\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 06, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/06\hfil On the instability of solitary-wave solutions]
{On the instability of solitary-wave solutions for
fifth-order water wave models}

\author[Jaime Angulo Pava\hfil EJDE--2003/06\hfilneg]
{Jaime Angulo Pava}

\address{Jaime Angulo Pava \hfill\break
 Department of Mathematics, IMECC-UNICAMP\\
C.P. 6065, CEP 13083-970-Campinas\\
S\~ao Paulo, Brazil}
\email{angulo@ime.unicamp.br}

\date{}
\thanks{Submitted August 13, 2002. Published January 10, 2003.}
\thanks{Partially supported by grant 99/02636-0 from
FAPESP (S\~ao Paulo-Brazil)}
\subjclass[2000]{35B35, 35B40, 35Q51, 76B15, 76B25, 76B55, 76E25}
\keywords{Water wave model, variational methods, solitary waves, instability}

\begin{abstract}
 This work presents new results about the instability of solitary-wave
 solutions to a generalized fifth-order Korteweg-deVries equation
 of the form
 \[
 u_t+u_{xxxxx}+bu_{xxx}=(G(u,u_x,u_{xx}))_x,
 \]
 where $ G(q,r,s)=F_q(q,r)-rF_{qr}(q,r)-sF_{rr}(q,r)$ for some
 $F(q,r)$ which is homogeneous of degree $p+1$ for some $p>1$. This
 model arises, for example, in the mathematical description of
 phenomena in water waves and magneto-sound propagation in plasma.
 The existence of a class of solitary-wave solutions is obtained by
 solving a constrained minimization problem in $H^2(\mathbb{R})$ which
 is based in results obtained by Levandosky. The instability of
 this class of solitary-wave solutions is determined for $b\neq0$,
 and it is obtained by making use of the variational
 characterization of the solitary waves and a modification of the
 theories of instability established by Shatah  \&  Strauss,
 Bona  \&  Souganidis  \&  Strauss  and Gon\c calves Ribeiro. Moreover, our
 approach shows that the trajectories used to exhibit instability
 will be uniformly bounded in $H^2(\mathbb{R})$.
\end{abstract}

\maketitle

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction} %sec. 1

In this work we study some instability properties of solitary-wave
solutions to a Hamiltonian generalized fifth-order
Korteweg-deVries equation of the form
\[
u_t+u_{xxxxx}+bu_{xxx}=(G(u,u_x,u_{xx}))_x,
\tag{1.1}
\]
where we assume that the nonlinear term has the form
\[
G(q,r,s)=F_q(q,r)-rF_{qr}(q,r)-sF_{rr}(q,r),
 \tag{1.2}
\]
for some $F(q,r)\in C^3(\mathbb{R}^2)$ which is homogeneous of degree $p+1$ for
some $p>1$. That is, we assume
\[
F(\lambda q, \lambda r)=\lambda^{p+1}F(q,r)
\tag{1.3}
\]
for all $\lambda\geq 0$ and $(q,r)\in \mathbb{R}^2$.

It is important to note that equation (1.1) arises as a model for
a variety of physical phenomena. For instance by choosing
$F(u,u_x)=-u^3$, we have that the corresponding model describes
the evolution of gravity-capillary water waves on a shallow layer
([14], [30]), as well as a chain of coupled nonlinear oscillators
([12]) and magneto-sound propagation in plasma ([16]). On the
other hand, for the water wave problem, namely, the irrotational
motion of an inviscid and incompressible fluid under the influence
of gravity, where we have the interaction of small amplitude and
long two-dimensional waves over a shallow horizontal bottom, we
observe that  Olver in [24] derived equation (1.1) as a
second-order expansion for unidirectional wave propagation with a
nonlinear term of the form $F(u,u_x)=uu_x^2-au^3$. Here the
parameters $a$ and $b$ depend on wave amplitude, wave length and
surface tension (see Benney [5] and Craig  \&  Groves [9] where
equation (1.1) with this special choice of $F$ has also been
studied).  Our study  of instability in this paper will consider
the cases (in the water wave framework) when the coefficient $b$
in (1.1) is proportional to $\tau-\frac13$, where $\tau$
represents the Bond number which is a nondimensionalized surface
tension. So, we have that $b$ in (1.1) may be negative, zero or
positive, which occurs in the respective limits $\tau\to
\frac13^{+}$, $\tau=\frac13$ (no surface tension) and $\tau\to
\frac13^{-}$. For a more detailed discussion about higher-order
water wave models equations and specific forms of the function
$G$, we refer the reader to Benney [5], Craig \& Groves [9],
Hunter \& Scheurle [14], Kichenassamy \& Olver [19], Olver [24], Ponce
[25] and the references therein.

Our main interest in this paper is to obtain conditions which will
imply that the orbit generated by a solitary-wave solution of
(1.1) will be unstable. By a solitary-wave solution we mean a
solution of (1.1) of the form $u(x,t)=\varphi(x+ct)$, where $c$
represents the speed of the wave and $\varphi(\xi)$ and its
derivatives tend to zero as the variable $\xi=x+ct$ approaches to
$\pm\infty$. By the orbit generated by the solitary-wave $\varphi$
we mean the set $ \Omega_{\varphi}=\{\varphi(\cdot+y)\;|\;y\in
\mathbb{R}\;\}$, {\it i.e.}, the set of all the translates of
$\varphi$. So, putting this form of $u$ in (1.1) and integrating
once, we see that $\varphi$ must satisfy the fourth-order ordinary
differential equation
\[
\varphi_{xxxx}+b\varphi_{xx}+c\varphi=G(\varphi,\varphi_x,\varphi_{xx}).
\tag{1.4}
\]
We observe that equation (1.4) has also been studied in other
contexts, such as travelling  waves equation for models of both a
elastic strut (Amick \& Toland [1]) and a suspension bridge
(McKenna \& Walter [23]). Now, the problem of existence of solutions
to (1.4) with specific form of $G$ has already been considered by
several authors, by example Weinstein [29], Kichenassamy \& Olver
[19], Kichenassamy [18] and Champneys \& Groves [8]. In fact, in
[29] a variational characterization of solutions of (1.4) in the
case $b<0$ and $F(u,u_x)=|u|^{p+1}$ has been proved, while  in
[19] is given a classification of admissible expression for $G$
which lead to explicit $sech^2$ solitary-wave. Finally, for a
general function $G$ such  as that established in (1.2),
Levandosky in [20] proved via the Concentration Compactness Method
(Lions [21], [22]) (see Theorem 2.2 below) the existence of a
class of solitary-wave solutions of (1.1) providing $c>b^2_{+}/4$,
where $b_{+}=max\{b,0\}$. In this point, we note that our interest
of study will be the class of solitary waves found by Levandosky
above.

Now, if we consider $G$ satisfying (1.2) and we assume the
existence of such a large range of solitary-wave solutions for
(1.1) found in [20], a natural issue arises: to determine whether
they are stable or unstable (see Definition 3.1 below) by the flow
of the fifth-order equation (1.1). This point was considered
initially in [20] in which  a definitive picture for the specific
case $b=0$ was obtained (see subsection 2.1 below for a review of
it). In this case, the method of proving the stability (or
instability) was based by making use of the variational
characterization of the solitary-wave solutions and the theory of
Grillakis \& Shatah \& Strauss established in [13]. So the use of the
function $d(c)=E(\varphi)+c Q(\varphi)$, which is associated with
the conserved quantities $E$ and $Q$ for (1.1), namely,
\[
E(f)=\frac12\int_{-\infty}^{\infty}\;[(f_{xx})^2-b(f_{x})^2
-2F(f,f_x)]\,dx,\quad
Q(f)=\frac12\int_{-\infty}^{\infty}\;f^2\,dx, \tag{1.5}
\]
was a main ingredient in the analysis of stability. In fact, as
well-known  the solitary waves with speed $c$ will be stable if
and only if $d$ is convex at $c$. An explicit formula to $d(c)$
(see (2.5) below) has been obtained if $b=0$ and so in this case
is possible to determine the values of $p$ for which we have
stability or instability, while for $b \neq 0$ it is not easy to
determine the behaviour of $d$. Here we will obtain a theory of
instability exactly when $b \neq 0$. We note that a recent theory
about the linear instability problem of solitary-wave solutions
for equation (1.1), with a more general admissible expressions for
$G$, has been established by Bridges \& Derks in [7] by using the
{\it sympletic Evans function} and the {\it sympletic Evans
matrix} for Hamiltonian evolutions equations.


In this paper we will show a new approach to study instability of
solitary-wave solutions to nonlinear Hamiltonian evolution
equations. Applying this method to equation (1.1), we can obtain
and improve the results established in [20] about the property of
instability of the $\varphi$-orbit, $\Omega_{\varphi}$, where the
solitary-wave solution $\varphi$ is obtained by solving a
constrained minimization problem (see Theorem 2.2 below). Our
approach makes use of the variational characterization of
solutions for (1.4) as well as of modifications made in the
theories   established by Grillakis \& Shatah \& Strauss in [13],
Bona \& Souganidis \& Strauss in [6] and Gon\c calves Ribeiro in [11].
More specifically,  we make an extension of some ideas from Gon\c
calves Ribeiro in [11] about the instability of stationary states
of a semi-linear Schr\"odinger equation  to nonlinear evolution
equations which are not semi-linear. Moreover, in our analysis we
avoid the use of the function $d$. The idea of our approach for
obtaining the instability of the set $\Omega_{\varphi}$ in
$H^2(\mathbb{R})$ will be to find a vector field $B$ (see (3.3) for
definition), which will be defined in a specific neighbourhood
$V(\Omega_{\varphi},\epsilon_0)$ of $\Omega_{\varphi}$, such that
the vector $B(\varphi)$  will produce an unstable direction if we
have that the {\it action} defined by $S(u)=E(u)+c\;Q(u)$ has the
Hessian at $\varphi$ strictly negative along $B(\varphi)$, namely
\[
\langle S''(\varphi)B(\varphi),B(\varphi)\rangle <0.
\tag{1.6}
\]
More exactly, we will obtain that the flow created by $B$ (see
(3.13) below) through $\varphi$ will produce a sequence of points
$\{u_n\}$ such that $u_n\to \varphi$ in $H^2(\mathbb{R})$ as $n\to
\infty$, and the flow generated by equation (1.1) starting in
these points $u_n$, it will escape from
$V(\Omega_{\varphi},\epsilon_0)$ in a finite time. So, as it is
shown in Theorem 4.2 below, if we choose the direction
$B(\varphi)=\varphi+2x\varphi'$ then condition (1.6) will imply
that if $F$ is  homogeneous in $r$ of degree $\beta$, $\beta\in
[0,p+1]$ ( and therefore homogeneous in $q$ of degree $\alpha$
where $\alpha+\beta=p+1$), then the conditions
\[
\begin{gathered}
 b=0\quad\text{and}\quad  \beta>\frac{9-p}{2},\quad\text{or}\\
 b<0\quad\text{and}\quad  \beta\geq \frac{9-p}{2},\quad\text{or}\\
 \beta>\frac{9-p}{2},\; b>0\quad\text{and}\quad  b\quad\text{small},\\
\end{gathered}
 \tag{1.7}
\]
imply that $\Omega_{\varphi}$ is unstable by the flow of equation
(1.1). Moreover, our approach also produces that the trajectories
used to exhibit instability are uniformly bounded in
$H^2(\mathbb{R})$ (see (3.23) below). Therefore, if we have a
local existence theory for equation (1.1) in $H^2(\mathbb{R})$
(see Assumption 2.1 below) then the trajectories in this case,
will be defined for all time and hence the mechanics that leads to
instability in our case is not produced by any singularity
formation or blow-up of solutions. In this point, we would like to
comment that our Assumption 2.1 related with the well-posedness
problem to equation (1.1) in $H^2(\mathbb{R})$,  is induced by the
difficulties that appear when we work with a general non-linear
term as $G(u,u_x,u_{xx})$ in (1.1) (see Section 2 to get examples
of local existence theory of solutions for equation (1.1)).
Finally, we do not know if in a general context the property of
instability can be produced by some singularity formation of
solutions of (1.1), namely, if there is a set of initial data
$u_0\in H^2(\mathbb{R})$ and $T^*=T^*(\|u_0\|_{H^2})$ with
$0<T^*<\infty$, such that the solution $u(t)$ of (1.1) with
$u(0)=u_0$ satisfies that $ \|u(t)\|_{H^2}\to +\infty$ as $t\to
{T^*}^{-}$.


We note that similarly  as occurred  in the approach made in [20],
we do not need in our analysis of instability to show  the
existence of exactly one negative eigenvalue simple and that zero
is a simple eigenvalue with eigenfunction $\varphi'$ associated to
the linear operator
\[
\begin{split}
S''(\varphi)=&\partial_x^4+b\;\partial_x^2+c-F_{qq}(\varphi,
\varphi')+\varphi' F_{qqr}(\varphi, \varphi')
+\varphi'F_{qrr}(\varphi, \varphi')\partial_x\\
&+\varphi''F_{qrr}(\varphi, \varphi')+\varphi''F_{rrr}(\varphi,
\varphi')
\partial_x+
F_{rr}(\varphi, \varphi')\partial_x^2,
\end{split}
\]
such as is required in the instabilities theories of [6] and [13].
We would like to add that, in general for our approach, we do not
know if there is a mechanics that shows that our choice of the
value of the vector field $B$ in $\varphi$, produces that the
restrictions for $b\neq 0$ in (1.7)  are sharp, a situation that
does not occur when we use an explicit form for the function
$d(c)$. In fact, our restrictions about $p$ in (1.7) when $b=0$
are sharp in the light of those obtained in [20] to prove
stability. In this point we would like to note that there is a
typo in Leavandosky's work [20] about the right formula to $d(c)$,
where we must say that the correct expression for $d(c)$ is that
given by (2.5) below. Finally, we think that a possible good
choice for $B(\varphi)$ in cases of evolution equations in
one-dimension ($(x,t)\in \mathbb{R}\times \mathbb{R}$) may be
$B(\varphi)=\varphi+2x\varphi'$, which has given sharp results in
the study of instability  as it has already happened in a recent
work of Angulo ([2]) about the  instability for solitary-wave
solutions of the following generalization of the Benjamin equation
(Benjamin [3], [4])
\[
u_t+(u^n)_x+l\mathcal{H} u_{xx}+u_{xxx}=0,
\]
where $l\in \mathbb{R}$, $\mathcal{H}$ is the Hilbert transform and $n\in
\mathbb{N}$, $n\geq 2$.

The plan of this note is as follows. Section 2 is devoted to give
a review on the results known about the stability and instability
of solitary-wave solutions to equation (1.1) when $b=0$ and
$F(q,r)$ is homogeneous in both $q$ and $r$. In this section we
also establish an assumption about the problem of local
well-posedness to equation (1.1) which will be used in our
instability theory, as well as, the existence of solutions for
equation (1.4) which are constructed by solving a constrained
minimization problem in $H^2(\mathbb{R})$. We finish this section
obtaining some regularities and asymptotics properties for
solitary-wave solutions of equation (1.1). Section 3 contains our
criterium of instability of solitary-wave solutions for (1.1) (see
Theorem 3.5 below). Finally in Section 4, we give our main result
of instability of the ${\varphi}$-orbit $\Omega_{\varphi}$, by the
flow of the fifth-order equation (1.1).

\subsection*{Notation} We denote by $\widehat f$
 the Fourier transform of $f$, which is defined as
 $\widehat f(\xi)=\int_{-\infty}^{\infty}\;f(x)e^{-i\xi x}\,dx$.
$|f|_{L^p}$ denotes the $L^p(\mathbb{R})$ norm of $f$, $1\leq p\leq
\infty$. In particular, $|\cdot|_{L^2}=\|\cdot\|$ and
$|\cdot|_{L^{\infty}}=|\cdot|_\infty$. The inner product of two
elements $f,g\in L^2(\mathbb{R})$ will be denote by $\langle
f,g\rangle$. We denote by $H^s(\mathbb{R})$, $s\in \mathbb{R}$,
 the Sobolev space of all $f$ (tempered distributions) for which the norm $
\|f\|_{H^s}^2=\int_{-\infty}^{\infty}\;(1+|\xi|^2)^{s}|\widehat
f(\xi)|^2\,d\xi$ is finite.

\section{Review and Preliminaries} %sec 2.

\subsection{Stability theory for equation (1.1) when $b=0$ (review)} %2.1

 The problem of existence and stability of solitary-wave solutions of (1.4) with
 $G$ established in (1.2), has been
studied recently by Levandosky in [20]. The results about both
stability and instability obtained in [20], were based essentially
in the possibility of obtaining a solution to (1.4) as the
Euler-Lagrange equation for a constrained minimization problem in
$H^2(\mathbb{R})$ (because of this approach we have the assumptions
established in (1.2)). More exactly,  we have the following main
results of [20]: It considers the functionals $I_{c,b}, K\in
C^2(H^2(\mathbb{R});\mathbb{R})$ defined by
\[
I_{c,b}(f)=\frac12\int_{-\infty}^{\infty}\;[
(f_{xx})^2-b(f_{x})^2+cf^2 ]\,dx \tag{2.1}
\]
and
\[
K(f)=\int_{-\infty}^{\infty}\; F(f,f_x)\,dx. \tag{2.2}
\]
Define for $\lambda>0$ and $c>b_{+}^2/4$, where $b_{+}=\max
\{b,0\}$,  the family of minimization problems
\[
M_c(\lambda)=\inf\{I_{c,b}(f):f\in H^2(\mathbb{R})\quad\text{and}\quad
K(f)=\lambda\}. \tag{2.3}
\]
Then using the Concentration Compactness Method (see Lions
[21,22]), it is shown in Theorem 2.3 in [20] (see also Theorem 2.2
below) that (2.3) has a solution which after a rescale (using that
$G$ is homogeneous) is a classical solution $\varphi_c$ of (1.4).
Now, with regard to the stability theory we have initially the
following classical function $d(c)$ (see
Grillakis \& Shatah \& Strauss [13] and Bona \& Souganidis \& Strauss [6])
defined for  $c>b_{+}^2/4$, by
\[
d(c)=E(\varphi_c)+c \;Q(\varphi_c)
\tag{2.4}
\]
where $\varphi_c$ is {\it any } solitary-wave solution of (1.4)
obtained via the minimization problem (2.3) and $E$, $Q$ are
defined by (1.5). We note that in terms of $E$, the evolution
equation (1.1) has the following Hamiltonian form
\[
u_t=-\partial_xE'(u),
\]
and the solitary-wave equation (1.4) takes the form
\[
E'(\varphi_c)+c\;Q'(\varphi_c)=0.
\]
Therefore, from the variational characterization of $\varphi_c$,
(2.1) and (2.2), it is obtained the following main form for
$d(c)$:
\[
d(c)=\frac{p-1}{p+1}\;I_{c,b}(\varphi_c)=\frac{p-1}{2}\;K(\varphi_c)=\frac{p-1}{2}\Big(
\frac{2M_c(1)}{p+1}\Big)^{(p+1)/(p-1)}.
\]
So, with the condition that $d''(c)>0$ implies stability, it is
obtained that the following set of solitary-wave solutions for
(1.1)
\[
\mathcal{G}_c=\;\{\varphi\in  H^2(\mathbb{R}):
2(p-1)I_c(\varphi_c)=(p^2-1)K(\varphi_c)= 2(p+1)d(c)\;\},
\]
is a set stable with respect to the flow generated by (1.1). We
note that this criterium of stability works sharply if we have a
good expression to $d(c)$. In fact, in the case when $b=0$ in
(1.1) and $F(q,r)$ is homogeneous in $r$ of degree $\beta$,
$\beta\in[0,p+1]$ (and therefore homogeneous in $q$ of degree
$\alpha$ where $\alpha+\beta=p+1$), it was found in [20] that for
all $c>0$
\[
d(c)=d(1)c^{\gamma}\quad\text{where}\quad
\gamma=\frac{3p-2\beta+5}{4(p-1)}, \tag{2.5}
\]
and therefore we have that the set $\mathcal{G}_c$ will be stable if
$2-\beta<\alpha<10-3\beta$. In particular, when $F(q,r)$ depends
only on $q$ we have stability of the set $\mathcal{G}_c$ for $1<p<9$.
In the case $b\neq 0$ in (1.1) an explicit formula for $d(c)$ is
not known yet.

Finally with regard to the problem of instability, it has been
established in [20] that if we suppose that ``{\it there exists a
choice $\varphi(c)$ which is $C^1$ as a map from
$(b_{+}^2/4,+\infty)$ to $H^2(\mathbb{R})$ such that $\varphi(c)\in
\mathcal{G}_c$}'' and $d''(c)<0$, then the set $
\Omega_{\varphi_c}=\{\varphi_c(\cdot+y)\;|\;y\in \mathbb{R}\;\}$ is
unstable by the flow of the evolution equation (1.1). So, for
$b=0$ and $F(q,r)$ homogeneous in $r$ of degree $\beta$,
$\beta\in[0, p+1]$, it follows from (2.5) that
$\Omega_{\varphi_c}$ will be unstable for $\alpha>10-3\beta$,
where $\alpha+\beta =p+1$. In particular, when $F(q,r)$ depends
only on $q$ we have instability of $\Omega_{\varphi_c}$ for $p>9$.
We observe that via our approach of instability, we do not need to
use the function $d$ and hence the existence of a smooth curve of
solitary-wave solutions depending on $c$ is not necessary in our
analysis.



\subsection{Cauchy problem and existence of solitary-wave
solutions for equation (1.1)} %2.2

\vskip.2cm

In this subsection, we establish an assumption about the
initial-value problem to (1.1) and  also give some results
concerning to the existence of solitary-wave solutions, as well
as, about regularities and asymptotic properties of these
solutions.

Here we need to make  an assumption more than an affirmation about
the well-posedness problem to equation (1.1) in the space
$H^2(\mathbb{R})$, that is, about existence, uniqueness, persistence
property and continuous dependence of the solution upon the
initial data. This assumption is naturally induced by the
difficulties which appear when we work with a general non-linear
term as $G(u,u_x,u_{xx})$ in (1.1). In fact, when $F(u,u_x)$
depends only on $u$  is possible to obtain a local well-posedness
(and global also) theory in $H^2(\mathbb{R})$ (see Kato [15], Saut
[26]). For more general nonlinearities, for example
$F(u,u_x)=uu_x^2-u^3$ (so that
$G(u,u_x,u_{xx})=-3u^2-u_x^2-2uu_{xx}$), Ponce in [25] has shown a
well-posedness theory in $H^s(\mathbb{R})$ with $s\geq 4$. Finally in
[17], Kenig \& Ponce \& Vega have shown that the following
higher-order nonlinear dispersive equation
\[
\partial_t u+\partial_x^{2j+1}u+P(u, \partial_x u,\dots, \partial_x^{2j}u)=0,
\quad x,t\in\mathbb{R},\;\;j \in\mathbb{N},
\]
where $P(\cdot)$ is a polynomial having no constant or linear terms, is locally well posed in the
weighted Sobolev spaces $H^s(\mathbb{R})\cap L^2(|x|^mdx)$ for some $s$ (in general sufficiently large)
and $m\in \mathbb{N}$.


\begin{assumption} \label{a21.}\rm
 For any $u_0 \in H^2(\mathbb{R})$ there
 exist $T=T(\|u_0\|_{H^2})>0$
and a unique solution $u(t)\equiv U(t)u_0\in C([-T,T]; H^2(\mathbb{R}))$ of (1.1) with $u(x,0)=u_0(x)$. Moreover, for $T_1<T$ the map
$u_0\to U(t)u_0$ is continuous from $H^2(\mathbb{R})$ to
$C([-T_1,T_1]; H^2(\mathbb{R}))$.
\end{assumption}

\noindent\textbf{Remark:} Since our analysis of instability is based in the
Sobolev space $H^2(\mathbb{R})$, Assumption 2.1 is sufficient to our
purpose. But we can also suppose a local well-posedness theory to
equation (1.1) in a general Hilbert space $Z$ such that $Z \subset
H^2(\mathbb{R})$ and the embedding $Z\hookrightarrow H^2(\mathbb{R})$ is
continuous, so we can deduce the existence of a continuous curve
$u(t)\in C([0,T]; H^2(\mathbb{R}))$ which is solution of (1.1).


Now we establish a result of existence of solitary-wave solutions
for (1.4) based in the minimization problem (2.3).

\begin{theorem} \label{thm2.2}
Suppose $G$ has the structure established in (1.2) and $F(q,r)$ is
homogeneous of degree $p+1$ for some $p>1$. Moreover, assume that there is some
 $f\in H^2 (\mathbb{R})$
such that $K(f)>0$ and $c>b_{+}^2/4$ $, $ where $b_{+}=\max
\{b,0\}$, then:
\begin{itemize}

\item[(i)] Any minimizing sequence $\{\psi_k\}_{k\in \mathbb{N}}$ for $M_c(\lambda)$ is
 relatively compact in $H^2(\mathbb{R})$ up to translations, i.e., there are sequences
 $\{\psi_{n_k}\}_{k\in \mathbb{N}}$ and $\{y_{n_k}\}_{k\in \mathbb{N}}\subset \mathbb{R}$ such that
 $\psi_{n_k}(\cdot-y_{n_k})$ converges strongly in
 $H^2(\mathbb{R})$ to some $\psi$, which is a minimum of $M_c(\lambda)$.

\item[(ii)] If the variational problem (2.3) has a solution, then there
exists a $\mu$ such that every corresponding minimum point
$\varphi$ of $M_c(\mu)$ satisfies
\[
\varphi_{xxxx}+b\varphi_{xx}+c\varphi=G(\varphi,\varphi_x,\varphi_{xx}).
\tag{2.6}
\]
Moreover, $\mu=\Big [\frac{M_c(1)}{p+1}\Big ]^{\frac{p+1}{p-1}}$.
\end{itemize}
\end{theorem}

\noindent\textbf{Proof}\quad The proof of existence of minimum is based on the
Concentration Compactness Method as it was shown in [20]. The
value of $\mu$ is obtained via the relations
\[
qF_q(q,r)+rF_r(q,r)=(p+1)F(q,r)
\]
and $M_c(\lambda)=\lambda^{2/(p+1)}M_c(1)$. \hfill$\square$


Now we will obtain some  regularity and asymptotic properties of
the solutions of equation (1.4) obtained via Theorem 2.2.
Initially, from the homogeneity of $F$ and (1.2) we have that
\[
|G(q,r,s)|\leq
C\big(|q|^p+|r|^p+|r||q|^{p-1}+|s|[|q|^{p-1}+|r|^{p-1}]). \tag{2.7}
\]
Since $\varphi\in H^{2}(\mathbb{R})$, we know from Sobolev's embedding
that both $\varphi$ and $\varphi'$ are in $L^{\infty}(\mathbb{R})\cap
L^{2}(\mathbb{R})\cap C_0(\mathbb{R})$ (where $C_0(\mathbb{R})$ is the set of
all continuous function on $\mathbb{R}$ that converges to zero at the
infinity) and so from (2.7), we get
$G(\varphi,\varphi',\varphi'')\in L^{2}(\mathbb{R})$. Since $\varphi$
is a solution of (2.6) we have then that $\varphi\in H^{4}(\mathbb{R})$ and so
\[
\varphi, \varphi', \varphi'', \varphi'''\in L^{\infty}(\mathbb{R})\cap
C_0(\mathbb{R}).
\tag{2.8}
\]
Moreover, from (1.2) it follows that
$G(\varphi,\varphi',\varphi'')\in C^{1}(\mathbb{R})$ and from
Sobolev's embedding  theorem it follows that
\[
|G(\varphi,\varphi_y,\varphi_{yy})|_{L^1}
\leq C \|\varphi\|_{H^2}.
\]

Now, our attention is turned to the asymptotic properties of
solutions for (2.6). Initially, we choose $c$ such that $min_{x\in
\mathbb{R}}\{c-b\xi^2+\xi^4\}>0$, that means, $c>b_{+}^2/4$ where
$b_{+}=\max \{b,0\}$, so we can write
\[
k(\xi)\equiv \frac{1}{\xi^4-b\xi^2+c}=\frac{1}{\xi^4+2\alpha^2
cos(2\theta) \xi^2+\alpha^4} \tag{2.9}
\]
where $\alpha=c^{1/4}$ and $cos(2\theta)=-\frac{b}{2\sqrt{c}}$ for
$-\frac{\pi}{2} <\theta<\frac{\pi}{2}$. Then from [10] we find
that there exists an even function $Z\in L^2(\mathbb{R})\cap L^1(\mathbb{R})$ such that its Fourier transform satisfies $\widehat
Z(\xi)=k(\xi)$, more precisely, we have
\[
Z(x)=\frac{\pi}{2}\alpha^{-1/3}e^{-\alpha |x|
\cos\theta}\sin(\theta+\alpha |x| \sin \theta)\csc(2\theta). \tag{2.10}
\]
Note that $Z\in C^\infty (\mathbb{R}-\{0\})$ and $Z^{(n)}\in L^1(\mathbb{R})$ for $n=1,2,3,\dots$. Now, by writing $\mathcal{M}=\partial_x^4
+b\partial_x^2$ we can rewrite equation (2.6) in the integral form
\[
\varphi(x)=(c+\mathcal{M})^{-1}G(\varphi,\varphi',\varphi'')(x)
=\int_{-\infty}^{\infty}\;Z(x-y)G(\varphi,\varphi_y,\varphi_{yy})\,dy.
 \tag{2.11}
\]
So, from Young's inequality on convolutions we get that for all
$n$
\[
\begin{split}
|\varphi^{(n)}|_{L^1}&\leq |Z^{(n)}|_{L^1}
|G(\varphi,\varphi_y,\varphi_{yy})|_{L^1}<\infty,\\
\|\varphi^{(n)}\|_{L^2}&\leq |Z^{(n)}|_{L^1}
\|G(\varphi,\varphi_y,\varphi_{yy})\|_{L^2}<\infty.
\end{split}
\]
Finally, from the following results: (2.8),
$G(\cdot,\cdot,\cdot)\in C^{1}(\mathbb{R}^3)$, $c>b_{+}^2/4$, and the
stable-manifold theorem (see Lemma 2.4 in [20]) we have that
$\varphi, \varphi', \varphi'', \varphi''' $
 decay exponentially to zero at $\pm \infty$.  Then using (2.11), (2.7) and (2.8), we obtain that for all $n$,
$\varphi^{(n)}$ also decay  exponentially to zero at $\pm \infty$.
Therefore,  we have shown the following Lemma.


\begin{lemma} \label{lm2.3}
Suppose that $\varphi$ is a solution of (2.6)
obtained via Theorem 2.2-(ii). Then $\varphi\in H^n(\mathbb{R})$,
$\varphi^{(n)}\in L^1(\mathbb{R})$, $n=0,1,2,\dots$, and there exist
$\sigma>0$ and constants $C_n>0$ such that
\[
|\varphi^{(n)}(x)|\leq C_n e^{-\sigma|x|},\quad\text{for all }x\in \mathbb{R}.
\]
Moreover, $x\varphi^{(n)}\in L^1(\mathbb{R})$ for $n\in \mathbb{N}$.
\end{lemma}


We finish this Section giving for $u(t)$, solution of (1.1), an
estimate of how fast its tail near infinity grows with $t$. This
estimate will be a key step in the proof of instability. We note
that this type of information for solutions of dispersive
evolution equations has already used as an essential ingredient in
studies about instability (see [6] and [28]).


\begin{theorem} \label{thm2.4}
Let $p\geq 2$ and suppose Assumption 2.1.
Assume  $u_0\in H^{2}(\mathbb{R})\cap L^{1}(\mathbb{R})$ and $\mathcal{M}
u_0\equiv (\partial_x^4+b\partial_x^2)u_0\in L^{1}(\mathbb{R})$. If
$u(t)$ is the solution of (1.1) with initial data $u(0)=u_0$, then
\[
\sup_{-\infty<x<\infty}\Big|\int_{-
\infty}^xu(y,t)\,dy\Big|\leq C\Big (1+t^{\theta/(1+\theta )}\Big)
\]
for $0\leq t<T_0$, where $T_0$ denotes the existence time for $u$.
Moreover, $\theta=2$ if $b\neq 0$,  $\theta=4$ if $b=0$ and the
constant $C$ depends only on $sup\;\{\|u(t)\|_{H^2}\;:\;0\leq
t<T_0\}$, $ |u_0|_{L^1}$ and $|\mathcal{M} u_0|_{L^1}$.
\end{theorem}

\noindent\textbf{Proof}\quad  The proof follows the same lines of the proof of
Theorem 4.3 in [28] or Lemma 5.12 in [20], and is based
essentially on the following estimate for the free group $U_0(t)$
associated to the linear evolution equation
\[
u_t+u_{xxxxx}+bu_{xxx}=0,\qquad u(0)=u_0,
\]
namely, for all $t>0$ we have
\[
|U_0(t)u_0|_\infty\leq C_0(1+t)^{-1/(1+\theta)}|u_0|_{L^1}
\]
where $\theta=2$ if $b\neq 0$ and $\theta=4$ if $b=0$.
\hfill$\square$

\section{Instability Theory for Equation (1.1)} %sec. 3

 In this section the attention is turned to establish
a criterium of instability for solitary-wave solutions associated
to equation (1.1), which are obtained via the variational problem
(2.3). By using the notations from [11], for any $X\subset
H^2(\mathbb{R})$ and $\delta>0$ we define $\mathcal{V} (X,\delta)$, the
$\delta$-neighbourhood of $X$
 in $H^2(\mathbb{R})$, by
\[
\mathcal{V} (X,\delta)=\cup_{v\in X} B_\delta(v)=\{g
\in H^2(\mathbb{R}) : \inf_{v\in X}\|v-g\|_{H^2}<\delta\}
\]
where $B_\delta(v)=\{u\in H^2(\mathbb{R}) : \|u-v\|_{H^2}<\delta\}$.
So, based in Assumption 2.1 we have the following definition of
stability.


\begin{definition} \label{def3.1} \rm
We say that $X\subset H^2(\mathbb{R})$ is
stable by the flow of (1.1) if and only if for any $\epsilon>0$
there exists $\delta>0$ such that for all $u_0 \in \mathcal{V}
(X,\delta)$,  the solution $u(t)$ of (1.1) with initial data
$u(0)=u_0$ satisfies $u(t)\in \mathcal{V} (X,\epsilon)$ for all $t\in
\mathbb{R}$. Otherwise, we say $X$ is unstable by the flow of (1.1).
\end{definition}

For each $y\in \mathbb{R}$, let $\tau_y$ be the translation operator
defined by $\tau_yv=v(\cdot+y)$. If $Y\subset L^p(\mathbb{R})$, we
denote $\Omega_{Y}=\{\tau_yv\;|\;y\in \mathbb{R}, v\in Y\}$. With this
notation we introduce for $\varphi$ solution of (1.4), the set
$\Omega_{\varphi}\equiv \{\tau_y \varphi\;|\;y\in \mathbb{R}\}$,
called the $\varphi$-orbit.

In the next, we give some basic Lemmas which will be used in the
proof of the criterium of instability   given in Theorem 3.5.

\begin{lemma} \label{lm3.2}
There exist $\epsilon_0>0$ and a unique
function $\Lambda:\mathcal{V} (\Omega_{\varphi},\epsilon_0)\to \mathbb{R}$,
which is a $C^2$-functional, such that $\Lambda(\varphi)=0$ and
such that for all $v\in \mathcal{V} (\Omega_{\varphi},\epsilon_0)$ and all
$w\in \Omega_{\varphi}$,
$\|v-\tau_{\Lambda(v)}\varphi\|\leq \|v-w\|$.
Moreover, for $y\in \mathbb{R}$
\[
\begin{gathered}
\Lambda(\tau_y v)=\Lambda(v)+y\\
\Lambda'(v)=-\;\frac{1}{\langle
v,\varphi''(\cdot+\Lambda(v))\rangle}\varphi'(\cdot+ \Lambda(v)).
\end{gathered} \tag{3.1}
\]
In particular, for $v\in \Omega_{\varphi}$ we have
\[
\begin{gathered}
\langle\Lambda'(v),v\rangle=0,\\
\Lambda'(v)=\frac{1}{\|\varphi'\|^2}v'.
\end{gathered} \tag{3.2}
\]
\end{lemma}

For the proof of this lemma see Theorem 3.1 in [11] or Lemma 3.3 in [2].


\noindent\textbf{Remark:} We note that the value $\Lambda(v)$, obtained in
Lemma 3.2, can be seen as the ``optimal" translation for $\varphi$
to be the best approximation of $v$ to the $\varphi$-orbit,
$\Omega_{\varphi}$, in the $L^2(\mathbb{R})$-norm .


Now, we define our main vector field in the study of instability.
We consider initially a function $\psi\in L^\infty(\mathbb{R})$ such
that $\psi' \in H^2(\mathbb{R})$ and $\varphi$ a nonzero-solution of
(1.4), then for $v\in V(\Omega_{\varphi},\epsilon_0)$ we define
$B_\psi (v)$ by the formula
\[
B_\psi(v)\equiv\tau_{\Lambda(v)}\psi' - \frac{\langle
v,\tau_{\Lambda(v)}\psi'\rangle} {\langle
v,\tau_{\Lambda(v)}\varphi''\rangle}\;\tau_{\Lambda(v)}\varphi''.
\tag{3.3}
\]
We note that the vector field  $B_\psi$ is an extension of the
formula (4.2) in [6] as well as of the formula (5.9) in [20] (a
similar vector field was also used in [2] and [11]). Now, for
$v\in \Omega_{\varphi}$ we can obtain an easy  geometric
interpretation of the vector $B_\psi (v)$ as being the derivative
of the `` orthogonal component of $\tau_{\Lambda(v)}\psi$ with
regard to $\tau_{\Lambda(v)}\varphi'$ ". In fact, let $y\in \mathbb{R}$ such that $v=\tau_y\varphi$. Then from Lemma 3.2 it follows
that $\Lambda(v)=y$. Now, even that $\psi\notin L^2(\mathbb{R})$ we
have from Lemma 2.3 that
\[
\frac{\langle v,\tau_{\Lambda(v)}\psi'\rangle} {\langle
v,\tau_{\Lambda(v)}\varphi''\rangle}\;\tau_{y}\varphi'=\frac{\langle
\tau_y\varphi,\tau_{y}\psi'\rangle} {\langle
\tau_y\varphi,\tau_{y}\varphi''\rangle}\;\tau_{y}\varphi'=\frac{\langle
\tau_y\varphi',\tau_{y}\psi\rangle}
{\|\tau_{y}\varphi'\|^2}\;\tau_{y}\varphi'\equiv P_\shortparallel.
\]
So, defining $Q_\perp\equiv \tau_{y}\psi-P_\shortparallel$ we
obtain that $\langle Q_\perp,\tau_{y}\varphi'\rangle =0$ and
$B_\psi(v)=\partial_x Q_\perp$.


Next, we have some basic properties  of the vector field
$B_\psi$.

\begin{lemma} \label{lm3.3}
For $\psi\in L^\infty(\mathbb{R})$ such that
$\psi', \psi'' \in H^2(\mathbb{R})$ and $\varphi$ nonzero-solution of
(1.4), we have
\begin{gather*}
 B_\psi:V(\Omega_{\varphi},\epsilon_0)\to H^2(\mathbb{R})\quad\text{is}\;
C^1\quad\text{with bounded derivative},\tag{3.4}\\
 B_\psi \quad\text{commutes with translations}, \tag{3.5}\\
 \langle B_\psi(v),v\rangle=0 \quad \text{for
all }v\in V(\Omega_{\varphi},\epsilon_0)
 ,\tag{3.6}\\
 \text{if } \langle\varphi,\psi'\rangle=0\quad\text{then}\quad
  B_\psi(\varphi)=\psi', \tag{3.7}\\
 \text{if }
\langle\varphi',\psi'\rangle=0\quad\text{then}\quad \langle
B_\psi(v),v'\rangle=0,
\quad \text{for all}\;v\in \Omega_{\varphi}. \tag{3.8}\\
\end{gather*}
\end{lemma}

\noindent\textbf{Proof}\quad The proof of statements (3.4), (3.5),
(3.6) follows the same lines from the proof of Lemma 3.5 in [2].
Statements (3.7) and (3.8) follow from equality
$\Lambda(\varphi)=0$ and from the formula
\[
B_\psi(\varphi)=\psi'+\frac{\langle\varphi,\psi'\rangle}
{\|\varphi'\|^2}\varphi''.
\tag{3.9}
\]
\hfill$\square$

Before establishing our instability criterium, we need to assure
the convergence of an integral which will be used in our analysis.

\begin{lemma} \label{lm3.4}
Let $p\geq 2$ and $\varphi$ a solution of (2.6) obtained via
Theorem 2.2-(ii) . Assume that $u_0\in
V(\Omega_{\varphi},\epsilon_0)$ such that $u_0, \mathcal{M}
u_0\equiv (\partial_x^4+b\partial_x^2)u_0 \in L^{1}(\mathbb{R})$.
If $u(t)$ is a solution of (1.1) which corresponds to the initial
data $u_0$ and $u(t)\in V(\Omega_{\varphi},\epsilon_0)$
 for $t\in [0,T_1]$,
then we have that for
$\psi(x)=\int_{-\infty}^x[\varphi(y)+2y\varphi'(y)]dy$ it follows
that
\[
\mathcal{A}_{\psi}(u(t))\equiv\int_{-\infty}^{\infty}\psi(x-
\Lambda(u(t)))u(x,t)\,dx<\infty,\quad\quad\text{for all }t\in
[0,T_1]. \tag{3.10}
\]
\end{lemma}

\noindent\textbf{Proof}\quad Let $H$ be the Heaviside function and
$\gamma=\int_{\mathbb{R}}[\varphi(y)+2y\varphi'(y)]dy$ (note that
$\gamma<\infty$ from Lemma 2.3) then
\[
\mathcal{A}_{\psi}(u(t))=\int_{-\infty}^{\infty}[\psi(x-\Lambda(u(t)))-\gamma
H(x- \Lambda(u(t)))]u(x,t)dx+\gamma \int_{\Lambda(u(t))}^\infty
u(x,t)dx.
\]
It follows then from Cauchy-Schwarz  inequality, (1.5) and Theorem
2.4 that
\[
|\mathcal{A}_{\psi}(u(t))|\leq \|\psi-\gamma
H\|\|u_0\|+|\gamma|C_0\Big (1+t^{\theta/(1+\theta)}\Big),
\]
and therefore
\[
|\mathcal{A}_{\psi}(u(t))|\leq K_1\Big (1+t^{\theta/(1+\theta)}\Big )
\tag{3.11}
\]
where $\theta$ is given by Theorem 2.4 and $K_1$ is a constant. To
show that $\psi-\gamma H\in L^2(\mathbb{R})$ it is sufficient to know
that $\int_\mathbb{R} |y|^{1/2}|\varphi(y)|+ |y|^{3/2}|\varphi'(y)|
dy<\infty$, which is true by Lemma 2.3. This proves the Lemma.
\hfill$\square$

Now we establish a sufficient condition of instability for
solitary-wave solutions of equation (1.1) which are obtained via
the variational problem (2.3).

\begin{theorem}[Criterium of Instability] \label{thm3.5}
Let $p\geq 2$ and $\varphi$ be a solution from (1.4) obtained via
Theorem 2.2-(ii) with $\lambda=\mu$.
 If there is  $\psi\in L^\infty(\mathbb{R})$ such that
 $\psi', \psi'' \in H^2(\mathbb{R})\cap L^1(\mathbb{R})$,
 $\psi''',\psi^{(v)}\in L^1(\mathbb{R})$, and it is chosen such that
 (3.11) is true with $0<\theta/(1+\theta)<1$ and
\[
\langle S''(\varphi)B_\psi(\varphi),B_\psi(\varphi)\rangle < 0
\tag{3.12}
\]
where $S(u)=E(u)+c\;Q(u)$, then there exist $\epsilon>0$ and a
sequence $\{u_0^j\}$ in $V(\Omega_{\varphi},\epsilon)$ satisfying:
\begin{itemize}
\item[(i)]  $u_0^j\to \varphi$ in $H^2(\mathbb{R})$ as $j\to
\infty$.
\item[(ii)]  For all $j\in \mathbb{N}$, $u(t)=U(t)u_0^j$ is
uniformly bounded in $H^2(\mathbb{R})$, but escapes from
$V(\Omega_{\varphi},\epsilon)$ in a finite time.
\end{itemize}
\end{theorem}

The proof of this theorem follows the same spirit of ideas as those
established in [11] and [2] (see also [27]), therefore we will
explain only the main points of the argument. We will divide the
proof in a few Lemmas.


\begin{lemma} \label{lm3.6}
Let $\varphi$ be a solution from (1.4) obtained via Theorem
2.2-(ii) with $\lambda=\mu$. Suppose that there is  $\psi\in
L^\infty(\mathbb{R})$ such that $\psi', \psi'' \in
H^2(\mathbb{R})\cap L^1(\mathbb{R})$, $\psi''',\psi^{(v)}\in
L^1(\mathbb{R})$ and the {\it action} $S$  satisfies (3.12). Then
there are positive numbers $\epsilon_3$ and $\sigma_3$ such that
\[
\text{for}\quad v_0\in V(\Omega_{\varphi},\epsilon_3),\quad
\text{there exists } s\in (-\sigma_3,\sigma_3):\quad
S(\varphi)\leq S(v_0)+ P(v_0)s
\]
where $P(u)=\langle S'(u), B_{\psi}(u)\rangle$.
\end{lemma}

\noindent\textbf{Proof}\quad  We consider for $v_0\in
V(\Omega_{\varphi},\epsilon_0)$, where $\epsilon_0$ is given by
Lemma 3.2, the initial-value problem
\[
\begin{gathered}
\frac{d}{ds} v(s)= B_\psi(v(s))\\
v(0)=v_0.
\end{gathered}
\tag{3.13}
\]
So, from (3.4) we have that (3.13) admits for each
 $v_0\in V(\Omega_{\varphi},\epsilon_0)$
a unique maximal solution
$v\in C^2((-\sigma, \sigma);V(\Omega_{\varphi},\epsilon_0))$,
where $v(0)=v_0$ and $\sigma=\sigma(v_0)\in (0,\infty]$. Moreover,
for each $\epsilon_1 < \epsilon_0$, there exists $\sigma_1>0$ such that
$\sigma(v_0)\geq \sigma_1$, for all $v_0\in V(\Omega_{\varphi},\epsilon_1)$.
 Hence, we can
define for fixed $\epsilon_1, \sigma_1$ the following dynamical
system
\begin{gather*}
\mathcal{W}: (-\sigma_1,\sigma_1)\times V(\Omega_{\varphi},\epsilon_1)\to
V(\Omega_{\varphi},\epsilon_0)\\
(s,v_0)\to \mathcal{W}(s)v_0,
\end{gather*}
where $s\to\mathcal{W}(s)v_0$ is the maximal solution of (3.13) with
initial data $v_0$. Now we establish some basic properties of the
flow $s\to\mathcal{W}(s)v_0$. In fact, from Lemma 3.3 one has that
$\mathcal{W}$ is a $C^1$-function, also we have that for each $v_0\in
V(\Omega_{\varphi},\epsilon_1)$ the function $s\in
(-\sigma_1,\sigma_1)\to \mathcal{W}(s)v_0 $ is $C^2$ and  the flow
$s\to\mathcal{W}(s)v_0$ {\it commutes with translations}. Finally, from the
relation
\[
\mathcal{W}(s)\varphi=\varphi+\int_0^s\tau_{\Lambda(\mathcal{W}(t)\varphi)}\psi'\,dt -\int_0^s D(t) \tau_{\Lambda(\mathcal{W}(t)\varphi)}\varphi''\; dt,
\]
(where the map $s\in (-\sigma_1,\sigma_1)\to D(s)$ is a continuous
function), Lemma 2.3 and the hypothesis on $\psi$, we have the
following consequences
\[
\mathcal{W}(s)\varphi,\; \partial_x^2\mathcal{W}(s)\varphi,\; \partial_x^4
\mathcal{W}(s)\varphi\in L^1(\mathbb{R})\quad\text{for all }s\in
(-\sigma_1,\sigma_1). \tag{3.14}
\]

Now, given $v_0\in V(\Omega_{\varphi},\epsilon_1)$ we get from
Taylor's Theorem that there is $\theta\in (0,1)$ such that $S(\mathcal{W}(s)v_0)=S(v_0)+ P(v_0)s+\frac12 R(\mathcal{W}(\theta s)v_0) s^2$,
where $ P$ and $R$ are functionals defined on
$V(\Omega_{\varphi},\epsilon_1)$ by $P(v)=\langle S'(v),B_\psi
(v)\rangle$ and $R(v)=\langle S''(v)B_\psi (v),B_\psi (v)\rangle +
\langle S'(v),B'_\psi (v) (B_\psi (v))\rangle $. Since $R$, $\mathcal{W}$ are continuous, $S'(\varphi)=0$ and $R(\varphi)<0$, then there
exist $\epsilon_2\in (0,\epsilon_1]$, $\sigma_2\in (0,\sigma_1]$
such that
\[
S(\mathcal{W}(s)v_0)\leq S(v_0)+ P(v_0)s\quad \text{for }v_0\in
B_{\epsilon_2}(\varphi),\quad s\in (-\sigma_2,\sigma_2). \tag{3.15}
\]
Since $\mathcal{W}(s)v_0$ {\it commutes with translations}, we obtain
from the relation
$V(\Omega_{\varphi},\epsilon_2)=\Omega_{B_{\epsilon_2}(\varphi)}$,
the extension
\[
\begin{gathered}
\text{for } v_0\in V(\Omega_{\varphi},\epsilon_2),\quad
 s\in (-\sigma_2,\sigma_2)\\
S(\mathcal{W}(s)v_0)\leq S(v_0)+ P(v_0)s.
\end{gathered} \tag{3.16}
\]
So that, for $v_0=\mathcal{W}(\tau)\varphi$  with $\tau\neq 0$ small
enough, we get
\[
S(\varphi)\leq S(\mathcal{W}(\tau)\varphi)- P(\mathcal{W}(\tau)\varphi)\tau.
\tag{3.17}
\]
Moreover, from (3.12)  the function $\tau\to S(\mathcal{W}(\tau)\varphi)$ has a strict local maximum  in $0$ and therefore
\[
S(\mathcal{W}(\tau)\varphi)<S(\varphi)\quad\text{for }\tau\in
(-\sigma_2,\sigma_2),\;\tau\neq 0, \tag{3.18}
\]
and so from (3.17) and (3.18) we have that for some $\sigma_3\leq
\sigma_2$
\[
P(\mathcal{W}(\tau)\varphi)<0,\quad \tau\in (0,\sigma_3).
\tag{3.19}
\]
On the other hand, we have that for $K(v)=\int_{-\infty}^{\infty}
F(v,v_x)\, dx$
\[
\langle K'(\varphi),B_\psi (\varphi)\rangle\neq 0, \tag{3.20}
\]
otherwise, $B_\psi (\varphi)$ would be tangent to $\mathcal{F}=\{v\in
H^2(\mathbb{R})|\; K(v)= \mu \}$ and then one would have $\langle
S''(\varphi)B_\psi(\varphi),B_\psi(\varphi)\rangle\geq 0$ (since
$\varphi$ minimizes $S$ on $\mathcal{F}$ by Theorem 2.2), but this is a
contradiction with (3.12). So, if we consider the function
\[
(v_0,s)\in V(\Omega_{\varphi},\epsilon_1)\times (-\sigma_1,\sigma_1)
\longrightarrow K(\mathcal{W}(s)v_0)=\int_{-\infty}^{\infty} F(\mathcal{W}(s)v_0,
\partial_x\mathcal{W}(s)v_0) \,dx,
\]
which is a $C^1$- mapping with $(\varphi,0)\to K(\varphi)=\mu$, it
follows from (3.20) that
\[
\frac{d}{ds}K(\mathcal{W}(s)v_0)\Big|_{(\varphi,0)}=\langle
K'(\varphi),B_\psi (\varphi)\rangle \neq 0,
\]
and therefore from the Implicit Function Theorem it follows that,
there exist $\epsilon_3\in (0,\epsilon_2)$ and $\sigma_3\in
(0,\sigma_2]$ such that for all $v_0\in B_{\epsilon_3}(\varphi)$,
there exists a unique $s\equiv s(v_0)\in (-\sigma_3,\sigma_3)$
such that $K(\mathcal{W}(s)v_0)=\mu$, {\it i.e.}  $\mathcal{W}(s)v_0 \in
\mathcal{F}$. Then, applying (3.15) to $(v_0,s(v_0))\in
B_{\epsilon_3}(\varphi)\times (-\sigma_3,\sigma_3)$ and using that
$\varphi$ minimizes $S$ on $\mathcal{F}$, we have that for $ v_0\in
B_{\epsilon_3}(\varphi)$ there exists $ s\in (-\sigma_3,\sigma_3)$
such that $S(\varphi)\leq S(v_0)+ P(v_0)s$. Therefore, using
again that {\it $\mathcal{W}(s)v_0$ commutes with translations}
 and $V(\Omega_{\varphi},\epsilon_2)=\Omega_{B_{\epsilon_2}(\varphi)}$ we
 obtain the extension
\[
\text{for each } v_0\in V(\Omega_{\varphi},\epsilon_3),\quad
\text{there exists}\; s\in (-\sigma_3,\sigma_3):\quad
S(\varphi)\leq S(v_0)+ P(v_0)s.
\]
This shows the Lemma. \hfill$\square$

Since  $B_{-\psi} (\varphi)=-B_{\psi} (\varphi)$ we
can assume from (3.20) that $ \langle K'(\varphi),B_\psi
(\varphi)\rangle < 0$. So, if we consider the continuous flow
$\tau\to \mathcal{W}(\tau)\varphi$ which is solution of (3.13) with
initial data $\varphi$, we will have  for $\tau>0$ and small
enough that we can get some $\delta$ small such that
\[
K(\mathcal{W}(\tau)\varphi)=K(\varphi)+\int_0^\tau \langle K'(\mathcal{W}(\xi)\varphi),B_\psi (\mathcal{W}(\xi)\varphi)\rangle
\,d\xi=\mu-\delta<\mu\,. \tag{3.21}
\]

\begin{lemma} \label{lm3.7}
Suppose $\varphi$, $\psi$ and $S$ satisfy the
same hypotheses as those established in Lemma 3.6 and $\psi$ is
chosen such that (3.11) is true with $0<\theta/(1+\theta)<1$.
Define
\[
\mathbb{D}=\{v\in H^2(\mathbb{R}) |\; S(v)<S(\varphi), \; K(v)<\mu\}\cap
\{v\in H^2(\mathbb{R})| \; P(v)<0\}\equiv \mathbb{B}\cap \mathcal{P} \tag{3.22}
\]
where $P(v)=\langle S'(v), B_{\psi}(v)\rangle$. Then we have:
\begin{itemize}
\item[(i)] $\mathcal{W}(\tau)\varphi\in \mathbb{D}$ for
  all $\tau\in (0,\sigma_3)$.
\item[(ii)] $\mathbb{B}$ is invariant by the flow $u(t)$
  of equation (1.1).
\item[(iii)] The flow $u(t)$ of (1.1) with initial
 data in $\mathbb{B}$ has a uniformly bounded
 trajectory in $H^2(\mathbb{R})$
\item[(iv)] For $\mathcal{A}_\psi$ defined in (3.10) we have
$\partial_t\mathcal{A}_\psi(u(t))=-P(u(t))$.
\end{itemize}
\end{lemma}

\noindent\textbf{Proof}\quad
$(i)$\; From (3.18), (3.19) and (3.21) we get that the
flow generated by (3.13) through $\varphi$ satisfies $ \mathcal{W}(\tau)\varphi\in \mathbb{D}$ for all $\tau\in (0,\sigma_3)$.

$(ii)$\; Let $u_0\in \mathbb{B}\subset H^2(\mathbb{R})$. Then from
Assumption 2.1 (local existence theory),  there exists $T_0>0$
such that $u(t)=U(t)u_0\in H^2(\mathbb{R})$ and  satisfies (1.1) for
all $t\in [0,T_0)$. So from (1.5) we have $S(U(t)u_0)=S(u_0)<
S(\varphi)$. Therefore from this last relation and the property of
minimization of $S$ on $\mathcal{F}=\{v\in H^2(\mathbb{R})|\; K(v)= \mu \}$
by $\varphi$, we have  that for all $t\in [0,T_0)$,
$K(U(t)u_0)\neq\mu$. Finally, since $t\to K(U(t)u_0)$ is
continuous on $[0,T_0)$ we obtain that $ K(U(t)u_0)<\mu$ for all $
t\in [0,T_0)$. This shows property (ii).

$(iii)$\; Let $u_0\in \mathbb{B}$. Then from the conservation
quantities in (1.5) by the flow of equation (1.1), we have that
\begin{align*}
\frac12 \int_{-\infty}^{\infty}[\partial_x^2 U(t)u_0]^2 -b
[\partial_x U(t)u_0]^2 +c [U(t)u_0]^2dx
&=S(U(t)u_0)+K(U(t)u_0)\\
&<S(\varphi)+\mu \tag{3.23}
\end{align*}
for all $t\in [0,T_0)$. So, if we have $b=0$ in (3.23) then a
simple interpolation argument shows that there is a constant
$M(\varphi,\mu)>0$ such that $\|\partial_x U(t)u_0\|\leq
M(\varphi,\mu)$ and therefore $\|U(t)u_0\|_{H^2}^2\leq
M(\varphi,\mu)$ for all $t\in [0,T_0)$. If $b>0$ then from
restriction $c>b^2/4$ we get
\[
\begin{split} &\frac12\Big(1-\frac{b}{2\sqrt c}\Big
)\;\int_{-\infty}^{\infty} [\partial_x^2 U(t)u_0]^2+c
[U(t)u_0]^2dx\\
&\leq \frac12 \int_{-\infty}^{\infty}[\partial_x^2 U(t)u_0]^2 -b
[\partial_x U(t)u_0]^2 +c [U(t)u_0]^2dx<S(\varphi)+\mu,
\end{split}
\]
and again an interpolation argument shows that
$\|U(t)u_0\|_{H^2}^2\leq M(\varphi,\mu)$ for all $t\in [0,T_0)$.
Note that this last fact shows that the flow $t\to U(t)u_0$ will
be a global solution for equation (1.1) if we have in fact a local
existence theory of solutions  in $H^2(\mathbb{R})$.

$(iv)$\; Let $u(t)$ be a solution of (1.1). Then as long as this
flow remains in $V(\Omega_{\varphi},\epsilon)$ (note that from
hypothesis $\mathcal{A}_\psi(u(t))<\infty $) we have that
\[
\begin{split}
\partial_t \mathcal{A}_\psi(u(t))&=\int_{-\infty}^{\infty} \Big(\langle \tau_{\Lambda(u(t))}\psi'(x),u(t)\rangle \Lambda'(u(t))+ \tau_{\Lambda(u(t))}\psi(x)\Big)\frac{\partial u}{\partial t}(x,t)\,dx\\
&=-
\langle\langle\tau_{\Lambda(u(t))}\psi'(x),u(t)\rangle\frac{d}{dx}\Lambda'(u(t))+
\tau_{\Lambda(u(t))}\psi'(x), E'(u(t))\rangle\\
&=-\langle B_\psi (u(t)),S'(u(t))\rangle + c \langle B_\psi
(u(t)),u(t)\rangle=-P(u(t)),
\end{split}
\tag{3.24}
\]
where in the last equality we have used the fact that $\langle
B_\psi(v),v\rangle=0$ for all $v\in V(\Omega_{\varphi},\epsilon)$.
This shows the Lemma. \hfill$\square$


\noindent\textbf{Proof of Theorem 3.5}
By $(i)$ in Lemma 3.7, let $\tau_j\in (0,\sigma_3)$
 such that $\tau_j\to 0$ as $j\to \infty$ and define $u_0^j\equiv
\mathcal{W}(\tau_j)\varphi$. Then $u_0^j\to\varphi$ in $H^2(\mathbb{R})$ as
$j\to \infty$. Since $u_0^j\in \mathbb{D}\subset \mathbb{B}$, it follows
from Lemma 3.7 that $t \to u(t)\equiv U(t)u_0^j$ is  uniformly
bounded for all $j$. So, to conclude the proof of our criterium of
instability we need only to verify that $t\to U(t)u_0^j$ escapes
from $V(\Omega_{\varphi},\epsilon_3)$ for some $\epsilon_3>0$ and
for all $j\in\mathbb{N}$ in a finite time. In fact, let $\epsilon_3>0$
determined in Lemma 3.6 and define
\[
T_j=\sup \{\tau >0 : U(t)u_0^j\in
V(\Omega_{\varphi},\epsilon_3),\quad\text{ for all } t\in (0,
\tau)\}.
\]
Then, it follows from Lemma 3.6 that for all $j\in \mathbb{N}$ and
$t\in (0,T_j)$ there exists $s=s_j(t)\in (-\sigma_3,\sigma_3)$
satisfying $ S(\varphi)\leq S(U(t)u_0^j)+ P(U(t)u_0^j)s=S(u_0^j)+
P(U(t)u_0^j)s$. Now, since $u_0^j\in \mathbb{D}$ then $t\to
U(t)u_0^j\in \mathcal{P}$ for $t\in (0,T_j)$, so we have that
\[
-P(U(t)u_0^j)\geq
\frac{S(\varphi)-S(u_0^j)}{\sigma_3}=\eta_j>0,\quad \text{for
all} \quad t\in (0,T_j). \tag{3.25}
\]
Now suppose that for some $j$, $T_j=+\infty$. Then from the
properties obtained by the flow $\tau\to \mathcal{W}(\tau)\varphi$ in
(3.14) and considering that (3.11) is true, we obtain from (3.24)
and (3.25) that $\mathcal{A}_\psi(U(t)u_0^j))\geq t \eta_j+ \mathcal{A}_\psi(u_0^j)$ for all $t\in(0,+\infty)$. Then from (3.11) it
follows
\[
K_1\geq\frac{t\eta_j+ \mathcal{A}_\psi(u_0^j)}{1+t^{\theta/(1+\theta)}}\quad \text {for
all }t\in (0,+\infty),
\]
which is a contradiction . Therefore $T_j<+\infty$, which means
that $u(t)=U(t)u_0^j$ eventually leaves
$V(\Omega_{\varphi},\epsilon_3)$. This proves the Theorem.
\hfill$\square$


\section{Instability of the Orbit $\Omega_\varphi$} %sec. 4

In this section we give conditions to assure inequality (3.12) and
so to obtain the instability of the $\varphi$-orbit,
$\Omega_\varphi$, with respect to equation (1.1). For that, we
start showing the following Lemma.

\begin{lemma} \label{lm4.1}
 Let $p\geq 2$ and suppose that $F(q,r)$,
homogeneous of degree $p+1$, satisfies relation (1.2). Consider
$\varphi$ a solution of (1.4) obtained via Theorem 2.2-(ii) with
$\lambda=\mu$. Then for
$\psi(x)=\int_{-\infty}^x[\varphi(y)+2y\varphi'(y)]dy$
 we get that
\[
\begin{split} &\langle S''(\varphi)B_\psi(\varphi),B_\psi(\varphi)\rangle\\
&= 8b\int_{-\infty}^{\infty}\;(\varphi')^2\,dx+
(9-p)(p-1)\int_{-\infty}^{\infty}\;F(\varphi, \varphi')\,dx \\
&\quad +4\int_{-\infty}^{\infty}\;[4\varphi'F_r(\varphi, \varphi')-\varphi\varphi'F_{qr}(\varphi, \varphi')-
2(\varphi')^2F_{rr}(\varphi, \varphi')]\,dx.
\end{split}\tag{4.1}
\]
\end{lemma}

\noindent\textbf{Proof}\quad
Since $\langle\psi', \varphi\rangle=0$ it follows
immediately from (3.9) that $B_\psi(\varphi)=\psi'$ and therefore
we need only to estimate the quantity $\langle
S''(\varphi)\psi',\psi'\rangle $. In fact, it denotes by $\mathcal{L}=S''(\varphi)$ the linear operator
\[
\begin{split}
\mathcal{L}=&\partial_x^4+b\;\partial_x^2+c-F_{qq}(\varphi, \varphi')+\varphi'F_{qqr}(\varphi, \varphi')
+\varphi'F_{qrr}(\varphi, \varphi')\partial_x\\
&+\varphi''F_{qrr}(\varphi, \varphi')+\varphi''F_{rrr}(\varphi, \varphi')\partial_x+
F_{rr}(\varphi, \varphi')\partial_x^2.
\end{split}
\tag{4.2}
\]
Initially, from the homogeneity of $F$ we obtain the relations
\[
\begin{gathered}
\varphi F_{q}(\varphi, \varphi')+\varphi' F_{r}(\varphi, \varphi')=
(p+1)F(\varphi, \varphi'),\\
\varphi^2 F_{qq}(\varphi, \varphi')+2\varphi\varphi' F_{qr}(\varphi, \varphi')
+(\varphi')^2 F_{rr}(\varphi, \varphi')=
p(p+1)F(\varphi, \varphi').
\end{gathered}
\tag{4.3}
\]
Now, using (1.2) and (1.4), we have from (4.2)
\[
\begin{gathered}
\mathcal{L} \varphi=F_{q}-\varphi'F_{qr}-
\varphi F_{qq}+ \varphi(F_{qr})_x+
\varphi'(F_{rr})_x\\
\mathcal{L}
(x\varphi')=-2b\varphi''-4c\varphi+4F_{q}-4\varphi'F_{qr}-2\varphi''F_{rr}
+(\varphi')^2 F_{qrr}+ \varphi'\varphi''F_{rrr}.
\end{gathered}
\tag{4.4}
\]
So, using the first equation in (4.4), integration by parts and (4.3) we get
that
\[
\begin{split} \langle\mathcal{L} \varphi,\varphi\rangle
&=\int_{-\infty}^{\infty}\; [\varphi F_{q}-
\varphi\varphi'F_{qr}-\varphi^2F_{qq}+\varphi^2 (F_{qr})_x
+\varphi \varphi'(F_{rr})_x]\,dx\\
&=\int_{-\infty}^{\infty}\; [\varphi F_{q}-
\varphi\varphi'F_{qr}-\varphi \varphi''F_{rr}-p(p+1)F]\,dx\\
&=\int_{-\infty}^{\infty}\;[ \varphi
F_{q}+\varphi'F_{r}-p(p+1)F]\,dx
=(p+1)(1-p)\int_{-\infty}^{\infty}\;F\,dx.
\end{split}
\tag{4.5}
\]
Now, from the first equation in (4.4), some integrations by parts and
the first equation of (4.3),
we estimate the following term
\[
\begin{split} \langle \mathcal{L} \varphi,x\varphi'\rangle
&=\int_{-\infty}^{\infty}\;[ -\varphi F_{q}-
2x\varphi(F_{r})_x-2x\varphi'(F_{q})_x-\varphi\varphi' F_{qr}-(\varphi')^2F_{rr}]\,dx\\
&=\int_{-\infty}^{\infty}\;[ -\varphi F_{q}+2x(F)_x+2(\varphi
F_{q}+\varphi' F_{r})
-\varphi\varphi' F_{qr}-(\varphi')^2F_{rr}]\,dx\\
&=(p-1)\int_{-\infty}^{\infty}\;F\,dx+
\int_{-\infty}^{\infty}\;[\varphi' F_{r}-\varphi\varphi'
F_{qr}-(\varphi')^2F_{rr} ]\,dx.
\end{split}
\tag{4.6}
\]
We are going to estimate the term $\langle \mathcal{L}
(x\varphi'),x\varphi'\rangle $. Initially from the second equation in (4.4)
and integration by parts we get
\[
\begin{split}
&\langle \mathcal{L} (x\varphi'),x\varphi'\rangle \\
&=\int_{-\infty}^{\infty}\;[b(\varphi')^2+4x\varphi'F_{q}-4x(\varphi')^2F_{qr}-
2x\varphi'\varphi''F_{rr}+x(\varphi')^2(F_{rr})_x+2c\varphi^2]\,dx\\
&=\int_{-\infty}^{\infty}\;[b(\varphi')^2+4x\varphi'F_{q}-4x\varphi'(F_{r})_x-(\varphi')^2F_{rr}+2c\varphi^2]\,dx\\
&=\int_{-\infty}^{\infty}\;[b(\varphi')^2-4F+4\varphi'F_{r}-(\varphi')^2F_{rr}+2c\varphi^2]\,dx.
\end{split}
\tag{4.7}
\]
We need now an expression for
$\int_{-\infty}^{\infty}2c\varphi^2\,dx$. In fact, multiplying
(1.4) by $x\varphi'$, integrating by parts several times and using
the relation
$$\int_{-\infty}^{\infty}\;x\varphi'\partial_x^4
\varphi\,dx= \frac{3}{2}
\int_{-\infty}^{\infty}\;(\varphi'')^2\,dx
$$
we have
\[
\int_{-\infty}^{\infty}\;\Big
[\frac32(\varphi'')^2-\frac{b}{2}(\varphi')^2+F-\varphi'
F_{r}-\frac{c}{2}\varphi^2\Big ]\,dx=0. \tag{4.8}
\]
Moreover, from (1.4) we obtain immediately that
\[
\int_{-\infty}^{\infty}\;(\varphi'')^2\,dx=\int_{-\infty}^{\infty}\;[b(\varphi')^2+(p+1)F-c\varphi^2]\,dx.
\tag{4.9}
\]
So, from (4.8) and (4.9) we get the main relation
\[
\int_{-\infty}^{\infty}\;\Big [b(\varphi')^2+\frac32 \varphi
F_q+\frac12 \varphi' F_r +F\Big ]\,dx=
\int_{-\infty}^{\infty}\;2c\varphi^2\,dx. \tag{4.10}
\]
Then, from (4.7) and (4.10) we obtain
\[
\langle \mathcal{L}
(x\varphi'),x\varphi'\rangle=\int_{-\infty}^{\infty}\;\Big
[2b(\varphi')^2 + \frac{3(p-1)}{2}F \Big
]\,dx+\int_{-\infty}^{\infty}\;[3\varphi' F_r -(\varphi')^2
F_{rr}]\,dx. \tag{4.11}
\]
Finally, from (4.5), (4.6) and (4.11) we get (4.1). This completes
the Proof. \hfill$\square$

With Theorem 3.5 and Lemma 4.1 we are ready to establish our
Theorem of instability for solitary-wave solutions associated to
the  fifth-order equation (1.1).

\begin{theorem}[Instability for $\Omega_{\varphi}$] \label{thm4.2}
 Let $p\geq 2$ and suppose that $F(q,r)$, homogeneous of degree $p+1$,
satisfies relation (1.2). Consider $\varphi$ a solution of (1.4)
obtained via Theorem 2.2-(ii) with $\lambda=\mu$. Then if
$F(\varphi, \varphi')$ is homogeneous in $\varphi'$ of degree
$\beta$, $ \beta\in[0,p+1]$, then the conditions
\begin{gather*}
 b=0\quad\text{and}\quad  \beta>\frac{9-p}{2},\quad\text{or}\\
 b<0\quad\text{and}\quad  \beta\geq \frac{9-p}{2},\quad\text{or}\\
 \beta>\frac{9-p}{2},\; b>0\quad\text{and}\quad  b\quad\text{small},
\end{gather*}
imply that the $\varphi$-orbit, $\Omega_{\varphi}$, is unstable by
the flow of (1.1).
\end{theorem}

\noindent\textbf{Proof}\quad From Lemma 3.4 we have that (3.11) is
true since we have $0<\theta/(1+\theta)<1$, and from Lemma 2.3 we
obtain the properties of regularity on
$\psi(x)=\int_{-\infty}^x[\varphi(y)+2y\varphi'(y)]dy$ required by
Theorem 3.5. So, we only need to verify condition (3.12) and
therefore from Lemma 4.1 we need to know when expression (4.1) is
negative. In fact, let $F(\varphi, \varphi')$ be homogeneous in
$\varphi'$ of degree $\beta$, $\beta\in[0,\ p+1]$. Since $F$
satisfies the relations $\varphi'F_{r}=\beta F$,
$(\varphi')^2F_{rr}=\beta(\beta-1) F$, $\varphi F_{q}=\alpha F$
and  $\varphi^2F_{qq}=\alpha(\alpha-1) F$, where
$\alpha+\beta=p+1$, we have initially from (4.1) and the second
equation in (4.3) that
\[
4\int_{-\infty}^{\infty}\;[4\varphi'F_r-\varphi\varphi'F_{qr}-
2(\varphi')^2F_{rr}]\,dx=4\beta(5-\beta-p)\int_{-\infty}^{\infty}\;F\,dx
\]
so from (4.1) we get
\[
\begin{aligned}
&\langle S''(\varphi)B_\psi(\varphi),B_\psi(\varphi)\rangle\\
&=\Big[(1-p)(p-9)+4(5-p)\beta-4\beta^2\Big]
\int_{-\infty}^{\infty}F(\varphi,\varphi')\,dx+
8b\int_{-\infty}^{\infty}\;(\varphi')^2\,dx,
\end{aligned} \tag{4.12}
\]
which is negative if either $b=0$ and $\beta>\frac{9-p}{2}$ or
$b<0$ and $\beta\geq \frac{9-p}{2}$.

Now if we consider $\beta>\frac{9-p}{2}$ and $b>0$ then it follows
from condition $c>b^2/4$, properties obtained for $\varphi$ and
Theorem 2.2, that
\[
\begin{split} b\int_{-\infty}^{\infty}\;(\varphi')^2\,dx&\leq
b\int_{-\infty}^{\infty}\;[(\varphi'')^2+ \varphi^2]\,dx\leq
\frac{b\quad\text{max}\{1,c\}}{c}
\int_{-\infty}^{\infty}\;[(\varphi'')^2+c
\varphi^2]\,dx\\
&\leq
\frac{4b\quad\text{max}\{1,c\}}{\sqrt{c}(2\sqrt{c}-b)}\;I_{c,b}(\varphi)=
\frac{4b\quad\text{max}\{1,c\}}{\sqrt{c}(2\sqrt{c}-b)}\;\mu^{\frac{2}{p+1}}M_c(1)\\
&=\frac{4b\quad\text{max}\{1,c\}}{\sqrt{c}(2\sqrt{c}-b)}\;
\Big(\frac{1}{p+1}\Big)^{\frac{2}{p-1}}[M_c
(1)]^{\frac{p+1}{p-1}}.
\end{split}
\tag{4.13}
\]
So, we get from (4.12) and (4.13) that
\[
\begin{split}
\langle S''(\varphi)B_\psi(\varphi),B_\psi(\varphi)\rangle
&\leq
\Big[[(9-p)(p-1)+4(5-p)\beta-4\beta^2]\Big(\frac{1}{p+1}\Big)^{\frac{p+1}{p-1}}\\
&\quad+
\frac{32b\quad\text{max}\{1,c\}}{\sqrt{c}(2\sqrt{c}-b)}\Big(\frac{1}{p+1}\Big)^{\frac{2}{p-1}}\Big]
[M_c(1)]^{\frac{p+1}{p-1}}\\
&\equiv K_0(b)[M_c(1)]^{\frac{p+1}{p-1}}.
\end{split}
\]
Therefore, if we choose $b$ small such that $K_0(b)<0$ then we
obtain from the last inequality that $\langle
S''(\varphi)B_\psi(\varphi),B_\psi(\varphi)\rangle<0$. This
finishes the proof. \hfill$\square$

\subsection*{Acknowledgments}  This paper was finished while
the author was a visitor to the Department of Mathematics of the
University of Texas at Austin. The author  also wants to express
his gratitude to Professor Jerry Bona for his interest in this
work.

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\end{document}