{LinEstGlo}, the property of $f$ established in \eqref{fprop}, 
and the H\"{o}lder inequality, we have
\begin{align*}
 \| \mathcal{F}(u)-\mathcal{F}(v)\|_{(p',\infty)}
&\leq C\int_0^{t}(t-\tau)^{-\frac{n(2-p)}{4p}}
 \| f(|u|)u-f(|v|)v\|_{(p,\infty )}d\tau \\
&\leq C \int_0^{t}(t-\tau)^{-\frac{n(2-p)}{4p}}
 \| |u-v|(|u|^{\alpha}+|v|^{\alpha})\|_{(p,\infty )}d\tau\\
& \leq C\int_0^{t}(t-\tau)^{-\frac{n(2-p)}{4p}}\|
u-v\|_{(p',\infty )}\big[ \| u\|_{(p',\infty
)}^{\alpha}+\| v \|_{(p',\infty)}^{\alpha}\big] d\tau.
\end{align*}
Since $\frac{1}{p}+\frac{1}{p'}=1$ and we used the H\"{o}lder inequality, 
 we obtain the restriction $p'=\alpha+2$. Hence
\begin{align*}
 &\| \mathcal{F}(u)-\mathcal{F}(v)\|_{(\alpha+2,\infty)}\\
&\leq C\int_0^{t}(t-\tau)^{-\frac{n\alpha}{4(\alpha+2)}}
 \| u-v\|_{(\alpha+2,\infty )}\big[ \| u\|_{(\alpha+2,\infty )}^{\alpha}
+\| v \|_{(\alpha+2,\infty)}^{\alpha}\big] d\tau\\
&\leq  C\sup_{t>0}t^{\sigma}\|
u-v\|_{(\alpha+2,\infty)}\sup_{t>0}t^{\alpha\sigma}
\big[ \|u\|_{(\alpha+2,\infty )}^{\alpha}+\| v\|_{(\alpha+2,\infty
)}^{\alpha}\big]t^{-\sigma}t^{1-\frac{n\alpha}{4(\alpha+2)}-\sigma\alpha}.
\end{align*}
From $1-\frac{n\alpha}{4(\alpha+2)}-\sigma\alpha=0$, we
conclude that
\begin{equation} \label{DesNon2aa}
\begin{aligned}
&t^{\sigma}\| \mathcal{F}(u)-\mathcal{F}(v)\|_{(\alpha+2,\infty )} \\
&\leq C\sup_{t>0}t^{\sigma}\| u-v\|_{(\alpha+2,\infty)}
\sup_{t>0}t^{\alpha\sigma}\big[ \| u\|_{(\alpha+2,\infty )}^{\alpha}
+\| v \|_{(\alpha+2,\infty )}^{\alpha} \big].
\end{aligned}
\end{equation}
Taking the supremum in \eqref{DesNon2aa} we conclude the proof of the
estimate \eqref{Global-estim}. The proof of \eqref{Global-estim2}
follows in a similar way.
\end{proof}

In the next lemma we estimate the nonlinear term $\mathcal{F}(u)$ in
the norm $\|\cdot \|_{\mathcal{G}_{\beta}^T}$, which is
crucial in order to obtain existence of local-in-time mild solutions.
 Here we  use the notation $A\lesssim B$ which means that there exists a 
constant $c>0$ such that $A\leq cB$.

\begin{lemma}\label{NonEstLoc}
Let $1\leq \alpha<\infty$, and 
$(1/p,1/{(\alpha+1)p})\in\Xi_0\setminus\partial\Xi_0$.

(1) If $\frac{n\alpha}{4p}<1$ and $A=\Delta^2$, then there exists a constant $C_3>0$ such that
\begin{equation}
\begin{aligned}
&\| \mathcal{F}(u)-\mathcal{F}(v)\|_{\mathcal{G}_{\beta}^T} \\
&\leq  C_3\sup_{-T<t<T}|t|^{\beta}\| u-v\|_{((\alpha+1)p,\infty)}
 \sup_{-T<t<T}|t|^{\beta\alpha}\Big[ \| u\|_{((\alpha+1)p,\infty )}^{\alpha}\\
&\quad +\| v\|_{((\alpha+1)p,\infty )}^{\alpha}\Big] T^{1-\beta(\alpha+1)},
\end{aligned} \label{local-estim1}
\end{equation}
for all $u,v$ such that the right hand side of \eqref{local-estim1} is finite.

(2) If $\frac{(2n-d)\alpha}{4p}<1$ and $A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}$, 
then there exists a constant $C_4>0$ such that
\begin{equation}
\begin{aligned}
&\| \mathcal{F}(u)-\mathcal{F}(v)\|_{\mathcal{G}_{\beta}^T} \\
&\leq  C_4\sup_{-T<t<T}|t|^{\beta}\| u-v\|_{((\alpha+1)p,\infty)}
 \sup_{-T<t<T}|t|^{\beta\alpha}\Big[ \| u\|_{((\alpha+1)p,\infty )}^{\alpha}\\
&\quad +\| v\|_{((\alpha+1)p,\infty )}^{\alpha}\Big] T^{1-\beta(\alpha+1)},
\end{aligned} \label{local-estim2}
\end{equation}
for all $u,v$ such that the right hand side of \eqref{local-estim2} is finite.
\end{lemma}

\begin{proof} 
We only prove the first inequality; the proof of the second one  
is analogous. Without loss of generality suppose that $t>0$. Then, 
from Lemma \ref{LinEstLoc}, the property of $f$ established in \eqref{fprop} 
and  the H\"{o}lder inequality, we obtain
\begin{align*}
 \| \mathcal{F}(u)-\mathcal{F}(v)\|_{(q,\infty)}
&\leq \int_0^{t}(t-\tau)^{-b_l}\| f(|u|)u-f(|v|)v \|_{(p,\infty )}d\tau\\
&\leq C\int_0^{t}(t-\tau)^{-b_l}\| |u-v|(|u|^{\alpha}+|v|^{\alpha})
 \|_{(p,\infty )}d\tau\\
& \leq C\int_0^{t}(t-\tau)^{-b_l}\| u-v\|_{(q,\infty)}
\big(\| u\|^{\alpha}_{(q,\infty)}+\| v\|^{\alpha}_{(q,\infty )}\big)d\tau.
\end{align*}
Since we used the H\"{o}lder inequality the next restriction appears
 $q=(\alpha+1)p$. Therefore,
\begin{align*}
&\| \mathcal{F}(u)-\mathcal{F}(v)\|_{((\alpha+1)p,\infty)}\\
&\lesssim \int_0^{t}(t-\tau)^{-\frac{n\alpha}{4p(\alpha+1)}}
 \| u-v\|_{((\alpha+1)p,\infty)}
 \big(\| u\|^{\alpha}_{((\alpha+1)p,\infty)}
 +\| v\|^{\alpha}_{((\alpha+1)p,\infty )}\big)d\tau\\
&\lesssim \sup_{0<t<T}t^{\beta}\| u-v\|_{((\alpha+1)p,\infty)}
 \sup_{0<t<T}t^{\alpha\beta}\big[ \| u\|_{((\alpha+1)p,\infty )}^{\alpha}
 +\| v\|_{((\alpha+1)p,\infty )}^{\alpha}\big] t^{1-\beta(\alpha+2)}.
\end{align*}
Hence,
\begin{align*}
&t^{\beta} \| \mathcal{F}(u)-\mathcal{F}(v)\|_{((\alpha+1)p,\infty)}\\
& \leq C\sup_{0<t<T}t^{\beta}\| u-v\|_{((\alpha+1)p,\infty)}\\
&\quad \times \sup_{0<t<T}t^{\beta\alpha}
 \big[ \| u\|_{((\alpha+1)p,\infty )}^{\alpha}
 +\| v\|_{((\alpha+1)p,\infty )}^{\alpha}\big] T^{1-\beta(\alpha+1)}.
\end{align*}
Taking supremum on $t$ in the last inequality, we obtain the desired result.
\end{proof}

\section{Local and global solutions}

In this section we prove some results of local and global well-posedness 
for the Schr\"{o}dinger equations with isotropic and anisotropic 
fourth-order dispersion in the setting of Lorentz spaces.

\subsection{Local-in-time solutions}

\begin{theorem}[Local-in-time solutions]
\label{LocalTheo} 
Let $1\leq \alpha<\infty$, and
$(1/p,1/{(\alpha+1)p})\in\Xi_0\setminus\partial\Xi_0$. Consider
$\frac{n\alpha}{4p}<1$ if $A=\Delta^2$, or
$\frac{(2n-d)\alpha}{4p}<1$ if
$A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}$. If $u_0\in
\mathcal{S}'(\mathbb{R}^n)$ such that
$\|G_{\epsilon,\delta}(t)u_0\|_{\mathcal{G}_{\beta}^T}$ is finite,
then there exists $0<T^{\ast }\leq T<\infty $ such that the initial
value problem \eqref{FoSch} has a mild solution $u\in
\mathcal{G}_{\beta}^{T^*}$, satisfying $u(t)\rightharpoonup u_0$ in
$\mathcal{S}'(\mathbb{R}^n)$ as $t\to 0^+$. The solution $u$
is unique in a given ball of $\mathcal{G}_{\beta}^{T^*}$, and the
data-solution map $u_0\mapsto u$ into $\mathcal{G}_{\beta}^{T^*}$ is
Lipschitz.
\end{theorem}

\begin{remark}\label{rem1} \rm

(i) (Large class of initial data) From the definition of the norm
$\|\cdot\|_{\mathcal{G}_{\beta}^T}$ and Lemma \ref{LinEstLoc},
if we take $u_0\in L^{(p,\infty)}$, the quantity
$\|G_{\epsilon,\delta}(t)u_0\|_{\mathcal{G}_{\beta}^T}$ is finite.

(ii) (Regularity) If the initial data  is such that
 \[\sup_{-T<t<T}| t|^\beta\| G_{\epsilon,\delta}(t)u_0\|_{(p(\alpha+1),d)}<\infty,\]
  for $1\leq d <\infty$, then the local mild solution satisifes
\[
\sup_{-T^*<t<T^*}| t|^\beta\| u\|_{(p(\alpha+1),d)}<\infty,
\]
(possibly reducing the time of existence $T^*$).

(iii) (Finite energy solutions) From Theorem \ref{LocalTheo}
some results of local existence in Sobolev spaces can be recovered.
For that, notice that $H^{s}(\mathbb{R}^{n})\hookrightarrow
 L^{(p(\alpha+1),\infty)}$, for  $s>0$ such that
$2<p(\alpha+1)\leq\frac{2n}{n-2s}$ if $n>2s$
($2<p(\alpha+1)\leq\infty$ if $n<2s$). Therefore, if $u_0\in H^s$,
then
\begin{align*}
\| G_{\epsilon,\delta}(t)u_0\|_{\mathcal{G}_{\beta}^T}
&\leq C\sup_{-T<t<T}|t|^\beta\| G_{\epsilon,\delta}(t)u_0\|_{H^s}\\
&\leq C\sup_{-T<t<T}|t|^\beta
\| u_0\|_{H^s}<\infty.
\end{align*}
Consequently, by Theorem \ref{LocalTheo} there exists  
a mild solution \;
$u:(-T^*,T^*)\to L^{(p(\alpha+1))}(\mathbb{R}^n)$
in $\mathcal{G}_{\beta}^{T^*}$. On
the other hand, for the same initial data $u_0\in
H^s(\mathbb{R}^n)$, suppose $v\in C([-T_0,T_0];H^s(\mathbb{R}^n))$
the unique energy finite solution
 for some $T_0$ small enough. By the embedding
$H^s\hookrightarrow L^{(p(\alpha+1),\infty)}$, we obtain that $v\in
\mathcal{G}_{\beta}^{T_0}$. Thus, taking $T_0$ small enough, the
uniqueness of solution given in Theorem \ref{LocalTheo}, implies
that $u=v$ on $[-T_0,T_0]$ and consequently, $u\in
C([-T_0,T_0];H^s)$.
\end{remark}

Before proving Theorem \ref{LocalTheo}, we enunciate a result related to 
the existence of radial solutions. First of all, we
recall that a solution $u$ in ${\mathcal{G}_{\beta}^T}$ is said to be 
radially symmetric, or simply radial, for a.e.
$0<| t| <T$, if $u(Rx,t)=u(x,t)$ a.e. $x\in\mathbb{R}^{n}$
for all $n\times n$-orthogonal matrix $R$. Then, we have the following corollary.

\begin{corollary} \label{cor1}
Under the hypotheses of Theorem \ref{LocalTheo},
if the initial data $u_0$ is radially symmetric, then the corresponding
solution $u$ is radially symmetric for a.e. $0<\left| t\right| <T$.
\end{corollary}


\begin{proof}[Proof of Theorem \ref{LocalTheo}] 
This proof  will be obtained as an
application of the Banach fixed point theorem. First, notice that by
hypothesis on the initial data, we have 
\begin{eqnarray*}
\|G_{\epsilon,\delta}(t)u_0\|_{\mathcal{G}_{\beta}^T}
:=\sup_{-T<t<T}|t|^{\beta}\|G_{\epsilon,\delta}(t)u_0\|_{(p(\alpha+1),\infty)}
\equiv\frac{K}{2}<\infty.
\end{eqnarray*}
We consider the mapping $\Upsilon$ defined by
\begin{equation}
\Upsilon(u(t))=G_{\epsilon,\delta}(t)u_0+i\int_0^t
G_{\epsilon,\delta}(t-\tau)f(|u(x,\tau)|)u(x,s)d\tau.
\end{equation}
Then, we  prove that $\Upsilon$ defines a contraction on
 $(B_{K},d)$ where $B_{K}$ denotes the closed ball 
$\{u\in \mathcal{G}_{\beta}^{T^*}: \|u\|_{\mathcal{G}_{\beta}^{T^*}}\leq K\}$ 
endowed with the complete metric $d(u,v)=
 \|u-v\|_{\mathcal{G}_{\beta}^{T^*}}$ for some $0<T^*\leq T$.
In fact, let us consider $0<T^*\leq T$ such that
$\tilde{C}K^{\alpha}{(T^*)}^{1-\beta(\alpha+1)}<\frac{1}{2}$ where
$\tilde{C}$ denotes the constant $C_3$ or $C_4$ in Lemma \ref{NonEstLoc}. Then,
from Lemma \ref{NonEstLoc} with $v=0$ we obtain
\begin{align*}
\| \Upsilon(u)\|_{\mathcal{G}_{\beta}^{T^*}} 
&\leq \|G_{\epsilon,\delta}(t)u_0\|_{\mathcal{G}_{\beta}^{T^*}}
+\|\mathcal{F}(u)\|_{\mathcal{G}_{\beta}^{T^*}} \\
&\leq \frac{K}{2}+ \tilde{C}K^{\alpha+1}({T^*})^{1-\beta(\alpha+1)}
\leq  \frac{K}{2}+ \frac{K}{2}=K,
\end{align*}
for all $u\in B_{K}$. Consequently, $ \Upsilon(B_K)\subset B_K$. Now,
assuming that $u,v\in B_{K}$, from Lemma \ref{NonEstLoc} we obtain
\begin{equation}
\begin{aligned}
\|\Upsilon(u(t))-\Upsilon(v(t))\|_{\mathcal{G}_{\beta}^{T^*}}
&=\| \mathcal{F}(u)-\mathcal{F}(v)\|_{\mathcal{G}_{\beta}^{T^*}} \\
&\leq  2\tilde{C} K^\alpha ({T^*})^{1-\beta(\alpha+1)}
 \|u-v\|_{\mathcal{G}_{\beta}^{T^*}}.
\end{aligned}\label{lo1}
\end{equation}
Thus, as
$\tilde{C}K^{\alpha}({T^*})^{1-\beta(\alpha+1)}<1/2$, the map
$\Upsilon$ is a contraction on $(B_K,d)$. Consequently, the Banach fixed
point theorem implies the existence of a unique solution $u\in
\mathcal{G}_{\beta}^{T^*}$. Through standard argument one can prove
that $u(t)\to u_0$ as $t\to0$, in the sense of
distributions \cite{LucEldPab}. On the other hand, in order to prove the local
Lipschitz continuity of the data-solution map, we consider $u,v$ two
local mild solutions with initial data $u_0, v_0$, respectively.
Then, as in estimate \eqref{lo1} we obtain
\begin{equation}
\begin{aligned}
\| u-v\|_{\mathcal{G}_{\beta}^{T^*}} 
&=  \|G_{\epsilon,\delta}(t)(u_0-v_0)\|_{\mathcal{G}_{\beta}^{T^*}}+\|
\mathcal{F}(u)-\mathcal{F}(v)\|_{\mathcal{G}_{\beta}^{T^*}} \\
&\leq  \|G_{\epsilon,\delta}(t)(u_0-v_0)\|_{\mathcal{G}_{\beta}^{T^*}}+ 2\tilde{C}
K^\alpha ({T^*})^{1-\beta(\alpha+1)}\|
u-v\|_{\mathcal{G}_{\beta}^{T^*}}.
\end{aligned}
\end{equation}
Since $2\tilde{C} K^\alpha ({T^*})^{1-\beta(\alpha+1)}<1$, 
from the above  inequality, the local Lipschitz continuity of the data-solution 
map holds.
\end{proof}

\begin{proof}[Proof of Corollary \ref{cor1}]
From the fixed point argument used in the
proof of Theorem \ref{LocalTheo}, we can see the local solution $u$ as the
limit in $\mathcal{G}^T_{\beta}$ of the Picard
sequence
\begin{equation}
 u_1 =G_{\epsilon, \delta}(t)(u_0){,}\quad
 u_{k+1} =u_1+\mathcal{F}(u_{k}),\quad  k\in  \mathbb{N}. \label{sequencee}
\end{equation}
Since the symbol of the group
$G_{\epsilon,\delta}(t)$ is radially symmetric for each
fixed $0<t<T$, it follows that $G_{\epsilon,\delta}(t)u_0$ is radial,
provided that $u_0$ is radial. Furthermore,
since the nonlinear term $\mathcal{F}(u)$ is radial when
$u$ is radial, an induction argument gives that the sequence 
$\{u_{k}\}_{k\in \mathbb{N}}$ given in \eqref{sequencee} is radial. 
Since pointwise convergence
preserves radial symmetry, and $\mathcal{G}_{\beta}^{T}$ implies (up
to a subsequence) almost everywhere pointwise convergence in the
variable $x$, for a.e. fixed $t\neq0$, it follows that $u(x,t)$ is
radially symmetric. 
\end{proof}


\subsection{Global-in-time solutions}

\begin{theorem}\label{GlobalTheo} 
Let $1\leq \alpha<\infty $  and  assume that
$(\alpha+1)\sigma<1$. Consider either
$\frac{n\alpha}{4(\alpha+2)}<1$ if $A=\Delta^2$, or
$\frac{(2n-d)\alpha}{4(\alpha+2)}<1$ if
$A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}$. Suppose further that  $
\xi>0$ and $M>0$ satisfy the inequality
$\xi+\widetilde{C}M^{\alpha+1}\leq M$ where
$\widetilde{C}=\widetilde{C}(\alpha, n)$ is the constant $C_1$ or $C_2$ in Lemma
\ref{EstNonGlo}. If $u_0\in \mathcal{D}_\sigma$, with
$\sup_{t>0}t^\sigma\|
G_{\epsilon,\delta}(t)u_0\|_{(\alpha+2,\infty)}<\xi$, then
the initial value problem \eqref{FoSch} has a unique global-in-time
mild solution $u\in \mathcal{G}^{\infty}_{\sigma}$ with $\|
u\|_{\mathcal{G}^{\infty}_\sigma}\leq M$, such that
 $\lim_{t \to 0} u(t)=u_0$ in distribution sense. Moreover, if $u,v$ are two
global mild solutions with respective initial data $u_0,v_0$, then
\begin{equation}
\| u-v\|_{\mathcal{G}^{\infty}_\sigma}\leq C \|
G_{\epsilon,\delta}(t)(u_0-v_0)\|_{\mathcal{G}^{\infty}_\sigma}.
\end{equation}
Additionally, if $G_{\epsilon,\delta}(t)(u_0-v_0)$ satisifes the
stronger decay
\[
\sup_{t>0}| t|^\sigma(1+| t|)^\varsigma\| G_{\epsilon}(t)(u_0-v_0)\|_{(\alpha+2,
\infty )}<\infty,
\]
 for some $\varsigma>0$ such that $\sigma(\alpha+1)+\varsigma<1$, then
\begin{equation} \label{stronger}
\begin{aligned}
&\sup_{t>0}| t|^\sigma(1+| t|)^\varsigma\| u(t)-v(t)\|_{(\alpha+2,\infty )}\\
&\leq C\sup_{t>0}| t|^\sigma(1+| t|)^\varsigma 
\| G_{\epsilon}(t)(u_0-v_0)\|_{(\alpha+2,\infty )}.
\end{aligned}
\end{equation}
\end{theorem}

\begin{remark} \rm

(i) (Regularity) In addition to the assumptions of Theorem \ref{GlobalTheo}, 
if we consider that the initial data satisifes
\[
\sup_{-\infty<t<\infty}t^\sigma \| G_{0,\delta}(t)u_0\|_{(\alpha+2,d)}<\infty
\]
for some $1\leq d <\infty$, then there exists $\xi_0$ such that if
\[
\sup_{-\infty<t<\infty}t^\sigma \| G_{0,\delta}(t)u_0\|_{(\alpha+2,d)}\leq \xi_0,
\]
then global solution provided in Theorem \ref{GlobalTheo} satisfies that
\[
\sup_{-\infty<t<\infty}t^\sigma \| u(t)\|_{(\alpha+2,d)}< \infty.
\]

(ii) (Radial solutions) As in Corollary \ref{cor1}, if the initial data $u_0$ 
is radially symmetric, then
the global-in-time solution $u$ is radially symmetric for a.e.
$t\neq0$.

(iii) (Asymptotic stability) Following the proof of \eqref{stronger}
 we can obtain that if $u,v$ are global mild solutions of  the Cauchy 
problem \eqref{FoSch} given by Theorem \ref{GlobalTheo}, with initial 
data $u_0,v_0\in \mathcal{D}_\sigma$ respectively, satisfying
\[
\lim_{t \to \infty} t^\sigma(1+ t)^\varsigma \|
G_{\epsilon}(t)(u_0-v_0)\|_{(\alpha+2,\infty )}=0,
\]
then $\lim_{t \to \infty}t^\sigma(1+ t)^\varsigma \|
u(t)-v(t)\|_{(\alpha+2,\infty )}=0$.

(iv) (biharmonic and anisotropic biharmonic global solutions) Theorem
\ref{GlobalTheo} gives existence of global mild solution for Cauchy
problem associated with equation \eqref{FoNLS} in the class $
\mathcal{G}^{\infty}_{\sigma}$. The proof was based on the time-decay 
estimate of the group $G_{0,\delta}(t)$ given in Lemma
\ref{LinEstGlo}. However, taking into account that if $\epsilon=0$
the time-decay estimate in Lemma \ref{LinEstLoc} holds true for all
$t\neq 0$, we are able to prove the existence of global-in-time mild
solutions for the Cauchy problem associated with equation
\eqref{FoNLS} in the class $G^\infty_{\sigma(p)}$ defined as the set
of Bochner measurable functions $u:(-\infty ,\infty )\to
L^{(p(\alpha+1),\infty )}$ such that
\[
\| u\|_{\mathcal{G}^{\infty}_{\sigma(p)}}=\sup_{-\infty
<t<\infty}|t|^{\sigma(p)}\| u(t)\|_{(p(\alpha+1),\infty )}<\infty,
\]
where 
\begin{equation}\label{Defsigmabb}
\sigma(p)=\begin{cases}
\frac{1}{\alpha}-\frac{n}{4p(\alpha+1)}, &\text{if } A=\Delta^2,\\
\frac{1}{\alpha}-\frac{2n-d}{4p(\alpha+1)}, &\text{if }
A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}.
\end{cases}
\end{equation}
Here $p,\alpha$ must verify $1\leq \alpha<\infty$,
$(1/p,1/{(\alpha+1)p})\in\Xi_0\setminus\partial\Xi_0$ and
$\frac{4p}{n\alpha}<1<\frac{4p(\alpha+1)}{n\alpha}$ if $A=\Delta^2$
or, $\frac{4p}{(2n-d)\alpha}<1<\frac{4p(\alpha+1)}{(2n-d)\alpha}$ if
$A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i.}$
\end{remark}

\begin{corollary}[Biharmonic and anisotropic biharmonic self-similar solutions] 
\label{self} \quad\\
Let $\epsilon=0$, $1\leq \alpha<\infty $  and  assume that
$(\alpha+1)\sigma<1$. Consider either
$\frac{n\alpha}{4(\alpha+2)}<1$  if $A=\Delta^2$, or
$\frac{(2n-d)\alpha}{4(\alpha+2)}<1$ if
$A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}$. Assume that the initial data $u_0$ 
is  a homogeneous function of degree $\frac{-4}{\alpha}$. 
Then the solution $u(t, x)$ provided by Theorem \ref{GlobalTheo} is self-similar, 
that is, $u(t, x) = \lambda^{\frac{4}{\alpha}}
 u(\lambda^4 t, \lambda x)$ for all $\lambda > 0$, almost everywhere for
 $x\in \mathbb{R}^n$ and $t>0$.
\end{corollary}

\begin{remark} \rm
 An admissible class of initial data for the existence of self-similar solutions 
in Corollary \ref{self} is given by  the set of functions
 $u_0(x)=P_m(x)| x|^{-m-\frac{4}{\alpha}}$ where $P_m(x)$ is a homogeneous 
polynomial of degree $m$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{GlobalTheo}]
It will be also obtained as an application of the Banach fixed point Theorem. 
We denote by $B_{M}$ the set of $u\in\mathcal{G}^{\infty}_{\sigma}$ such that
\[
\| u \|_{\mathcal{G}^\infty_\sigma}\equiv\sup_{-\infty<t<\infty}|t|^\sigma
\| u(t)\|_{(\alpha+2,\infty )}\leq M,
\]
endowed with the complete metric
 $d(u,v)=\sup_{-\infty<t<\infty}|t|^\sigma\| u(t)-v(t)\|_{(\alpha+2,\infty )}$. 
We will show that the mapping 
\begin{equation}
\Upsilon(u(t))=G_{\epsilon,\delta}(t)u_0+i\int_0^t
G_{\epsilon,\delta}(t-\tau)f(|u(x,\tau)|)u(x,s)d\tau,
\end{equation}
is a contraction on $(B_{M},d)$. From the assumptions on the initial
data and Lemma \ref{EstNonGlo} (with $v=0$), we have (for all $u\in B_M$)
\begin{align*}
\| \Upsilon(u)\|_{\mathcal{G}^\infty_\sigma} 
& \leq  \| G_{\epsilon,\delta}(t)u_0\|_{\mathcal{G}^\infty_\sigma}
+\| \mathcal{F}(u)\|_{\mathcal{G}^\infty_\sigma} \\
&\leq  \xi + \tilde{C}\| u\|^{\alpha+1}_{\mathcal{G}^\infty_\sigma} \\
& \leq  \xi +\tilde{C} M^{\alpha+1}\leq M,
\end{align*}
because $M$ and $\xi$ satisfy $\xi+\tilde{C} M^{\alpha+1}\leq M$.
Thus, $\Upsilon$ maps $B_M$ itself. On the other hand, 
Lemma \ref{EstNonGlo}, we obtain
\begin{equation}
\| \Upsilon(u)-\Upsilon(v)\|_{\mathcal{G}^\infty_\sigma} 
\leq \| \mathcal{F}(u)-\mathcal{F}(v)\|_{\mathcal{G}^\infty_{\sigma}} 
\leq 2\tilde{C}M^\alpha \| u-v\|_{\mathcal{G}^\infty_{\sigma}}.\label{f1}
\end{equation}
Since $\tilde{C}M^{\alpha}<1$, it follows that $\Upsilon$ is a contraction
on $(B_{M},d)$  and consequently, the Banach fixed point theorem implies 
the existence of a unique global solution $u\in \mathcal{G}^\infty_\sigma$. 
To prove the continuous dependence of the mild
solutions with respect to the initial data, it suffices to observe
that \eqref{f1} implies that
\begin{equation*}
\| u-v\|_{\mathcal{G}^\infty_{\sigma}}
\leq \| G_{\epsilon,\delta}(t)u_0-G_{\epsilon,\delta}(t)v_0\|_{\mathcal{G}_{\sigma}}
+CM^\alpha \| u-v\|_{\mathcal{G}^\infty_{\sigma}}.
\end{equation*}
Thus, as $\tilde{C}M^\alpha<1$, then 
$\| u-v\|_{\mathcal{G}^\infty_{\sigma}}
\leq C\| G_{\epsilon,\delta}(t)u_0-G_{\epsilon,\delta}
 (t)v_0\|_{\mathcal{G}^\infty_{\sigma}}$.
Finally,  to prove the stronger decay, notice that
\begin{equation}
\begin{aligned}
&t^\sigma(1+ t)^\varsigma\| u(t)-v(t)\|_{(\alpha+2,\infty )} \\
&\leq  C\sup_{t>0} t^\sigma(1+ t)^\varsigma \| G_{\epsilon,\delta}(t)
 (u_0-v_0)\|_{(\alpha+2,\infty )}
+ t^\sigma(1+ t)^\varsigma \| \mathcal{F}(u)-\mathcal{F}(v)\|_{(\alpha+2,\infty )}.
\end{aligned}\label{stn1}
\end{equation}
Since $\| u \|_{\mathcal{G}^\infty_\sigma}, \| v
\|_{\mathcal{G}^\infty_\sigma}\leq M$, using the change of
variable $\tau\mapsto \tau t$ and noting that
$(1+t)^\varsigma(1+t\tau)^{-\varsigma}\leq t^\varsigma
(t\tau)^{-\varsigma}$ for $\tau\in [0,1]$, we obtain
\begin{equation}
\begin{aligned}
& t^\sigma(1+ t)^\varsigma\| \mathcal{F}(u)-\mathcal{F}(v)\|_{(\alpha+2,\infty )}\\
&\leq t^\sigma(1+t)^\varsigma\int_0^t(t-\tau)^{-\frac{n\alpha}{4(\alpha+2)}}
 \tau^{-\sigma(\alpha+1)}(1+\tau)^\varsigma
\\
&\times (\tau^\sigma(1+\tau)^\varsigma\| u(\tau)-v(\tau)\|_{(\alpha+2,\infty )})
\big[ \tau^\sigma\| u(\tau)\|_{(\alpha+2,\infty )}^{\alpha}+\tau^\sigma
\| v(\tau) \|_{(\alpha+2,\infty)}^{\alpha}\big] ds
\\
&\leq 2M^\alpha\int_0^1 (1-\tau)^{-\frac{n\alpha}{4(\alpha+2)}}
 \tau^{-\sigma(\alpha+1)} (1+t)^\varsigma  (1+ t\tau)^{-\varsigma}
((t\tau)^\sigma(1+(t\tau))^\varsigma \\
&\quad \times \| u(t\tau)-v(t\tau)\|_{(\alpha+2,\infty )})ds \\
&\leq 2M^\alpha\int_0^1(1-\tau)^{-\frac{n\alpha}{4(\alpha+2)}}
\tau^{-\sigma(\alpha+1)} \tau^{-\varsigma} ((t\tau)^\sigma
 (1+(t\tau))^\varsigma\\
&\quad\times \| u(t\tau)-v(t\tau)\|_{(\alpha+2,\infty )})d\tau.
\end{aligned}\label{st2}
\end{equation}
Therefore, by denoting $A=\sup_{t>0}t^\sigma(1+ t)^\varsigma\|
u(t)-v(t)\|_{(\alpha+2,\infty )}$, from \eqref{stn1} and
\eqref{st2} we obtain
\begin{align*}
A&\leq C\sup_{t>0} t^\sigma(1+ t)^\varsigma
 \| G_{\epsilon,\delta}(t)(u_0-v_0)\|_{(\alpha+2,\infty )}\\
&\quad +\left (2M^\alpha\int_0^1(1-\tau)^{-\frac{n\alpha}{4(\alpha+2)}}
 \tau^{-\sigma(\alpha+1)} \tau^{-\varsigma} d\tau\right) A.
\end{align*}
Choosing $M$ small enough such that
$2M^\alpha\int_0^1 (1-\tau)^{-\frac{n\alpha}{4(\alpha+2)}}\tau^{-\sigma(\alpha+1)}
\tau^{-\varsigma} d\tau<1$, we conclude the proof.
\end{proof}

\begin{proof}[Proof of Corollary \ref{self}]
 We recall that by the fixed point argument used in the
proof of Theorem \ref{GlobalTheo}, the solution $u$ is the
limit in $\mathcal{G}_{\sigma}^{\infty}$ of the Picard
sequence
\begin{equation}
u_1  =G_{0,\delta}(t)u_0,\quad  u_{k+1} =u_1+\mathcal{F}(u_{k}),\quad
  k\in \mathbb{N}. \label{sequencee1}
\end{equation}
Notice that the initial data $u_0$ satisfying 
$ u_0(\lambda x)=\lambda^{-\frac{4}{\alpha}}u_0(x) $ belongs to the class
$\mathcal{D}_\sigma$ (see \cite[Corollary 2.6]{LucEldPab}).
Since $\epsilon=0$, we  obtain
\begin{equation}
u_1(\lambda x,\lambda^{4}t)=\lambda^{-\frac{4}{\alpha}}u_1(x,t) \label{aux-scal2}
\end{equation}
and then $u_1$ is invariant by the scaling
\begin{equation}\label{sc}
u(x,t)\to u_\lambda(x,t):=\lambda^{\frac{4}{\lambda}}u(\lambda x,\lambda^4 t),
\quad  \lambda>0.
\end{equation}
 Moreover, the nonlinear term
$\mathcal{F}(u)$ is invariant by scaling \eqref{sc} when $u$ is
also. Therefore, we can employ an induction argument in order to
obtain that all elements $u_{k}$ have the scaling invariance
property \eqref{sc}. Because the norm of $\mathcal{G}_{\alpha
}^{\infty}$ is scaling invariant, we obtain that the limit $u$ also is
invariant by the scaling transformation $u\to u_\lambda$, as
required.
\end{proof}

\section{Vanishing dispersion limit}

This section is devoted to the analysis of the solutions of \eqref{FoSch} 
as the second order dispersion vanishes. More exactly, we study the convergence, 
$\epsilon\to 0$, of the solutions of the Cauchy problem
\begin{equation}\label{FoSchedl}
\begin{gathered}
i\partial_tu+\epsilon \Delta u+\delta A u+\lambda |u|^\alpha u=0, \quad
 x\in \mathbb{R}^{n},\; t\in \mathbb{R}, \\
u(x,0)=u_0(x), \quad x\in \mathbb{R}^{n}, 
\end{gathered}
\end{equation}
to the solutions of
\begin{equation}\label{FoSche=0}
\begin{gathered}
i\partial_tu+\delta A u+\lambda|u|^\alpha u=0, \quad
 x\in \mathbb{R}^{n},\; t\in \mathbb{R}, \\
u(x,0)=u_0(x), \quad x\in \mathbb{R}^{n}. 
\end{gathered}
\end{equation}
in the framework of  the $H^2(\mathbb{R}^n)$ space.
Throughout this section we consider $\alpha$ as a positive even integer. 
Before to establish our main results, we give some preliminary
facts. First, we recall the following conserved quantities of \eqref{FoSchedl}:
\begin{gather}\label{cc1}
M(u)=\| u\|^2_{L^2(\mathbb{R}^n)}; \\
\label{cc2}
E_{\epsilon,\delta,\lambda}(u)=\delta\|\Delta u\|_{L^2}^2
-\epsilon \|\nabla u\|^2_{L^2} 
+\frac{2\lambda}{\alpha+2}\|u\|^{\alpha+2}_{L^{\alpha+2}}, \quad
 \text{if } A=\Delta^2; \\
\label{cc3}
E_{\epsilon,\delta,\lambda}(u)=\delta\sum_{i=1}^d\|u_{x_ix_i}\|_{L^2}^2
-\epsilon \|\nabla u\|^2_{L^2} +\tfrac{2\lambda}{\alpha+2}
\|u\|^{\alpha+2}_{L^{\alpha+2}}, \quad \text{if }  
A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}. 
\end{gather}
According to the signs of the pair $(\delta,\lambda)$, we have two cases:
 Case 1: $\delta \lambda>0$ and $\epsilon\in \mathbb{R}$.
 Case 2:  $\delta \lambda<0$ and $\epsilon\in \mathbb{R}$.
Thus we have the next result.

\begin{proposition}\label{cant} 
Fix $\delta=\pm 1$, $\lambda=\pm 1$ and let
 $u_\epsilon\in C([-T,T];H^2(\mathbb{R}^n))$ be the local solution of
\eqref{FoSchedl} with initial data $u_0\in H^2(\mathbb{R}^n)$ and $A=\Delta^2$. 
Assume that
\begin{itemize}
\item $(\epsilon,\delta,\lambda)$ is as in Case 1 or
\item $(\epsilon,\delta,\lambda)$ is as in Case 2,  $n\alpha <8$, 
$\frac{n\alpha}{4(\alpha+2)}\leq 1$, if $n\neq 2,4$, and 
$0\leq \frac{n\alpha}{4(\alpha+2)}<1$ if $n= 2,4$.
\end{itemize}
Then the following estimate holds
\begin{equation}
\| u_\epsilon(t)\|_{H^2(\mathbb{R}^n)}\leq C(\|u_0\|_{H^2}, \|u_0\|_{L^{\alpha+2}}).
\end{equation}
\end{proposition}

\begin{proof} 
First we consider Case 1. Using the conserved quantities of \eqref{FoSchedl} 
given in \eqref{cc1}-\eqref{cc2}, we obtain
\begin{equation} \label{FirIne}
\begin{aligned}
&\| u_\epsilon(t)\|^2_{L^2}+\| \Delta u_\epsilon(t)\|^2_{L^2}\\
&=M(u_0)+\delta^{-1}E_{\epsilon,\delta,\lambda}(u_0)+\delta^{-1}\epsilon\| \nabla
u_\epsilon\|^2_{L^2}
-\frac{2\delta^{-1}\lambda}{\alpha+2}\|u_{\epsilon}\|^{\alpha+2}_{L^{\alpha+2}}\\
&\leq  M(u_0)+\delta^{-1}E_{\epsilon,\delta,\lambda}(u_0)+\delta^{-1}\epsilon\| \nabla u_\epsilon(t)\|^2_{L^2}.
\end{aligned}
\end{equation}
At this point we have to consider two subcases. If $\delta^{-1}\epsilon<0$,
taking $0<| \epsilon|<\frac{1}{2}$, we arrived at
\[
\| u_\epsilon(t)\|^2_{L^2}+\| \Delta u_\epsilon(t)\|^2_{L^2}
\leq  M(u_0)+\delta^{-1}E_{\epsilon,\delta,\lambda}(u_0)
\leq  M(u_0)+E_{-\frac{1}{2},1,\delta^{-1}\lambda}(u_0).
\]
On the other hand, if  $\delta^{-1}\epsilon>0$,  from \eqref{FirIne} we  have
\begin{align*}
\|u_{\epsilon}(t)\|^2_{H^2}
&\leq C(\| u_\epsilon(t)\|^2_{L^2}+\| \Delta u_\epsilon(t)\|^2_{L^2}) \\
&\leq  CM(u_0)+CE_{0,1,\delta^{-1}\lambda}(u_0)+\delta^{-1}\epsilon
C\| u_\epsilon(t)\|^2_{H^2}.
\end{align*}
Again, consider $0<| \epsilon|<\frac{1}{2C}$ to arrive at
 \[
\|u_{\epsilon}(t)\|^2_{H^2}\lesssim M(u_0)+E_{0,1,\delta^{-1}\lambda}(u_0).
\]
In both subcases we obtain the desired result.
\smallskip

Now, we consider the Case 2. Consider the restrictions $n\alpha<8$, 
$0\leq \frac{n\alpha}{4(\alpha+2)}\leq 1$ if $n\neq 2,4$, and 
$0\leq \frac{n\alpha}{4(\alpha+2)}<1$ if $n= 2,4$. 
Thus, by applying the Douglas-Niremberg and Young inequalities we obtain
\begin{equation} \label{EstNor1}
\begin{aligned}
& \| u_\epsilon(t)\|^2_{L^2}+\| \Delta u_\epsilon(t)\|^2_{L^2} \\
&=M(u_0)+\delta^{-1}E_{\epsilon,\delta,\lambda}(u_0)+\delta^{-1}\epsilon\| \nabla
u_\epsilon(t)\|^2_{L^2}
 -\frac{2\delta^{-1}\lambda}{\alpha+2}\|u_{\epsilon}(t)\|^{\alpha+2}_{L^{\alpha+2}} \\
&\leq  M(u_0)+\delta^{-1}E_{\epsilon,\delta,\lambda}(u_0)
 +\delta^{-1}\epsilon\| \nabla u_\epsilon(t)\|^2_{L^2}
  +C_1\|u_\epsilon(t)\|^{\frac{n\alpha}{4}}_{H^2}\|u_\epsilon(t)\|^{\alpha+2
 -\frac{n\alpha}{4}}_{L^2}\\
&=M(u_0)+\delta^{-1}E_{\epsilon,\delta,\lambda}(u_0)
 +\delta^{-1}\epsilon\| \nabla u_\epsilon(t)\|^2_{L^2}
 +C_1\|u_\epsilon(t)\|^{\frac{n\alpha}{4}}_{H^2}\|u_0\|^{\alpha+2
 -\frac{n\alpha}{4}}_{L^2} \\
&\leq M(u_0)+\delta^{-1}E_{\epsilon,\delta,\lambda}(u_0)
 +\delta^{-1}\epsilon\| \nabla u_\epsilon(t)\|^2_{L^2}
  +C_1\mu_0 \|u_\epsilon(t)\|^2_{H^2} \\
&\quad +C(\mu_0) \|u_0\|^\kappa_{L^2},
\end{aligned}
\end{equation}
with $\kappa=\frac{8(\alpha+2)-8n\alpha}{8-n\alpha}$.
Taking $0<\mu_0 <\frac{1}{2C_1}$, from \eqref{EstNor1} we obtain
\begin{equation}\label{EstNor2}
\|u_\epsilon(t)\|^2_{H^2}\lesssim  M(u_0)
 +\delta^{-1}E_{\epsilon,\delta,\lambda}(u_0)+\delta^{-1}
 \epsilon\| \nabla u_\epsilon(t)\|^2_{L^2}+C(\|u_0\|_{L^2}).
\end{equation}
Again, we have two subcases. If $\delta^{-1}\epsilon<0$, it is easy
to see that for $0<|\epsilon|<\frac{1}{2}$,
\begin{equation}\label{EstNor3}
\|u_\epsilon(t)\|^2_{H^2}\lesssim  M(u_0)
 +E_{-\frac{1}{2},1, \delta^{-1}\lambda}(u_0)+C(\mu_0, \|u_0\|_{L^2}).
\end{equation}
Finally, if $\delta^{-1}\epsilon>0$, we use that
$\delta^{-1}\epsilon\| \nabla u_\epsilon(t)\|^2_{L^2}
\leq \frac{1}{2}\|  u_\epsilon(t)\|^2_{H^2}$ for
 $0<|\epsilon|<\frac{1}{2}$ in \eqref{EstNor2} to obtain again
inequality \eqref{EstNor3}.
\end{proof}

Now we are in a position to establish our main result of this section.

\begin{theorem}\label{TheCon1}
 Consider  $u_{\epsilon}$ and $u$  in $C([-T,T];H^2(\mathbb{R}^n))$, 
the solutions of \eqref{FoSchedl} and \eqref{FoSche=0} respectively,  
with common initial data  $u_0\in H^2(\mathbb{R}^n)$ and $A=\Delta^2$. 
Here  $[-T,T]$ is the common interval of local existence for  
$u_{\epsilon}$ and $u$. Suppose $n<4$, if $\delta \lambda<0$ assume that 
 $n\alpha <8$, $\frac{n\alpha}{4(\alpha+2)}\leq 1$, if $n\neq 2$, and 
$0\leq \frac{n\alpha}{4(\alpha+2)}<1$ if $n= 2$. Then
\[
\lim_{\epsilon\to 0} \|u_{\epsilon}(t)-u(t)\|_{H^2}=0,
\]
for all $t\in [-T,T]$.
\end{theorem}

\begin{remark} \label{AniDisp} \rm
A version of Theorem \ref{TheCon1} for the anisotropic dispersion
case,
$A=\sum_{i=1}^d\partial_{x_ix_ix_ix_i}$, by replacing the norm
convergence in $H^2$ by the natural norm
$H^2(\mathbb{R}^d)H^1(\mathbb{R}^{n-d})$, is not clear. In fact, we
are not able to bound $\| \nabla u_\epsilon\|_{L^2}$  or
 $ \| u_\epsilon\|^2_{H^1}+\sum_{i=1}^d\|  u_{\epsilon_{x_ix_i}}\|^2_{L^2}$ 
in terms of the conserved quantities associated  to \eqref{FoSchedl} and 
 independently of $\epsilon$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{TheCon1}]
 As usual, the mild  solutions associated with \eqref{FoSche=0}
satisfy the integral equation
\begin{equation}\label{IntEque=0}
u(x,t)=G_{0,\delta}(t)u_0(x)+i\int_0^tG_{0,\delta}(t-\tau)f(|u(x,\tau)|)
u(x,\tau)d\tau,
\end{equation}
where $G_{0,\delta}$ is defined in \eqref{DefGe} with $\epsilon=0$.
Computing the difference between the integral
equations \eqref{IntEqu} and \eqref{IntEque=0} we obtain 
\begin{align*}
&\|u_{\epsilon}(t)-u(t)\|_{H^2}\\
&\leq \|[G_{\epsilon,\delta}(t)-G_{0,\delta}(t)]u_0\|_{H^2}\\
&\quad +\|\int_0^tG_{\epsilon,\delta}(t-\tau)|u_{\epsilon}(\tau)|^\alpha u_{\epsilon}
 (\tau)d\tau-\int_0^tG_{0,\delta}(t-\tau)|u(\tau)|^\alpha u(\tau)d\tau\|_{H^2}\\
&\leq  \int_0^t\|G_{\epsilon,\delta}(t-\tau)
[|u_{\epsilon}(\tau)|^\alpha u_{\epsilon}(\tau)
 -|u(\tau)|^\alpha u(\tau)]\|_{H^2}d\tau
 + \|[G_{\epsilon,\delta}(t)-G_{0,\delta}(t)]u_0\|_{H^2}\\
&\quad +\int_0^t\|[G_{\epsilon,\delta}(t-\tau)-G_{0,\delta}(t-\tau)]
 |u(\tau)|^\alpha u(\tau)\|_{H^2}d\tau
\end{align*}
Since $G_{\epsilon,\delta}(t)$ is a unitary group on $H^2$, from last 
inequality we obtain
\begin{equation} \label{IneDif1}
\begin{aligned}
&\|u_{\epsilon}(t)-u(t)\|_{H^2}\\
&\leq \int_0^t \|[|u_{\epsilon}(\tau)|^\alpha u_{\epsilon}(\tau)
 -|u(\tau)|^{\alpha}u(\tau)]\|_{H^2}d\tau
 + \|[G_{\epsilon,\delta}(t)-G_{0,\delta}(t)]u_0\|_{H^2}\\
&\quad +\int_0^t\|[G_{\epsilon,\delta}(t-\tau)-G_{0,\delta}(t-\tau)]
 |u(\tau)|^\alpha u(\tau)\|_{H^2}d\tau\\
&\quad \leq \int_0^t \||u_{\epsilon}(\tau)-u(\tau)
 |(|u_{\epsilon}(\tau)|^{\alpha}
 +|u(\tau)|^{\alpha})\|_{H^2}d\tau
 + \|[G_{\epsilon,\delta}(t)-G_{0,\delta}(t)]u_0\|_{H^2}\\
&\quad +\int_0^t\|[G_{\epsilon,\delta}(t-\tau)-G_{0,\delta}(t-\tau)]|u(\tau)|^\alpha
u(\tau)\|_{H^2}d\tau.
\end{aligned}
\end{equation}
From \eqref{IneDif1} and Proposition \ref{cant} we have
\begin{equation}
\begin{aligned}
\|u_{\epsilon}(t)-u(t)\|_{H^2}
&\leq C\int_0^t \|u_{\epsilon}(\tau)-u(\tau)\|_{H^2}d\tau
 + \|[G_{\epsilon,\delta}(t)-G_{0,\delta}(t)]u_0\|_{H^2}\\
&\quad +\int_0^t\|[G_{\epsilon,\delta}(t-\tau)
 -G_{0,\delta}(t-\tau)]|u(\tau)|^\alpha u(\tau)\|_{H^2}d\tau.
\end{aligned}
\end{equation}
From the Gronwall inequality we arrived at
\[
\|u_{\epsilon}(t)-u(t)\|_{H^2}
\leq \Psi_{\epsilon, \delta}(t)+C\int_0^t
\Psi_{\epsilon, \delta}(\tau)e^{C(t-\tau)}d\tau,
\]
where
\begin{align*}
\Psi_{\epsilon, \delta}(t)
&= \|[G_{\epsilon,\delta}(t)-G_{0,\delta}(t)]u_0\|_{H^2}\\
&\quad +\int_0^t\|[G_{\epsilon,\delta}(t-\tau)-G_{0,\delta}(t-\tau)]|
 u(\tau)|^\alpha u(\tau)\|_{H^2}d\tau.
\end{align*}
Note that because  $\alpha$ is a positive integer, we have
\begin{align*}
\Psi_{\epsilon, \delta}(t)&\leq \|u_0\|_{H^2}
 +\int_0^t\| |u(\tau)|^\alpha u(\tau)\|_{H^2}d\tau\\
&\leq \|u_0\|_{H^2}  +\int_0^t\|u(\tau)\|^{\alpha+1}_{H^2}d\tau\\
&\leq  \|u_0\|_{H^2}+t\|u_0\|^{\alpha+1}_{H^2}.
\end{align*}
Thus $|\Psi_{\epsilon, \delta}(\tau)e^{C(t-\tau)}|\lesssim e^{C(t-\tau)}$.
Since $e^{C(t-\tau)}\in L^1(0,T)$, to obtain our result we just have
to show that $\Psi_{\epsilon, \delta}(t)\to 0$ as $\epsilon\to 0$,
for any $t\in[0,T]$.
 First, observe that
\[
\|[G_{\epsilon,\delta}(t)-G_{0,\delta}(t)]u_0\|^2_{H^2}
=\int_{\mathbb{R}^n}\langle \xi\rangle^4|e^{-it\epsilon|\xi|^2}-1|^2|
\widehat{u_0}(\xi)|^2d\xi.
\]
Since
\[
\langle \xi\rangle^4|e^{-it\epsilon|\xi|^2}-1|^2|\widehat{u_0}(\xi)|^2
\lesssim \langle \xi\rangle^4|\widehat{u_0}(\xi)|^2\quad\text{in }
L^1(\mathbb{R}^n)\]
and $\langle \xi\rangle^4|e^{-it\epsilon|\xi|^2}-1|^2|\widehat{u_0}(\xi)|^2\to
0$, as $\epsilon\to 0$, a.e. on $\mathbb{R}^n$, by the
Lebesgue dominated convergence theorem  we have
\[
\lim_{\epsilon\to 0} \|[G_{\epsilon,\delta}(t)-G_{0,\delta}(t)]u_0\|_{H^2}=0\,.
\]
From Proposition \ref{cant} we obtain
\begin{align*}
\|[G_{\epsilon,\delta}(t-\tau)-G_{0,\delta}(t-\tau)]|u(\tau)|^\alpha
u(\tau)\|_{H^2}&\leq \||u(\tau)|^\alpha u(\tau)\|_{H^2}
\leq \|u(\tau)\|^{\alpha+1}_{H^2}\\
& \lesssim [C(\|u_0\|_{H^2}, \|u_0\|_{L^{\alpha+2}})]^{\alpha+1}.
\end{align*}
Moreover, $\|[G_{\epsilon,\delta}(t-\tau)-G_{0,\delta}(t-\tau)]|u(\tau)|^\alpha
u(\tau)\|_{H^2}\to 0$, as $\epsilon\to 0$; then we
arrived at
\[
\lim_{\epsilon\to 0}\int_0^t\|[G_{\epsilon,\delta}(t-\tau)
-G_{0,\delta}(t-\tau)]|u(\tau)|^\alpha u(\tau)\|_{H^2}d\tau=0,
\]
which completes the proof.
\end{proof}

\subsection*{Acknowledgments}
The first author was supported by VIE-UIS, Proyecto C-2015-01.

\begin{thebibliography}{99}

\bibitem{AceAngTur} A. Aceves, C. De Angelis, S. Turitsyn;
 \emph{Multidimensional solitons in fiber arrays,} 
Optim. Lett. 19 (1995), 329-331.

\bibitem{CaLuEld} C. Banquet, L. C. F. Ferreira, E.J. Villamizar-Roa;
 \emph{On the Schr\"{o}dinger-Boussinesq system with singular initial data,} 
J. Math. Anal. Appl.  400 (2013), 487-496.

\bibitem{CaLuEld1} C. Banquet, L. C. F. Ferreira, E. J. Villamizar-Roa;
 \emph{On the existence and scattering theory for the Klein-Gordon-Schr\"{o}dinger
 system in an infinite $L^2$-norm setting,} Annali di Matematica Pura ed Applicata, 
194 (2015), 781-804.

\bibitem{BL} J. Bergh, J. L\"{o}fstr\"{o}m;
 \emph{Interpolation Spaces}, Springer-Verlag, Berlin-New York, 1976.

\bibitem{WC} T. Cazenave, F. Weissler;
 \emph{Asymptotically self-similar global solutions of the nonlinear 
Schr\"{o}dinger and heat equations,} Math. Z. 228 (1998), 83-120.

\bibitem{Cui1} S. Cui;
\emph{Pointwise estimates for a class of oscillatory integrals and 
related $L^p-L^q$ estimates,} J. Fourier Anal. Appl. 11 (2005) 441-457.

\bibitem{Cui2} S. Cui;
 \emph{Pointwise estimates for a class of oscillatory integrals and 
related $L^p-L^q$ estimates II: Multi-dimension case,} 
J. Fourier Anal. Appl. 6 (2006) 605-626.

\bibitem{GuoCui5b} S. Cui, A. Guo;
\emph{Well-posedness of higher-order nonlinear Schr\"{o}dinger equations 
in Sobolev spaces $H^s(\mathbb{R}^n)$ and applications,} 
Nonlinear Anal. 67 (2007) 687-707.

\bibitem{Dysthe} K. Dysthe;
\emph{Note on a modification to the nonlinear Schr\"{o}dinger equation 
for application to deep water waves,} 
Proc. R. Soc. Lond. Ser. A 369 (1979) 105-114.

\bibitem{Dudley} J. M. Dudley, C. Finot, D. J. Richardson, G. Millot;
\emph{Self-similarity in ultrafast nonlinear optics}
Nature Phys. 3 (9) (2007), 597-603.

\bibitem{Fermann} M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, 
J. D. Harvey, \emph{Self-similar
propagation and amplification of parabolic pulses in optical fibers}, Phys. Rev.
Lett. 84 (2000), 6010-6013.

\bibitem{Fer-Bousq} L. C. F. Ferreira;
 \emph{Existence and scattering theory for Boussinesq type equations with 
singular data,} J. Differential Equations, 250 (2011), 2372-2388.

\bibitem{LucEld1} L. C. F. Ferreira, E. J.  Villamizar-Roa;
 \emph{Self-similarity and asymptotic stability for coupled nonlinear 
Schr\"{o}dinger equations in high dimensions}, Phys. D 241 (2012), 534-542.

\bibitem{FPV} L. C. F. Ferreira, G. Planas, E. J. Villamizar-Roa;
\emph{On the nonhomogeneous Navier-Stokes system with Navier friction boundary 
conditions,} SIAM J. Math. Anal.  45  (2013),  no. 4, 2576-2595.

\bibitem{LucEldPab} L. C. F. Ferreira, E. J. Villamizar-Roa,  P. Braz e Silva;
 \emph{On the existence of infinite energy solutions for nonlinear 
Schr\"{o}dinger equations,} Proc. Amer. Math. Soc. 137 (2009), 1977-1987.

\bibitem{Guo} C. Guo;
\emph{Global existence of solutions for a fourth-order nonlinear 
Schr\"{o}dinger equation in $n+1$ dimensions,} Nonlinear Anal.
 73 (2010), 555-563.

\bibitem{Guo6} C. Guo;
 \emph{Global existence and asymptotic behavior of the Cauchy problem for
 fourth-order Schr\"{o}dinger equations with combined power-type
nonlinearities}, J. Math. Anal. Appl., 392 (2012), 111-122.


\bibitem{GuoCui4} A. Guo, S. Cui;
 \emph{Global existence of solutions for a  fourth-order nonlinear 
Schr\"{o}dinger equation,} Appl. Math. Lett. 19 (2006), 706-711.

\bibitem{GuoCui3} A. Guo, S. Cui;
 \emph{On the Cauchy problem of fourth-order nonlinear Schr\"{o}dinger equations,}
 Nonlinear Anal. 66 (2007), 2911-2930.

\bibitem{GuoCui} A. Guo, S. Cui;
 \emph{On the Cauchy problem of fourth-order nonlinear Schr\"{o}dinger equations,} 
Nonlinear Anal. 66 (2007), 2911-2930.

\bibitem{GuoCui5} C. Guo, S. Cui;
 \emph{Well-posedness of the Cauchy problem of high dimension non-isotropic
 fourth-order Schr\"{o}dinger equations in Sobolev spaces,} 
Nonlinear Anal. 70 (2009), 3761-3772.

\bibitem{Fibich} G. Fibich, B. Ilan,  G. Papanicolaou;
\emph{Self-focusing with fourth-order dispersion,} SIAM J. Appl. Math.,
 62 (2002), 1437-1462.

\bibitem{Hirota} R. Hirota;
 \emph{ Direct Methods in Soliton Theory,} Springer, Berlin, 1980.

\bibitem{Ivano} B. Ivano, A. Kosevich;
 \emph{Stable three-dimensional small-amplitude soliton in magnetic materials,} 
Sov. J. Low Temp. Phys. 9 (1983), 439-442.

\bibitem{Iftimie} D. Iftimie, F. Sueur;
 \emph{Viscous boundary layers for the Navier-Stokes equations with the Navier 
slip conditions,} Arch. Ration. Mech. Anal.  199  (2011),  no. 1, 145–175.

\bibitem{Kar} V.  Karpman;
 \emph{Stabilization of soliton instabilities by higher-order dispersion: 
fourth order nonlinear Schr\"{o}dinger-type equations,} Phys. 
Rev. E, 53 (1996), 1336-1339.

\bibitem{KarSha} V. Karpman, A. Shagalov;
 \emph{Stability of soliton described by nonlinear Schr\"{o}dinger 
type equations with higher-order dispersion,}  Phys. D 144 (2000), 194-210.

\bibitem{Miao} C. Miao, G. Xua, L. Zhao;
 \emph{Global well-posedness and scattering for the defocusing energy-critical
 nonlinear Schr\"{o}dinger equations of fourth order in
dimensions $d\geq 9$,} J. Differential Equations 251 (2011), 3381-3402.

\bibitem{Miao1} C. Miao, G. Xu, L. Zhao;
 \emph{Global well-posedness and scattering for the focusing energy-critical
 nonlinear Schr\"{o}dinger equations of fourth order in the radial case,}
 J. Differential Equations 246 (2009), 3715-3749.

\bibitem{Pau2} B. Pausader;
 \emph{Global well-posedness for energy critical fourth-order Schr\"{o}dinger 
equations in the radial case,} Dyn. Partial Differ. Equ. 4 (2007), 197-225.

\bibitem{Pau} B. Pausader;
 \emph{The cubic fourth-order  Schr\"{o}dinger equation,} J. Funct. Anal., 
256 (2009), 2473-2515.

\bibitem{Pau1} B. Pausader;
  \emph{The focusing energy-critical fourth-order Schr\"{o}dinger equation 
with radial data, Discrete Contin. Dyn. Syst. Ser. A}, 24
(2009), 1275-1292.

\bibitem{Segata} J. Segata;
 \emph{Modified wave operators for the fourth-order nonlinear Schr\"{o}dinger-type
 equation with cubic nonlinearity}, Math. Methods Appl. Sci., 26 (2006), 1785-1800.

\bibitem{SteinLibro1} E. Stein;
 \emph{Harmonic analysis: Real-variable methods orthogonality and oscillatory 
integrals,} Princeton University  Press, New Yersey, 1993.

\bibitem{EldJean} E. J. Villamizar-Roa,  J. E. P\'{e}rez-L\'{o}pez;
 \emph{On the Davey-Stewartson system with singular initial data,} 
Comptes Rendus Mathematique, 350 (2012), 959-964.

\bibitem{Wang} Y. Wang;
 \emph{Nonlinear fourth-order Schr\"{o}dinger equations with radial data,}
 Nonlinear Anal. 75 (2012), 2534-2541.

\bibitem{WenFan} S. Wen, D. Fan;
\emph{Spatiotemporal instabilities in nonlinear Kerr media in the presence 
of arbitrary higher order dispersions,} J. Opt. Soc. Amer. B 19 (2002), 1653-1659.

\bibitem{ZhaGuoSheWei} X. Zhao, C. Guo, W. Sheng, X. Wei;
 \emph{Well-posedness of the  fourth-order perturbed Schr\"{o}dinger type 
equation in non-isotropic Sobolev spaces,} J. Math. Anal. Appl. 382 (2011), 97-109.

\bibitem{ZhuYanZha} S. Zhu, H. Yang, J. Zhang;
 \emph{Blow-up of rough solutions Nonlinear to the  fourth-order nonlinear 
Schr\"{o}dinger equation,} Nonlinear Anal. 74  (2011), 6186-6201.

\end{thebibliography}

\end{document}