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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 108, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/108\hfil Generalized solutions]
{Generalized solutions to the gKdV equation}

\author[M. M. Melo\hfil EJDE-2010/108\hfilneg]
{Maurilio Marcio Melo}

\address{Maur\'ilio M\'arcio Melo \newline
Instituto de Matem\'atica e Estat\'istica,
Universidade Federalde Goi\'as,
Caixa Postal 131, 74001-970 -- Goi\^ania, Brazil}
\email{melo@mat.ufg.br}

\thanks{Submitted October 16, 2009. Published August 5, 2010.}
\subjclass[2000]{35D05, 35Q20}
\keywords{gKdV equation; generalized solution;  singular 
initial condition}

\begin{abstract}
 In this article we study the Cauchy problem in
 $\mathcal{G}_2((0,T)\times \mathbb{R})$
 (the algebra of generalized functions, in the sense of Colombeau)
 for the generalized Korteweg-de Vries equation, with
 initial condition $\varphi \in \mathcal{G}_2(\mathbb{R)}$,
 which contains $H^s(\mathbb{R})$, for $s\in \mathbb{R}$.
\end{abstract}

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

\section{Introduction}

In \cite{biaober1} and \cite{biaober}, Biagioni and
Oberguggenberger have shown that the nonlinear theory of the
generalized functions, introduced by Colombeau \cite{colombeau1},
can be used to deal with the Cauchy problem for the nonlinear
evolution equations. Following this approach, in this article we
study the Cauchy problem
\begin{equation}
u_t+u_{xxx}+a(u)u_x=0,\quad u(0)=\varphi ,  \label{cam}
\end{equation}
where $a(u)=u^3$.

Equations of the form \eqref{cam} are known as generalized
Korteweg-de Vries (gKdV), as opposed to the ordinary KdV, when
$a(u)=u$, and modified KdV (mKdV), when $a(u)=u^2$. The KdV
equation was derived by Korteweg-de Vries as a model for long
waves propagating in a channel. Subsequently, the mKdV equation
has been showed relevant in a number of different physical
systems. In fact, a large class of hyperbolic models has been
reduced to these equations. Another reason to study them is their
relation with inverse scattering theory.

The space chosen to deal with this problem is the Colombeau
algebra $\mathcal{G}_2(\Omega )$, $\Omega =(0,T)\times \mathbb{R}$
which we will describe in section 2. The KdV and mKdV equations
were studied in the same context in \cite{biaober} and
\cite{charlesm}, respectively. The authors obtained results of
existence and uniqueness of solutions in $\mathcal{G}_2(\Omega )$
to the Cauchy problem for these equations and initial condition in
$\mathcal{G}_2(\mathbb{R})$.

The KdV and mKdV equations have an infinite number of conserved
quantities, see \cite{mgk}. But, in general, if $a(u)\neq
u,u^2$, this fact is not true. This property was used in the
proof of the existence of solutions to \eqref{cam} for the cases
$a(u)=u,u^2$, see \cite{biaober} and \cite{charlesm}.

In \cite{melo2}, results of existence and uniqueness of solutions
were established in $\mathcal{G}_2((0,T)\times \mathbb{R})$ to the
Cauchy problem for the equation
\begin{equation}
u_t-30u^2u_x+20u_xu_{xx}+10uu_{xxx}-u_{xxxxx}=0, \label{k3u}
\end{equation}
which belongs to the Lax hierarchy \cite{lax} for the KdV
equation.

In \cite{melo3}, the Cauchy problem was studied for the equation
\begin{equation}
u_t=\sum_{\alpha_{1}}\partial ^{\alpha_{1}}G_{m}(u),
\label{kdvn}
\end{equation}
where $\alpha_{1}\in \mathbf{N}^{n}$ and $G_{m}(u)$ is the
gradient in $u$ of the functional $F_{m}(u)$, where $F_{m}(u)$ is
constant along solutions to the KdV equation in dimension $n$
\[
u_t=u\sum_{i=1}^{n}\frac{\partial u}{\partial
x_i}+\sum_{i,j=1}^{n}\frac{
\partial ^3u}{\partial x_i\partial ^2x_j}.
\]

We observe that the function $a(u)=u^3$, satisfies the
one-sided growth condition
\begin{equation}
\limsup_{|u|\to \infty }|u|^{-4}a(u)\leq 0,  \label{cond}
\end{equation}
which allows global solutions to problem \eqref{cam} in $H^s$,
$s>3/2$, according to \cite{kato}. If $a(u)=u^4$ the
problem \eqref{cam}, in Sobolev spaces, is considered as a
critical case by various reasons: there are no known global
results in the usual Sobolev space; for some initial condition in
$H^{1}$,  Martel and Merle have shown in \cite{merle}, the
finite time blow up; finally, the 4$^{th}$ power is the only one
for which the problem \eqref{cam} has no continuous dependence on
the initial condition when the time for existence of solutions
depend on $L^2$-norm of $\varphi $, see \cite{birnir}.

In \cite{biaio}, \cite{charles} and \cite{melo} other evolution
equations such as Benjamin-Ono (BO), Smith (S), Cubic
Schr\"odinger, Lax hierarchy, are studied in Colombeau's algebras.

The paper is organized as follows: In Section 2, we introduce
notation and some definitions.

In section 3, we study problem \eqref{cam} in the case
$a(u)=u^3$. In Lemmas \ref{lemma1} and \ref{lemma2}, we
establish estimates in which we use the results obtained by Kato
in \cite[Lemmas 4.2 and 4.3]{kato}. These estimates will be used
in the proof of Theorem \ref{teo1}, of existence and uniqueness of
solutions to the Cauchy problem \eqref{cam} with $a(u)=u^3$.

In Section 4, we give a sketch of the proof establishing that the
solution to \eqref{cam} in $\mathcal{G}_2((0,T)\times
\mathbb{R})$, given in the Theorem \ref{teo1}, is related to the
solution obtained by Kato \cite{kato}. We also state the result of
existence and uniqueness of solutions in
$\mathcal{G}_2((0,T)\times \mathbb{R})$ for \eqref{cam} with
$a(u)$ satisfying condition \eqref{cond}. Finally, in Remark
\ref{nuici}, we show what a soliton described by gKdV equation
leads to an example of a nonzero solution to equation of
\eqref{cam} in the Colombeau algebra $\mathcal{G}(
\mathbf{(}0,T)\times \mathbb{R})$, whose restriction to $t=0$ is
zero in $\mathcal{G}(\mathbb{R})$. This shows that we do not have
uniqueness of solutions to the problem \eqref{cam} with initial
condition in $\mathcal{G}( \mathbb{R})$.

\section{Notation and some basic definitions}

Let $\Omega $ be an open subset of $\mathbb{R}^{n}$, $s\in
\mathbb{R}$. We denote by $H^s(\Omega \mathbf{)}$ the usual
Sobolev space $L^2$-type; i.e., $H^s=J^{-s}L^2$, with norm
$\| u\|_s=\| J^su\|_0=(J^su,J^su)^{1/2}$, where
$J=(1-\Delta )^{1/2}$; $\Delta $ is the Laplacian, $\| \cdot \|
_0$ is the norm in $L^2$ and $(\cdot ,\cdot )$ its inner
product; $H^{\infty }(\Omega )=\cap_{k\in \mathbb{Z}}H^{k}(\Omega
)$, $H^{-\infty }(\Omega )=\cup_{k\in \mathbb{Z}}H^{k}(\Omega )$,
$[J^s,M_u]=J^sM_u-M_uJ^s$ is the commutator operator, where
$M_u $ is the multiplication by $u$ operator and $\
\mathcal{D}'(\Omega ) $ is the distribution space.

 Next we exhibit the algebra where we study the Cauchy problem
for the gKdV equation, the space $\mathcal{G}_2(\Omega )$ (algebra
of the generalized functions type Colombeau modeled in the space
$L^2$).

Let $I=(0,1)$ and $\Omega \subset $ $\mathbb{R}^{n}$ be an open set.
We set
\begin{gather*}
\mathcal{E}_2[\Omega ]=(H^{\infty }(\Omega ))^{I}
=\{ \widehat{u} :I\to H^{\infty }(\Omega \mathbf{),\;}\varepsilon \in
I\to \widehat{u}_{\varepsilon }\in H^{\infty }(\Omega )\} ,
\\
\begin{aligned}
\mathcal{E}_{M,2}[\Omega ]
=\big\{&\widehat{u}\in \mathcal{E}_2[\Omega
]:\forall \alpha \in N^{n},\;\exists N>0\text{ and
}C>0,\\
&\text{such that }\| \partial ^{\alpha }
\widehat{u}_{\varepsilon }\|_0\leq
C\varepsilon ^{-N},\text{ for small }\varepsilon \big\},
\end{aligned}
\\
\begin{aligned}
\mathcal{N}_2[\Omega ]
=\big\{&\widehat{u}\in \mathcal{E}_{M,2}[\Omega
]: \forall \alpha \in N^{n}\text{ and }M>0, \exists
C>0,\\
&\text{such that }\| \partial ^{\alpha }
\widehat{u}_{\varepsilon }\|_0\leq C\varepsilon ^{M},
\text{ for small }\varepsilon \big\}.
\end{aligned}
\end{gather*}

We observe that $\mathcal{E}_{M,2}[\Omega ]$ is an algebra with
partial derivatives and $\mathcal{N}_2[\Omega ]$ is an ideal of
$\mathcal{E}_{M,2}[\Omega ]$ which is invariant under
derivatives.

The Colombeau's algebra modelled in $L^2$ is defined as
the quotient space
\[
\mathcal{G}_2[\Omega ]=\frac{\mathcal{E}_{M,2}[\Omega
]}{ \mathcal{N}_2[\Omega ]}.
\]
Its elements $u,v,\dots$ are called generalized
functions in $\Omega $. The multiplication and derivatives in
$\mathcal{G}_2[\Omega ] $ are defined on the representatives.

If in the definition of $\mathcal{G}_2$ we consider the space
$H^{\infty }$ replaced by $W^{\infty ,\infty }=\cap_{k\in
\mathbb{Z}}W^{k,\infty }$, we obtain $\mathcal{G}$ (algebra of
generalized functions defined by Colombeau, see \cite{biagioni1}).

\begin{remark}[\cite{biaober}]\label{obs1} \rm
There is an embedding of $H^{-\infty }(\mathbb{R}^{n})$ into
$\mathcal{G}_2(\mathbb{R}^{n})$ obtained in the following way: we
fix $\rho \in S(\mathbb{R}^{n})$ such that
\[
\int_{\mathbb{R}^{n}}\rho (x)dx=1,\quad
\int_{\mathbb{R}^{n}}x^{\alpha
}\rho (x)dx=0,\quad \forall \alpha \in \mathbf{N}^{n},\;
 |\alpha |\geq 1.
\]
Let $\iota :w\to (w\ast \rho_{\varepsilon })_{\varepsilon }$,
where $\rho_{\varepsilon }(x)=\frac{1}{\varepsilon ^{n}}\rho
(x/\varepsilon)$. This defines a linear injection of
$H^{-\infty }(\mathbb{R} ^{n})$ into
$\mathcal{E}_{M,2}[\mathbb{R}^{n}]$, which induces an embedding
$H^{-\infty }(\mathbb{R}^{n})$ into $\mathcal{G}_2(\mathbb{R} ^{n})$;
so we can see $H^{\infty }(\mathbb{R}^{n})$ as a subalgebra of
$\mathcal{G}_2(\mathbb{R}^{n})$.
\end{remark}

\begin{definition}\label{def2} \rm
For $u\in \mathcal{G}_2((0,T)\times \mathbb{R})$,
 the restriction of $u$ to $\{0\}\times \mathbb{R}$ is the
class of $\widehat{u}_{\varepsilon }(0,\cdot)$ in
$\mathcal{G}_2(\mathbb{R})$, where $\widehat{u}_{\varepsilon }$
is a representative of $u$. We denote this
class by $u|_{\{t=0\}}$ or $u(0)$.
\end{definition}

\begin{definition}\label{def3} \rm
We say that $u\in \mathcal{G}_2(\mathbb{R}^{n})$ is
\emph{associated with the distribution} $w\in H^{-\infty
}(\mathbb{R}^{n})$ if there is a representative $\widehat{u}$ of
$u$ such that $\widehat{u}_{\varepsilon }(\cdot )\to w$ in
$\mathcal{D}'(\mathbb{R} ^{n})$ as $\varepsilon \to 0$. We denote
it by $u\approx w$. We say that $u$, $v$ $\in
\mathcal{G}_2(\mathbb{R}^{n})$ are associated if $u-v\approx 0$.
\end{definition}

\begin{definition}\label{def4} \rm
We say that $u\in \mathcal{G}_2(\Omega )$ is of
$r-(\log)^{1/j}$-type, $2\leq r\leq \infty $,
$j\geq 1$, if it has a representative
$\widehat{u}\in \mathcal{E}_{M,2}[\Omega ]$ such that
\[
\| \widehat{u}_{\varepsilon }\|_{L^{r}}\leq C(|\log \varepsilon
|^{1/j}),\quad \text{ for small }\varepsilon ,
\]
and $r$-bounded-type if
\[
\| \widehat{u}_{\varepsilon }\|_{L^{r}}\leq C,\quad
\text{for small }\varepsilon .
\]
\end{definition}

\begin{remark}\label{rmk2.2} \rm
We also observe a nonlinear property of generalized functions: if
$F\in O_{M}(\mathbb{R}^{l})$; i.e., $F$ is a smooth function and,
together with all its derivatives, grows at most like some power
of $|x|$ as $|x|\to \infty $, then we can define
$F(u_{1},u_2,\dots ,u_{l})\in \mathcal{G}_2(\Omega )$
for $u_i\in \mathcal{G}_2(\Omega )$, $i=1,\dots ,l$,
(see \cite{biagioni1}).
\end{remark}

\begin{definition}\label{solucao}\rm
Let $P(u,\partial ^{\alpha }u)$ be a polynomial in $u$ and
its derivatives. We say that $u$ is a solution to the problem
\begin{gather*}
u_t=P(u,\partial ^{\alpha }u)\quad\text{in }\mathcal{G}_2((0,T)\times
\mathbb{R}), \\
u|_{\{t=0\}}=g\quad\text{in }\mathcal{G}_2(\mathbb{R}),
\end{gather*}
if for every representative
$\widehat{u}\in \mathcal{E}_{M,2}[(0,T)\times \mathbb{R}]$ of $u$ and
$\widehat{g} \in \mathcal{E}_{M,2}[\mathbb{R}]$ of $g$, there are
$\widehat{N}\in $ $\mathcal{N}_2[(0,T)\times \mathbb{R}]$ and
$\widehat{\eta }\in $ $\mathcal{N}_2[\mathbb{R}]$ such that
\begin{gather*}
\widehat{u}_t=P(\widehat{u},\partial ^{\alpha
}\widehat{u})+\widehat{N}\quad\text{in }(0,T)\times \mathbb{R}, \\
\widehat{u}|_{\{t=0\}}=\widehat{g}+\widehat{\eta}\quad
\text{in }\mathbb{R}
\end{gather*}
\end{definition}

We observe that the time interval in this definition is the same
for all representatives. Also, we observe that the problem for
representatives is the classical problem.
For some properties of generalized functions see
\cite{biagioni1,biaober1,colombeau1,egorov,garetto,
grosser,obergu}.

\section{Generalized solutions}

Next, in the proof of Lemma \ref{lemma1}, we use the conservation
laws for the equation of \eqref{cam} obtained by Kato in
\cite[Th 4.2 and eq. 27] {kato}. We note that $T$ in this result is
independent of the initial condition.

\begin{lemma}\label{lemma1}
If $u_{\varepsilon }\in \mathcal{C}((0,T);H^s(\mathbb{R} ))$, with
$s>3/2$, is the solution of the problem
\begin{equation}
u_t+u_{xxx}+u^3u_x=0,\quad u(0)=\varphi_{\varepsilon }\in
H^s( \mathbb{R}),  \label{cauchy}
\end{equation}
given by \cite[Theorem 4.1]{kato}, then
\begin{gather}
\| u_{\varepsilon }(t,\cdot )\|_0=\| \varphi_{\varepsilon }\|
_0,  \label{es0} \\
\| \partial_xu_{\varepsilon }(t,\cdot )\|_0\leq
Cm_{1}^{7}(\varepsilon ),\quad \text{for small }\varepsilon ,
\label{es1} \\
\| \partial_x^2u_{\varepsilon }(t,\cdot )\|_0\leq
C(m_2(\varepsilon ))^{49}\exp (cT(m_0(\varepsilon ))^{5/2}),\quad
\text{for small }\varepsilon ,  \label{es3}
\end{gather}
where $m_{k}(\varepsilon )=\max \{ 1,\| \varphi_{\varepsilon
})\|_{k}\}$, $k\in \mathbf{N}$ and $C=C(T)$.
\end{lemma}

\begin{proof} From \cite[eq.18]{kato} we have
\eqref{es0}. From \cite[eq.19 and 27]{kato} we have
respectively,
\begin{equation}
\| \partial_xu_{\varepsilon }(t,\cdot )\|
_0^2-(a_2(u_{\varepsilon }(t.\cdot ),1)=\| \varphi_{\varepsilon
}'\|_0^2-(a_2(\varphi_{\varepsilon }),1)\text{ } \label{est2}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\frac{d}{dt}[\| \partial_x^2u_{\varepsilon }(t,\cdot )\|
_0^2-\frac{5}{3}(u_{\varepsilon }(t,\cdot )^3\partial
_xu_{\varepsilon }(t,\cdot ),\partial_xu_{\varepsilon }(t,\cdot
))] \\
&=\frac{1}{2}((\partial_xu_{\varepsilon }(t,\cdot
))^{5},1)+5(u_{\varepsilon }(t,\cdot ))^{5}(\partial
_xu_{\varepsilon }(t,\cdot ))^3,1),
\end{aligned} \label{est4}
\end{equation}
where $a_2(\lambda )=\lambda ^5/10$.
For the rest of this article, we
omit the subscript $\varepsilon $ and $(t,\cdot )$ in our
notation. Then, from \eqref{est2} we have
\[
\| \partial_xu\|_0^2=\frac{1}{10}\int u^{5}dx+\| \varphi '\|
_0^2-\frac{1}{10}\int \varphi ^{5}dx.
\]
Thus,
\begin{align*}
\| \partial_xu\|_0^2
&\leq \frac{1}{10}\| u\|_{L^{\infty
}}^3\| u\|_0^2+\| \varphi '\|_0^2+\frac{1}{10}\| \varphi
\|_{L^{\infty }}\| \varphi \|_0^4
\\
&\leq c(\| \partial_xu\|_0^{\frac{3}{2}}\| u\|
_0^{\frac{7}{2}}+\| \varphi '\|_0^2+\| \varphi '\|
_0^{\frac{1}{2} }\| \varphi \|_0^{1/2}\| \varphi \|_0^4
\\
&\leq c(\| \partial_xu\|_0^{\frac{3}{2}}\| u\|
_0^{\frac{7}{2}}+\| \varphi '\|_0^2+\| \varphi '\|_0+\|
\varphi
\|_{1}^{5}) \\
&\leq c(\delta \| \partial_xu\|_0^2+c(\delta )\| \varphi \|
_0^{14}+\| \varphi '\|_0^2+\| \varphi '\|_0+\| \varphi \|
_{1}^{5}),
\end{align*}
since by Gagliardo-Nirenberg $\| u\|_{L^{\infty }}\leq \|
\partial_xu\|_0^{1/2}\| u\|_0^{1/2}$ see \cite{gl};
and by Young's inequalities $ab\leq
\delta a^{p}+c(\delta )b^{q}$. Taking $\delta =1/(2c)$ we
obtain \eqref{es1}. The right-hand side of \eqref{est4} can be
estimated by
\begin{align*}
&c(\| \partial_xu\|_{L^{\infty }}^3\|
\partial_xu\|_0^2+\| u\|_{L^{\infty
}}^{5}\| \partial_xu\|_{L^{\infty }}\|
\partial
_xu\|_0^2) \\
&\leq c(\| \partial_xu\|_0^{\frac{7}{2}}\|
\partial_x^2u\|_0^{\frac{3}{2}}+\| u\|
_0^{5/2}\| \partial_xu\|_0^{5}\|
\partial_x^2u\|_0^{1/2}) \\
&\leq c(\| \partial_xu\|_0^{14}+\|
\partial_x^2u\|_0^2+\| u\|
_0^{\frac{5}{2} }(\| \partial_xu\|_0^{\frac{20}{3}}+\|
\partial_x^2u\|_0^2)).
\end{align*}
Therefore,
\begin{align*}
&\frac{d}{dt}[\| \partial_x^2u\|_0^2-\frac{5}{3}
(u^3\partial_xu,\partial_xu)] \\
&\leq c(\| \partial_xu\|_0^{14}+\|
\partial_x^2u\|_0^2+\| u\|
_0^{5/2}(\| \partial_xu\|_0^{20/3}+\|
\partial_x^2u\|_0^2)) \\
&\leq c(\| \partial_xu\|_0^{14}+\| u\|_0^{5}+\|
\partial_xu\|_0^{40/3}+(1+\| u\|_0^{5/2})\|
\partial_x^2u\|_0^2)).
\end{align*}
Integrating  this inequality from $0$ to $t\leq T$, and using
\eqref{es0} and \eqref{es1}, we obtain
\begin{align*}
&\| \partial_x^2u\|_0^2-\frac{5}{3}
(u^3\partial_xu,\partial_xu) \\
&\leq \| \varphi ''\|_0^2-\frac{5}{3} (\varphi
^3\varphi ',\varphi ')+cT(m_{1}(\varepsilon ))^{98}+c(1+\|
\varphi \|_0^{5/2})\int_0^{t}\|
\partial_x^2u\|_0^2.
\end{align*}
This implies that
\[
\| \partial_x^2u\|_0^2\leq c[\| \varphi \|
_0^{\frac{3}{2}}\| \partial_xu\|_0^{ \frac{7}{2}}+\| \varphi
''\|_0^2+\| \varphi \|
_{1}^{5}+T(m_{1}(\varepsilon ))^{98}+(m_0(\varepsilon
))^{5/2}\int_0^{t}\|
\partial_x^2u\|_0^2],
\]
 or
\[
\| \partial_x^2u\|_0^2\leq c[\| \varphi \|_0^3+\|
\partial_xu\|_0^{7}+\| \varphi ''\|_0^2+\| \varphi \|_{1}^{5}+T(m_{1}(\varepsilon
))^{98}+(m_0(\varepsilon ))^{5/2}\int_0^{t}\|
\partial_x^2u\|_0^2],
\]
or, using \eqref{es1}, we have the inequality
\[
\| \partial_x^2u\|_0^2\leq C(T)(m_2(\varepsilon
))^{98}+c(m_0(\varepsilon ))^{5/2}\int_0^{t}\|
\partial_x^2u\|_0^2]\text{.}
\]
Then, by Gronwall's lemma, we have \eqref{es3}.
\end{proof}

Next, in the proof of Lemma \ref{lemma2}, we follow the
idea in \cite{biaio1} and use the inequalities, obtained by
Kato and Ponce in \cite{kato2}, valid for $s>0$ and
$1<p<\infty$:
\begin{equation}
\| [ J^s,M_{f}]g\|_{L^{p}}\leq c(\|
\partial_xf\|_{L^{\infty }}\| J^{s-1}g\|
_{L^{p}}+\| J^sf\|_{L^{p}}\| g\|_{L^{\infty }}), \label{kp}
\end{equation}
and as a consequence,
\begin{equation}
\| J^s(fg)\|_{L^{p}}\leq c(\| f\|_{L^{\infty }}\| g\|
_{L^{p}}+\| f\|_{L^{p}}\| g\|_{L^{\infty }}). \label{kp1}
\end{equation}

\begin{lemma}\label{lemma2}
If $u_{\varepsilon }\in
\mathcal{C}((0,T);H^s(\mathbb{R} ))$ with $s\geq 2$, is the
solution of  \eqref{cauchy}, we have
\begin{equation}
\| u_{\varepsilon }(t,\cdot )\|_s\leq \| \varphi_{\varepsilon
}\|_s\exp [cT(m_2(\varepsilon ))^{35}\exp cT(m_0(\varepsilon
))^{5/2}].  \label{est5}
\end{equation}
\end{lemma}

\begin{proof}
Applying $J^s$ to equation of \eqref{cauchy},
multiplying by $J^su$ and integrating in $\mathbb{R}$, give
\begin{align*}
\frac{1}{2}\frac{d}{dt}\| u(t)\|_s^2
&=-(J^s(u^3\partial_xu),J^su) \\
&=-([J^s,M_{u^3}]\partial_xu+u^3J^s\partial_xu,J^su) \\
&\leq \| [J^s,M_{u^3}]\partial_xu\|_0\| J^su\|
_0+\frac{3}{2}|(u^2\partial_xuJ^su,J^su)|.
\end{align*}
Using  \eqref{kp} with $f=u^3$, $g=\partial_xu$
and $p=2$, we obtain
\begin{align*}
\frac{1}{2}\frac{d}{dt}\| u(t)\|_s^2
&\leq  c(\| \partial_xu^3\|_{L^{\infty }}\|
J^{s-1}\partial_xu\|_0+\| J^su^3\|_0\|
\partial_xu\|_{L^{\infty }})\| J^su\|_0 \\
&\quad +\frac{3}{2}\| u\|_{L^{\infty }}^2\|
\partial_xu\|_{L^{\infty }}\| J^su\|
_0^2.
\end{align*}
Using \eqref{kp1} with $f=u$,  $g=u^2$ and $p=2$, we obtain
\begin{align*}
\frac{1}{2}\frac{d}{dt}\| u(t)\|_s^2
&\leq c[\| 3u^2\partial_xu\|_{L^{\infty }}\|
J^su\|_0 \\
&\quad +c(\| u\|_{L^{\infty }}\| u^2\|_0+\| u\|_0\| u^2\|
_{L^{\infty }})\|
\partial_xu\|_{L^{\infty }}]\|
J^su\|_0 \\
&\quad +\frac{3}{2}\| u\|_{L^{\infty }}^2\|
\partial_xu\|_{L^{\infty }}\| J^su\|_0^2 \\
&\leq c[\| u\|_{L^{\infty }}^2\|
\partial_xu\|_{L^{\infty }}\| J^su\|_0+(\|
u\|_{L^{\infty }}^2\| u\|_0 \\
&\quad +\| u\|_0\| u\|_{L^{\infty }}^2)\|
\partial_xu\|_{L^{\infty }}]\| J^su\|_0 \\
&\quad+\frac{3}{2}\| u\|_{L^{\infty }}^2\|
\partial
_xu\|_{L^{\infty }}\| J^su\|_0^2 \\
&\leq c\| u\|_{L^{\infty }}^2\|
\partial_xu\|_{L^{\infty }}\| u\|
_s^2\leq c\| u\|_0\| \partial_xu\|_0^{3/2}\|
\partial_x^2u\|_0^{1/2}\| u\|_s^2.
\end{align*}
Therefore,
\begin{align*}
\frac{1}{2}\frac{d}{dt}\| u(t)\|_s^2
&\leq c\| u\|_0\| \partial_xu\|_0^{3/2}\|
\partial_x^2u\|_0^{\frac{1}{2}
}\| u\|_s^2 \\
&\leq c\| u\|_0(\| \partial_xu\|_0^3+\|
\partial_x^2u\|_0)\| u\|_s^2.
\end{align*}
Integrating from $0$ to $t\leq T$ and using Gronwall's lemma we
have
\[
\| u(t)\|_s^2\leq \| \varphi \|_s^2\exp [cT\| \varphi \|
_0\sup_{t\in [ 0,T]}(\| \partial_xu(t)\|_0^3+\|
\partial_x^2u(t)\|_0)].
\]
Then, from \eqref{es1} and \eqref{es3} we have \eqref{est5}.
\end{proof}

\begin{theorem}\label{teo1}
If $\varphi \in \mathcal{G}_2(\mathbb{R})$ is such
that $\varphi , \varphi ', \varphi ''$ are
$2$-$(\log (\log ))^{1/5}$-type; i.e, $\varphi $ have a
representative $\widehat{\varphi }$ such that
\begin{equation}
\| D^{\alpha }\widehat{\varphi }_{\varepsilon }\|_0
\leq C\Big(\log \big(\log (1/\varepsilon )\big)\Big)^{1/5},\quad
\text{for small }\varepsilon ,\; \alpha =0,1,2,  \label{unic}
\end{equation}
then, for all $T>0$, there is a unique solution $u\in
\mathcal{G}_2((0,T)\times \mathbb{R})$ for the problem
\begin{equation}
u_t+u_{xxx}+u^3u_x=0,\quad u(0)=\varphi .  \label{cauchy2}
\end{equation}
\end{theorem}

\begin{proof}
For $\varepsilon $ small enough, we have
$(\log (\log (1/\varepsilon)))^{1/5}
<\big(\log (1/\varepsilon)\big)^{1/70}$, thus condition
\eqref{unic} on $\widehat{\varphi }$ ensues
\begin{gather}
\| \widehat{\varphi }_{\varepsilon }\|_2
\leq C\Big( \log(1/\varepsilon )\Big)^{1/70}, \label{co} \\
\| \widehat{\varphi }_{\varepsilon }\|_0
\leq C( \log (\log (1/\varepsilon )))^{1/5},\quad
\text{for small } \varepsilon . \label{co1}
\end{gather}
For each $\varepsilon >0$, let $\widehat{u}_{\varepsilon }$ be the
solution of \eqref{cauchy} with $u(0)=\widehat{\varphi
}_{\varepsilon }$, given by \cite[corollary 4.7]{kato}. From
\eqref{est5}, \eqref{co} and \eqref{co1} we have
\[
\| u_{\varepsilon }(t,\cdot )\|_s
\leq \| \varphi_{\varepsilon}\|_s\exp [cT(\log(1/\varepsilon))
^{1/2}\exp cT(\log (\log(1/\varepsilon)))^{1/2}].
\]
Since
\[
\Big(\log \big(\log(1/\varepsilon)\big)\Big)^{1/2}\geq
2cT,\quad \text{for }\varepsilon \text{ small  enough,}
\]
we get
\[
\| u(t)\|_s\leq \| \varphi \|_s\varepsilon ^{-N},\quad N>0,
\]
which proves that $(\widehat{u}_{\varepsilon })_{\varepsilon }\in
\mathcal{E}_{M,2}[(0,T)\times \mathbb{R}]$. Thus class $u\in
\mathcal{G}_2((0,T)\times \mathbb{R})$, whose
representative is $(\widehat{u}_{\varepsilon })_{\varepsilon }$
is, by construction, solution to problem \eqref{cauchy2}.

For the uniqueness, let $u$ and
$v$ in $\mathcal{G}_2(\mathbf{(}0,T)\times \mathbb{R})$ be two solutions
of \eqref{cauchy2} with respective representatives
$\widehat{u}_{\varepsilon }$ and $\widehat{v}_{\varepsilon }$,
then, according to Definition \ref{solucao}, there exists
$\widehat{N} \in \mathcal{N}_2[(0,T)\times \mathbb{R}]$ and
$\widehat{n}\in \mathcal{N}_2[\mathbb{R}]$ such that, if $w=u-v$,
we have (we omit $\varepsilon $ and hat in our notation)
\begin{equation}
w_t+u^3w_x+(u^3-v^3)v_x+w_{xxx}=N,\quad
w(0)=n. \label{un}
\end{equation}

By changing representatives, we may assume that $n=0$.
By \cite[proposition 3.4(ii)]{garetto}, see also \cite{grosser},
it is sufficient show that
\begin{equation}
\| w(t)\|_0^2\leq C\varepsilon ^{q},\quad \text{for all } q.
\label{uni}
\end{equation}
Multiplying \eqref{un} by $w$ and integrating over $\mathbb{R}$,
we obtain
\begin{align*}
\frac{d}{dt}\| w(t)\|_0^2
&\leq  c(\| u(t)\|_{L^{\infty}}^2+\| v(t)\|_{L^{\infty
}}^2)(\| \partial_xu(t)\|_{L^{\infty }} \\
&\quad +\| \partial_xv(t)\|_{L^{\infty }})\| w(t)\|_0^2+\| N\|
_0\| w\|_0.
\end{align*}
Gronwall's lemma implies that  for $0\leq t\leq T$,
\begin{align*}
\| w(t)\|_0^2
&\leq \| N\|_0\| w\|_0\exp [T\sup_{0\leq t\leq
T}(\| u(t)\|_{L^{\infty }}^2 \\
&\quad +\| v(t)\|_{L^{\infty }}^2)\big( \Vert
\partial_xu(t)\|_{L^{\infty }}+\| \partial
_xv(t)\|_{L^{\infty }}\big)].
\end{align*}
Sobolev's embbeding implies
\begin{align*}
\| w(t)\|_0^2
&\leq \| N\|_0\| w\|_0\exp [T\sup_{0\leq t\leq T}(\|
u(t)\|_s^2 \\
&\quad +\| v(t)\|_s^2)(\| \partial_xu(t)\|_{s-1}+\|
\partial_xv(t)\|_{s-1})],
\end{align*}
or
\[
\| w(t)\|_0^2\leq \| N\|_0\| w\|_0\exp [cT\sup_{0\leq t\leq
T}(\| u(t)\|_s^3+\| v(t)\|_s^3)].
\]
Thus, from \eqref{es1}, \eqref{es3} and \eqref{est5} and since
$N\in \mathcal{N}_2[(0,T)\times \mathbb{R}]$, we obtain \eqref{uni}.
\end{proof}

\section{Other results}

\begin{remark} \label{rmk4.1} \rm
We observe that, following the same technique and using the result
by Kato in \cite[Corollary 4.7, Lemma 3.1 and Lemma A.6]{kato} it
is possible to show that a similar result holds for problem
\eqref{cam}: If $\varphi \in \mathcal{G}_2(\mathbb{R})$ and its
derivatives are $2$-bounded-type; i.e, $\varphi $ has a
representative $\widehat{\varphi }$ such that
\[
\| \widehat{\varphi }_{\varepsilon }\|_{k}\leq C,\quad
\text{for small }\varepsilon
\]
for all $k\in \mathbf{N}$ and $a(u)$ satisfies \eqref{cond}, then
for all $T>0$, there is a unique solution $u\in
\mathcal{G}_2(\mathbf{(}0,T)\times \mathbb{R})$ of \eqref{cam}
which is also of $2$-bounded-type. More precisely,
\[
\sup_{t\in [ 0,T]}\| \widehat{u}_{\varepsilon }(t)\|
_{k}\leq \widetilde{a}(\| \widehat{\varphi }_{\varepsilon }\|
_{k-1})\| \widehat{\varphi }_{\varepsilon }\|_{k},
\]
where $\widetilde{a}$ is a monotone increasing function depending
only on $a. $ We also observe that $a(u)=u^{r}$, $r<4$, satisfies
\eqref{cond}.
\end{remark}

The following result shows that the generalized solution to the
Cauchy problem \eqref{cauchy2}, is associated with the respective
classical solution $v\in C([0,T];H^s(\mathbb{R}))$ given in
\cite[Corollary 4.7] {kato}.

\begin{proposition} \label{assoc copy(1)}
If $\varphi \in H^s\mathbf{(R}),\;s\geq 2$,
then the solution of problem \eqref{cauchy2}, with initial data
$\iota (\varphi )\in \mathcal{G}_2(\mathbb{R})$ is associated with
the respective classical solution $v\in
C([0,T];H^s\mathbf{(R}))$ given in \cite[Corollary 4.7] {kato}.
\end{proposition}

\begin{proof}[Sketch of proof] It is based on the fact that
\[
\| \rho_{\varepsilon }\ast \varphi \|_s=\| J^s(\rho
_{\varepsilon }\ast \varphi )\|_0=\| \rho_{\varepsilon }\ast
J^s\varphi \|_0\leq \| \rho_{\varepsilon }\|_{L^{1}}\|
\varphi \|_s
\]
is bounded independently of $\varepsilon $, thus we have a unique
generalized solution to \eqref{cauchy2}. The continuous dependence
theorem given by \cite[Theorem 4.6]{kato} gives the association
result.
\end{proof}

\begin{remark} \label{rmk4.2} \rm
The $\delta$-Dirac distribution is in
$\mathcal{G}_2(\mathbb{R})$,  (see Remark \ref{obs1}). If
we replace the embedding of $H^{-\infty } \mathbf{(R})$ into
$\mathcal{G}_2(\mathbb{R})$ in Remark \ref{obs1} by $\iota (w)$,
whose representative is given by $\widehat{w}_{\varepsilon }=\
\widehat{\iota (w)}_{\varepsilon }=w\ast \rho_{h(\varepsilon )}$,
we obtain that the net $\widehat{\delta }_{\varepsilon }=\delta
\ast \rho_{h(\varepsilon )}=\rho_{h(\varepsilon )}$, with
$\widehat{\delta }_{\varepsilon }(x)=\rho_{h(\varepsilon
)}(x)=\frac{1}{h(\varepsilon )}\rho ( \frac{x}{h(\varepsilon )})$,
is a representative of the generalized function $\iota (\delta
)\mathbf{.}$ It is possible to choose $h(\varepsilon )$, such that
condition \eqref{unic} holds; i.e., $\| \widehat{\delta }
_{\varepsilon }\|_2\leq C(\mathbf{\log (\log }\frac{1}{
\varepsilon }))^{1/5}$. Indeed, since $\| \widehat{\delta
}_{\varepsilon }\|_2^2=\| \widehat{\delta }_{\varepsilon }\|
_0^2+\| \widehat{\delta }_{\varepsilon }''\|_0^2$, we obtain
\[
\| \widehat{\delta }_{\varepsilon }\|_2^2=\int_{
\mathbb{R}}\frac{1}{h^2(\varepsilon )}\rho
^2(\frac{x}{h(\varepsilon )}
)dx+\int_{\mathbb{R}}\frac{1}{h^{6}(\varepsilon )}[\rho ''( \frac{x}{h(\varepsilon
)})]^2dx
=\frac{c_{1}}{h(\varepsilon )}+\frac{c_2}{h^{5}(\varepsilon )}.
\]
By the Implicit Function Theorem, we can choose $h(\varepsilon )$
such that
$\frac{c_{1}}{h(\varepsilon )}+\frac{c_2}{h^{5}(\varepsilon )}
=C(\log (\log(1/\varepsilon)))^{2/5}$. Therefore, problem
\eqref{cauchy2} with initial condition $\varphi =\iota (\delta )$
has a unique solution $u\in \mathcal{G}_2(\mathbf{(}0,T)\times
\mathbb{R})$.
\end{remark}

\begin{remark} \label{nuici} \rm
We observe that  \eqref{cauchy2} has a
solitary wave solution, see \cite{linares}:
\[
u(t,x)=[10c\sec
h^2(\frac{3}{2}\sqrt{c}(x+x_0-ct))]^{1/3}.
\]
Taking $c=x_0=\frac{1}{\varepsilon }$, as in the proof given in
\cite{biaober}, we have that the generalized function $u$ with a
representative given by
\[
\widehat{u}_{\varepsilon }(t,x)=[\frac{10}{\varepsilon }\sec
h^2(\frac{3}{2 }\frac{1}{\sqrt{\varepsilon
}}(x+\frac{1}{\varepsilon }-\frac{1}{\varepsilon
}t))]^{1/3},
\]
is a nonzero solution of the equation of \eqref{cauchy2} which
belongs to algebra $\mathcal{G}(\mathbf{(}0,T)\times \mathbb{R})$,
as defined in \cite{biagioni1}. The restriction of $u$ to $t=0$
vanishes in $\mathcal{G}( \mathbb{R})$, yielding the non-uniqueness
of solutions in $\mathcal{G}( \mathbf{(}0,T)\times \mathbb{R})$ to
problem \eqref{cauchy2}. Indeed, introducing the notation $\xi
=\frac{3}{2}\frac{1}{\sqrt{\varepsilon }}(x+ \frac{1}{\varepsilon
}-\frac{1}{\varepsilon }t)$, we can check that each derivative of
$\widehat{u}_{\varepsilon }$ is of the form $\sum
a_{mn}\varepsilon ^{-j}\sec h^{m+\frac{2}{3}}(\xi )\tan h^{n}(\xi
)$, where $m\geq 0,\ n\geq 1$ and $j\geq 1$. Then for each $r$,
the absolute value of the $r$-derivative of
$\widehat{u}_{\varepsilon }(0,\cdot )$ is bounded by
\[
c\varepsilon ^{-j}|\sec h^{2/3}(\frac{3}{2}\frac{1}{\sqrt{
\varepsilon }}(x+\frac{1}{\varepsilon })|,
\]
where $j=j(r)\in \mathbf{N}$ and $c=c(r)>0$. Since
\[
|\sec h^{2/3}(\frac{3}{2}\frac{1}{\sqrt{\varepsilon
}}(x+\frac{1}{ \varepsilon })|\leq |\sec
h^{2/3}(\frac{1}{\sqrt{\varepsilon }} )|\leq
2^{2/3}\exp (-\frac{2}{3}\frac{1}{\sqrt{\varepsilon }}),
\]
for $x+\frac{1}{\varepsilon }\geq \frac{2}{3}$, then all
derivatives of  $\widehat{u}_{\varepsilon }(0,\cdot )$ are bounded
from above by any positive power of $\varepsilon $, thus \
$u(0,\cdot )$ is zero in $\mathcal{G}( \mathbb{R})$. On the other
hand, $\widehat{u}_{\varepsilon }(1,0)=\sqrt[3]{ 10/\varepsilon
}\to \infty $ as $\varepsilon \to 0$. Therefore $u$ is not equal
to zero in $\mathcal{G}(\mathbf{(}0,T)\times \mathbb{R})$, if
$T>1$.
\end{remark}

\subsection*{Acknowledgements}
The author wants to thank H. Biagioni (in memory) and
R. I\'{o}rio, for suggestions in the proof of Lemma \ref{lemma2}
and Theorem \ref{teo1}. The author also wants thank the referee
for suggestions in Remarks \ref{rmk4.2} and \ref{nuici}.
This research was partially supported by the PROCAD/CAPES.


\begin{thebibliography}{00}

\bibitem{biagioni1} {H. A. Biagioni};
\emph{A Nonlinear Theory of Generalized Functions,}
 Lecture Notes in Math., \textbf{1421}, Springer,
Berlin, 1990.

\bibitem{biaio} {H. A. Biagioni and R. J., I\'{o}rio Jr.};
Generalized solutions of the Benjamin-Ono and Smith
equations, \emph{J. Math. Anal. Appl.}, \textbf{182}, (1994),
pp. 465-485.

\bibitem{biaio1} {H. A. Biagioni and R. J., I\'{o}rio Jr.};
Generalized solutions of the generalized KDV equation,
\emph{notes unpublished}.

\bibitem{biaober1} {H. A. Biagioni, and M. Oberguggenberger};
Generalized solutions to the Burgers' equation,
\emph{J. Differential Equations}, \textbf{97} (1992), pp. 263-287.

\bibitem{biaober} {H. A.} {Biagioni, and M. Oberguggenberger};
Generalized solutions to the Korteweg-de Vries and the regularized
long-wave equations, \emph{SIAM J. Math. Anal}., \textbf{23}
(1992), pp. 923-940.

\bibitem{birnir} B. Birnir, C. Kenig, G. Ponce, N.
Svanstedt and V. Vega;
On the ill-posedness of the IVP for the generalized
Korteweg-de Vries and nonlinear Schr\"{o}dinger equation,
\emph{J. London Math.}, \textbf{53} (1996), pp. 551-559.

\bibitem{charles} {C. Bu};
Generalized solutions to the cubic Schr\"{o}dinger equation,
\emph{Nonlinear Analysis, Theory, Methods
and Applications}, \textbf{27} (1996), pp. 769-774.

\bibitem{charlesm} {C. Bu};
Modified KdV equation with generalized
functions as initial values, \emph{J. Math. Phys}., \textbf{36} (1996),
pp. 3454-3460.

\bibitem{colombeau1} {J. F. Colombeau};
 \emph{Elementary Introduction to New Generalized Functions},
North-Holland Math. Studies, \textbf{113}, Amsterdam, 1985.

\bibitem{egorov} {Yu. V. Egorov};
A contribution to the new generalized functions,
 \emph{Russian Math. Survey}, \textbf{45} (1990), pp. 1-49.

\bibitem{gl} {A. Friedman};
\emph{Partial Differential Equations, }
Holt, Rinehart and Winston, New York, 1976.

\bibitem{garetto} {C. Garetto};
Topological structures in Colombeau
algebras II: investigation into the duals of
$\mathcal{G}_{c}(\Omega ),\mathcal{G}(\Omega ),
\mathcal{G}_{s}(\mathbb{R}^{n})$,
//arxiv.org/abs/math.FA/0408278, pp. 1-25.

\bibitem{grosser} M. Grosser, M. Kunzinger, M.
Oberguggenberger and R. Steinbauer;
\emph{Geometric theory of generalized
functions, vol \textbf{537} of Mathematics and its Applications,}
Kluwer, Dordrecht, 2001.

\bibitem{kato} {T}. {Kato};
On the Cauchy problem for
the (generalized) Korteweg-de Vries Equation,
\emph{Studies in Appl. Math. Suppl. Studies}, \textbf{8}
(1983), pp. 93-128.

\bibitem{kato2} {T}. {Kato, G. Ponce};
Commutator estimates and the Euler and Navier-Stokes equations,
 \emph{Comm. Pure Appl. Math.}, \textbf{41}, (1988), pp. 891-907.

\bibitem{lax} {P. D. Lax,} Integral of nonlinear equations of
evolution and solitary waves,
\emph{Comm. Pure Appl. Math.,} \textbf{21}
(1968), pp. 467-490.

\bibitem{melo2} {M. M. Melo};
Generalized solutions to fifth-order
evolution equation, \emph{J. Math. Anal. Appl., }\textbf{259}
 (2001), pp. 25-45.

\bibitem{melo} {M. M. Melo};
 Generalized solutions to the KdV
hierarchy, \emph{Nonlinear Analysis: Theory, Methods and
Applications, } \textbf{37} (1999), pp. 363-388.

\bibitem{melo3} {M. M. Melo};
 Generalized solutions to the KdV
hierarchy in 2-dimension, \emph{Matem\'{a}tica
Contempor\^{a}nea,} \textbf{27}, (2004), pp. 147-168.

\bibitem{merle} {F. Merle};
 Existence of blow-up solutions in the
energy space for the critical generalized KdV equation.
\emph{J. Amer. Math. Soc.}, \textbf{14}, (2001), pp. 555-578.

\bibitem{mgk} {R. Miura, M.,} {C. S. Gardner, and M. D. Kruskal};
Korteweg-de Vries equation and generalizations II: Existence of
conservation laws and constants of motion, \emph{J. Math. Phys.},
\textbf{9} (1968), pp. 1204-1209.

\bibitem{obergu} {M. Oberguggenberger};
\emph{Multiplications of Distributions and Applications to
Partial Differential Equations}, Pitman
Research Notes in Mathematics Series, \textbf{259}, Longman,
New York, 1992.

\bibitem{linares}  G. Ponce and F. Linares;
 \emph{Introduction to Nonlinear Dispersive Equations}. IMPA,
Rio de Janeiro, 2004.

\end{thebibliography}

\end{document}