\magnification=\magstep1
\input amstex
\documentstyle{amsppt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\pf{\hfill\hfill\qed}
\def\BbbB{\Bbb B}
\def\BbbR{\Bbb R}
\def\calb{\Cal B}
\def\calc{\Cal C}
\def\call{\Cal L}
\def\calo{\Cal O}
\def\wa{\widehat{a}}
\def\wf{\widehat{f}}
\def\wg{\widehat{g}}
\def\wk{\widehat{k}}
\def\wu{\widehat{u}}
\def\wv{\widehat{v}}
\def\ww{\widehat{w}}
\def\intfty{\int_{-\infty}^{\infty}}
\def\intt{\int_{0}^{t-1}}
\def\carrot{\,\hat{}\;}
\def\supk{\underset k\to\sup}
\def\into{\text{Int}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\rightheadtext{Asymptotic Behavior of Solutions}
\leftheadtext{Bona Promislow Wayne}
\NoBlackBoxes
\topmatter
\title
On the Asymptotic Behavior of Solutions to Nonlinear,
Dispersive, Dissipative Wave Equations
\endtitle
\author
Jerry Bona\footnote{Department of Mathematics, The Pennsylvania State
University, University Park, PA 16802.}$^,$
\footnote{Applied Research Laboratory, The Pennsylvania State University,
University Park, PA 16802.}
Keith Promislow$^1$ and
Gene Wayne$^1$
\endauthor
\abstract
The renormalization techniques for determining the long-time
asymptotics of nonlinear parabolic equations pioneered by
Bricmont, Kupiainen and Lin are shown to be effective in
analyzing nonlinear wave equations featuring both dissipation
and dispersion.  These methods allow us to recover recent 
results of Dix in a way which is both transparent and has
interesting prospects for generalization.
\endabstract
\endtopmatter
\TagsOnRight
\document
\baselineskip 20pt
\specialhead
{I.\;\;Motivation and Statement of the Main Results}
\endspecialhead

This study is focussed on the long-time behavior of solutions of
the damped, non-linear wave equation
$$
u_t-Mu+u_{xxx}+u^p u_x=0,\qquad (x\in\BbbR,\;t>1),
\tag1.1
$$
with initial data
$$
u(\cdot,1)=f_0(\cdot),\hskip 1in(x\in\BbbR).
\tag1.2
$$
Here $M$ is a Fourier-multiplier operator, which in Fourier
transformed variables has the form $\widehat{Mu}(k)=-|k|^{2\beta}\wu$,
where $\frac12<\beta\leq 1$, and $p\geq 2$. The initial-value
problem (1.1)--(1.2) is always locally well posed, but for larger
values of $p$ and large initial data, it appears that it may not be
globally well posed (see Bona {\it et al.} 1993).  If attention is
restricted to small initial data, then (1.1)--(1.2) is globally
well posed and it is not difficult to see in this case that solutions
decay to the zero function as $t$ becomes unboundedly large.  It is
our purpose here to determine the detailed structure of this 
evanescence. 

This program has already been carried out in the elegant paper of Dix
(1992).  We will show how some of Dix's results follow readily from a
suitable adaptation of the renormalization techniques put forward by
Bricmont {\it et al.} (1993) in the context of nonlinear parabolic
equations.  While the results obtained in this way are no more telling
than those of Dix, they are here obtained in a transparent and
appealing way.  Furthermore, this method appears to be a promising
avenue of approach for the derivation of more refined aspects of
the long-time asymptotics of solutions.

The principle result derived herein is easily motivated by
consideration of the linear initial-value problem
$$
\gathered
v_t-Mv+\eta v_{xxx}=0,  \\
v(x,1)=f_0(x),
\endgathered
\tag1.3
$$
with parameter $\eta$, which is formally solved by the formula
$$
v(x,t)=S(t)f_0=S_{\eta}(t)f_0=\intfty e^{-(|k|^{2\beta}+i\eta k^3)
(t-1)}e^{-ikx}\wf_0(k)dk,
\tag1.4
$$
where $\wf_0(k)=\intfty e^{ikx}f_0(x)dx$ is the Fourier transform of
$f_0$.  For large $t$, the kernel in (1.4) is very small except where
$|k|$ is near $0$.  For such values of $k$, $|k|^{2\beta}\gg \eta
|k|^3$, and hence if $\wf_0$ is sufficiently regular, 
$$
\align
v&\approx\intfty e^{-|k|^{2\beta}(t-1)}e^{-ikx}(\wf_0(0)+k\wf_0\,'(0)+
\calo|k^2|)dk \\
&\simeq \wf_0(0)\intfty e^{-ikx}e^{-|k|^{2\beta}(t-1)}dk. \tag1.5
\endalign
$$
Defining
$$
f^*(x)=\intfty e^{-ikx}e^{-|k|^{2\beta}}dk,
\tag1.6
$$
and making the change of variables $k'=k\,t^{\frac{1}{2\beta}}$ in (1.5)
yields
$$
v(x,t)\simeq\frac{\wf_0(0)}{t^{\frac{1}{2\beta}}}f^*(x/t^{\frac{1}{2\beta}})
\tag1.7
$$
as $t\to\infty$.  Our main result, here stated informally to provide
the reader with a concrete goal, shows both that the approximations
made to reach (1.7) are justified, and that they remain so even in the
context of the non-linear initial-value problem (1.1)--(1.2).

\proclaim
{Theorem 1}  
For a sufficiently small and smooth initial condition $f_0$, and for
$1/2<\beta\leq 1$, $p\geq 2$, and $\epsilon>0$, there exist constants
$A,C_2,C_{\infty}>0$ depending only on the given data such that
$$
\align
&\left\|u(\cdot,t)-
\frac{A}{t^{\frac{1}{2\beta}}}
f^*(\cdot/t^{\frac{1}{2\beta}})\right\|_{L^2}
\leq 
\frac{C_2}{t^{\left(\frac{3}{4\beta}-\epsilon\right)}} \tag1.8a \\
&\left\|u(\cdot,t)-\frac{A}{t^{\frac{1}{2\beta}}}f^*(\cdot/t^{\frac{1}{2\beta}})\right\|_{L^{\infty}}
\leq \frac{C_{\infty}}{t^{\left(\frac{1}{\beta}-\epsilon\right)}} \tag1.8b
\endalign
$$
where $f^*$ is given in (1.6).
\endproclaim

\remark
{Remark}  A careful look at the proof in Section 3 will convince
the reader that the theory could be formulated in a sharper way,
in which for a given $\beta$, the decay results in (1.8) are valid
for any $p\geq 2\beta$.  In applications of (1.1) to physical problems,
the dependent variable is real-valued and consequently non-integer
values of $p$ are seen to be somewhat artificial.  If non-integer
$p$ are contemplated, a complex-valued version of the theory for 
(1.1)--(1.2) would be required.  In consequence, we have eschewed
the extra precision that might be possible in favor of the simpler
development that is available for integer values of $p$.
\endremark

The plan of the paper is straightforward.  In Section 2 detailed
consideration is given to the linear initial-value problem (1.3).  
This discussion leads naturally to the introduction of a particular
weighted norm that turns out to be very useful in the analysis.  
The theory for the linear problem is both instructive and useful
when attention is turned to the nonlinear problem in Section 3.  
The paper concludes with a short summary and suggestions for
further inquiry.


\specialhead
{II.\;\;Norms and the Linear Problem}
\endspecialhead

The approach taken is to apply the renormalization group ideas of
Bricmont {\it et al.} (1993) to the equation (1.1).  It is convenient
to start with the linear equation (1.3) since the theory for this
simple situation already deviates from that of Bricmont {\it et al.}
(1993), because the main ideas appear in bold relief, avoiding the
technicalities associated with the nonlinear term, and because use
will be made of the linear theory in studying the nonlinear 
initial-value problem.  

Let $\calb$ be the Banach space of functions $f$ such that
their Fourier transform $\wf$ lies in
$\calc^1(\BbbR)$ , and for which the norm
$$
\|f\|=\supk (1+|k|^3)|\wf(k)|+\supk
|\wf\,'(k)|
\tag2.1
$$
is finite.  Note that this norm controls both the 
$L_{2}$- and $L_{\infty}$-norms of $f$.  If $f$ belongs to
$\calb$, then $f$ lies in $W^{3,1}(\BbbR)$ and $xf$ is a member
of $H^1(\BbbR)$.

The renormalization group map is now introduced.  
Taking the solution $v$ given by (1.4), define
$$
v_L(x,t)=L\,v(Lx,L^{2\beta}t)
\tag2.2
$$
for $L\geq 1$.  The (linear) renormalization group map
is denoted $R_{L,\eta}$ and its action on a function $f$ is
$$
(R_{L,\eta }f)(x)=v_L(x,1),
\tag2.3
$$
where $v$ is the solution of (1.4) with initial data $f$, $v_L$ is as
in (2.2), and the second subscript on $R$ connotes the coefficient of
the dispersive term $v_{xxx}$ in (1.3).  Direct calculation shows that
$v_L$ satisfies
$$
\partial_t v_L=Mv_L-L^{2\beta-3}\eta \,\partial_x^3 v_L.
\tag2.4
$$
Thus the rescaling accomplished in the definition of $R_{L,\eta}$ 
diminishes the dispersive term if $\beta<3/2$ and $L$ is large. 
Moreover, because of the semi-group property of the evolution
equation, it transpires that
$$
R_{L^n,1}=R_{L,\alpha^{n-1}}\circ R_{L,\alpha^{n-2}}\circ\cdots\circ
R_{L,\alpha}\circ R_{L,1},
\tag2.5
$$
where $\alpha=L^{2\beta-3}$.

Observe that the putative similarity function $f^*$ defined in
(1.6), which according to the intuition provided by (1.5) captures
the long-term asymptotics of solutions of (1.3), is a fixed point
$$
R_{L,0}f^*=f^*
\tag2.6
$$
of the renormalization group map $R_{L,0}$ without dispersion.
Assuming that the $\alpha$ in (2.5) is less than one, which will
be true if $\beta<\frac32$ and $L>1$, it is a reasonable conjecture
that $R_{L,\alpha^n}$ converges to $R_{L,0}$ as $n$ becomes 
unboundedly large.  It is possible then that successive applications
of $R_{L,\alpha^n}$ for $n$ large drive one toward $f^*$.  Thus
for a given initial datum $f$, the right-hand side of (2.5) applied
to $f$ may, in the right circumstances, converge to $f^*$.  This
in turn means that $R_{L^n,1}f\to f^*$ and this result, properly
interpreted, is exactly what we are after.

The latter observation leads to a search for conditions under which
$R_{L,\alpha}$ is a contractive mapping.

\proclaim
{Lemma 1}  Let the power $\beta$ in the dissipative operator 
be positive and suppose $g\in\calb$ satisfies $\wg(0)=0$.  Then there
exists a positive constant $C=C(\beta)$ such that
$$
\|R_{L,\eta } g\|\leq C(\beta) L^{-1}\|g\|.\qquad\qquad
\tag2.7
$$
\endproclaim
\noindent The constant $C(\beta)$ is independent of $\eta\in[0,1]$.
\demo
{Proof}
The solution of (1.3) with initial condition $g$, written in 
Fourier-transformed variables, is
$$
\wv(k,t)=e^{-(|k|^{2\beta}+ik^3)(t-1)}\wg(k),
$$
and therefore
$$
\widehat{R_{L,\eta }(g)} (k)=\widehat{Lv}(Lx,L^{2\beta})=
\wv\left(\frac{k}{L},L^{2\beta}\right)=
e^{-(|k|^{2\beta}+iL^{2\beta-3}\eta k^3)(1-L^{-2\beta})}
\wg\left(\frac{k}{L}\right).
$$
Since $\wg(0)=0$ and $\wg\in C^1(\BbbR)$, the Mean-Value Theorem
implies that for any $k$, there is a point $\xi=\xi_k$ with
$|\xi_k|\leq k/L$ such that
$$
\left|\wg\left(\frac{k}{L}\right)\right|\leq
\left|\frac{k}{L}\right|\left|\wg'(\xi_k)\right|\;.
$$
In consequence, we have that
$$
\aligned
\supk(1+|k|^3)
\left|\widehat{R_{L,\eta}g}(k)\right|
&\leq \frac{1}{L}\supk (1+|k|^3)|k|
e^{-|k|^{2\beta}}|\wg'(\xi_k)| \\
&\leq\frac{1}{L}C(\beta)\|g\|.
\endaligned
$$
Similarly, one determines that
$$
\aligned
\frac{d}{dk}\left(\widehat{R_{L,\eta}g}\right)(k)&=
\bigg\{-(2\beta|k|^{2\beta-1}+3i L^{2\beta-3}\eta k^2)(1-L^{-2\beta})\wg(k/L) \\
&\left.\ \ \ \ \ +\frac{1}{L}\wg'\left(\frac{k}{L}\right)\right\} 
e^{-(|k|^{2\beta}+iL^{2\beta-3}\eta k^3)}(1-L^{-2\beta}) \\
&\leq\frac{1}{L}C(\beta)\left|\wg'\left(\frac{k}{L}\right)\right| \leq
\frac{1}{L}C(\beta)\|g\|,
\endaligned
$$
and the result follows.  \pf
\enddemo
\vskip .1in

It will also be useful to understand the action of $R_{L,\alpha}$ on
the fixed point $f^*$ as $L$ becomes large.  To this end, we state and
prove another lemma.

\proclaim    
{Lemma 2} Suppose that $\frac12<\beta<1$.  Then there is a constant
$L_0=L_0(\beta)$ and a constant $C=C(\beta)$ such that for any
$L>L_0$, one has 
$$ 
\|R_{L,\alpha^n}f^*-f^*\|\leq C\frac{1}{L^n},
\tag2.8
$$
for $n=1,2,\dots$, where $\alpha=L^{2\beta-3}$ and $f^*$ is defined in
(1.6).
\endproclaim

\demo
{Proof}  First  notice that
$$
\widehat{(R_{L,\alpha^n} f^*)}(k)=e^{-(|k|^{2\beta}+i\alpha^n k^3)
(1-L^{-2\beta})}e^{-|k|^{2\beta}L^{-2\beta}},
$$
from which is follows that
$$
\widehat{(R_{L,\alpha}f^*)}(k)-\wf^*(k)=
e^{-|k|^{2\beta}}(e^{-i\alpha^n k^3(1-L^{-2\beta})}-1).
$$
Attention is now turned to estimating the quantity
$$
\supk (1+|k|^3)|\widehat{(R_{L,\alpha^n}f^*)}
(k)-\wf^*(k)|.
$$
The estimate of the last quantity is made in two parts.  Suppose
first that $k,L$ and $n$ are such that $|k|^3<\alpha^{-n\mu}$ 
for some fixed positive $\mu<1$.  Then it is easily seen that
$$
|e^{-i\alpha^n k^3(1-L^{-2\beta})}-1|\leq C\;\alpha^{n(1-\mu)},
\quad\text{ while }\quad(1+|k|^3)e^{-|k|^{2\beta}}\leq C(\beta)
$$
for all $k$.  If on the other hand $k,L$ and $n$ are such that
$|k|^3\geq\alpha^{-n\mu}$, then
$$
|e^{-i\alpha^n k^3(1-L^{-2\beta})}-1|\leq 2 
$$
and, therefore,
$$
\aligned
\underset{|k|^3\geq\alpha^{-n\mu}}\to\sup
(1+|k|^3)e^{-|k|^{2\beta}}&\leq 
e^{-\frac12\alpha^{\frac{-2\beta n\mu}{3}}}
\supk(1+|k|^3)e^{-\frac12|k|^{2\beta}} \\
&\leq C(\beta)e^{-\frac12L^{\frac{(3-2\beta)2\beta n\mu}{3}}} \\
&\leq C\frac{1}{L^n}
\endaligned
$$
\vskip .2in
\noindent for $L$ large enough since $\beta,\mu>0$ and $3-2\beta>0$.  If
$\mu$ is restricted to lie in the interval $\left(0,
\frac{2-2\beta}{3-2\beta}\right]$, then
$$
\alpha^{n(1-\mu)}=L^{(2\beta-3)(1-\mu)n}\leq\left(\frac{1}{L}\right)^n.
$$
In consequence of the above inequalities, it is seen that there
are constants $C$ and $L_0$ depending only on $\beta$ such that
$$
\supk(1+|k|^3)\big|(R_{L,\alpha^n}f^*)\;{\widehat{}}\;(k)-
\wf^*(k)\big|\leq C\left(\frac{1}{L}\right)^n
$$
for $L\geq L_0$.  A similar set of inequalities shows that
$$
\supk\left|\frac{d}{dk}\left(\widehat{R_{L,\alpha^n} f^*}(k)
-\wf^*(k)\right)\right|\leq C\left(\frac{1}{L}\right)^n
$$
for $L$ large, where $C$ again depends only on $\beta$.

The lemma is thereby established.  \pf
\enddemo

\remark
{Remark}  If $\beta=1$, the foregoing reasoning lead to an 
estimate of the form
$$
\|R_{L,\alpha^n}f^*-f^*\|\leq 
C\left(\frac{1}{L}\right)^{(1-\epsilon)n},
$$
valid for any $\epsilon>0$, where $C$ depends on $\epsilon$
as well as $\beta$.
\endremark
\vskip .3in

These two lemmas provide the tools needed to show convergence of
the composition of the renormalization group maps.  More precisely,
suppose $f=f_0\in\calb$ to be given and define $f_n$ for $n\geq 1$
by the formula
$$
f_n=R_{L,\alpha^n}\circ R_{L,\alpha^{n-1}}\circ\cdots\circ
R_{L,1}f_0.
\tag2.9
$$
We aim to show that $\{f_n\}_{n=0}^{\infty}$ is a convergent
sequence in $\calb$ and to obtain the rate of convergence.

Turning to the just mentioned task, for each $n=1,2,\dots$, 
write
$$
f_n=A_n f^*+g_n
$$
where $\widehat{g_n}(0)=0$ so that $A_n=\widehat{f_n}(0)$.
Straightforward computations show that 
$$
\aligned
f_{n+1}&=R_{L,\alpha^{n+1}}f_n=A_n R_{L,\alpha^{n+1}}f^*+
R_{L,\alpha^{n+1}}g_n \\
&=A_n f^*+A_n(R_{L,\alpha^{n+1}}f^*-f^*)+R_{L,\alpha^{n+1}}g_n.
\endaligned
$$
It is easy to verify that $\widehat{R_{L,\eta}f}(0)=\widehat{f}(0)$
(see Proposition 2.1 in Bona {\it et al.} 1987).  Consequently,
one has $A_n=A=\wf_0(0)$ for all $n$ and
$$
g_{n+1}=A(R_{L,\alpha^{n+1}}f^*-f^*)+R_{L,\alpha^{n+1}}g_n.
$$
Applying Lemmas 1 and 2 leads to the estimate 
$$
\|g_{n+1}\|\leq A\frac{C}{L^{n+1}}+\frac{C}{L} \|g_n\|
$$
valid for all $n\geq 1$, where $C$, which is larger than 1 without
loss of generality, depends only on $\beta$.  A simple
induction then shows that
$$
\|g_{n+1}\|\leq (n+1)A\left(\frac{C}{L}\right)^{n+1}\|g_0\|\leq
(n+1)\left(\frac{C}{L}\right)^{n+1}\|f_0\|,
$$
from which is follows at once that for all $n$,
$$
\|f_n-A\,f^*\|\leq n\left(\frac{C'}{L}\right)^n\|f_0\|.
\tag2.10
$$
Recalling from the definition (2.3), that
$$
f_n(x)=L^n \,v\,(L^n x,L^{2\beta n}),
$$
where $v$ is the solution of (1.3) with initial value 
$v(\cdot,1)=f_0$, and setting $t=L^{2\beta n}$, we see that
$$
f_n(x)=t^{\frac{1}{2\beta}}\,v\,(xt^{\frac{1}{2\beta}},t)
$$
and that
$$
n\,L^{-n}\leq C(\log t)t^{-\left(\frac{1}{2\beta}\right)}.
$$
We have thus established the following proposition.

\proclaim
{Proposition 1}
For $\frac12<\beta<1$ and any initial condition $f_0\in\calb$, there
exists a constant $C>0$ and a time $t_0>0$ such that for $t\geq t_0$, 
the solution $v$ of (1.3) with $\eta =1$ satisfies
$$
\|v(\cdot t^{\frac{1}{2\beta}},t)-A\,t^{-\frac{1}{2\beta}}
f^*(\cdot)\|\leq
C(t^{-\frac{1}{\beta}}\log t)\|f_0\|,
\tag2.11
$$
where $A=\wf_0(0)$ and $f^*$ is defined in (1.6).
\endproclaim

\remark
{Remarks}  The proof only showed convergence for the sequence 
of times $t_n=L^{2n\beta}$.  However, the result is self improving
if one notes that as in Bricmont {\it et al.} (1993), the analysis
is unchanged if $L^{2\beta}$ is replaced by $\tau L^{2\beta}$ 
throughout for $1\leq\tau\leq L^{2\beta}$.  One thereby infers
(2.11) for all $t>L^{2\beta}$. 

The above analysis may be adapted to the case $0<\beta\leq\frac12$ in
the linear case.  One has to compensate for the non-differentiability
of $|k|^{2\beta}$ for $\beta$ in this range by modifying the norm
defining the space $\calb$, replacing the $C^1$-norm on $\widehat{f}$
by an appropriate H\"older norm.
\endremark

\specialhead
{III.\;\;The Non-Linear Problem}
\endspecialhead

Attention is returned to the nonlinear problem (1.1)--(1.2) with the
aim of showing that the term $u^p u_x$, $p\geq 2$, does not affect the
results of Proposition 1 provided $\|f_0\|$ is sufficiently small.

Let $u$ be a solution of (1.1)--(1.2) and consider the renormalized
version
$$
u_L(x,t)=Lu(Lx,L^{2\beta}t)
\tag3.1
$$
of $u$.  The renormalized solution $u_L$ satisfies the evolution
equation
$$
\partial_t u_L=Mu_L-L^{2\beta-3}\partial_x^3 u_L-L^{2\beta-p-1}
u_L^p\partial_x u_{_{L}}.
\tag3.2
$$
Introduce the notation $\alpha=L^{2\beta-3}$ as in Section 2, 
$\gamma=L^{2\beta-p-1}$, and let $R_{L,\eta ,\rho}$ denote the 
renormalization map (3.1) corresponding to the 
semi-group for equation (1.1) with coefficient $\eta$ for the dispersive
term and $\rho$ for the non-linear term.  Thus if $f\in\calb$ is
given, then $(R_{L,\eta,\rho}f)(x)$ is $u_L(x,1)$ where $u_L$ is
as in (3.1) and $u$ is the solution of the initial-value problem
$$
\gathered
u_t+\rho u^p u_x+\eta u_{xxx}-Mu=0, \\
u(x,0)=f(x),
\endgathered
\tag3.3
$$
for $x\in\BbbR$ and $t\geq 0$.  Note that
$(R_{L,1,1}f_0)(x)=Lu(Lx,L^{2\beta})$ where $u$ solves (1.1)--(1.2).

To proceed with the program that was effective in Section 2 for the
linear problem, estimates on the Fourier transform of the nonlinear
term in the evolution equation are needed.  The following lemma is
helpful in this regard.

\proclaim
{Lemma 3}  Let $p\geq 1$ be an integer.  Then there is a constant $C$
depending only on $p$ such that for any $f\in\calb$ and all $k\in\BbbR$,
$$
\align
|\widehat{f^{p+1}}(k)|&\leq \frac{C}{1+|k|^3}\|f\|^{p+1} \tag3.4a \\
|\widehat{f^{p}f_x}(k)|&\leq \frac{C}{1+k^2}\|f\|^{p+1} \tag3.4b \\
|\frac{d}{dk}\widehat{f^{p}f_x}&(k)|\leq C\|f\|^{p+1}. 
\tag3.4c 
\endalign
$$
\endproclaim

\demo
{Proof}  These straightforward estimates follow immediately upon 
writing
$$
\gathered
\widehat{f^{p+1}}(k)=\widehat{f}*\cdots*\widehat{f}(k)
=\int_{\BbbR^p}\widehat{f}
(k-(k_1+\cdots+k_p))\widehat{f}(k_1)\cdots \widehat{f}(k_p)
dk_1\cdots dk_p,  \\
\widehat{f^p f_x}(k)=\widehat{f}*\cdots*\widehat{f}_x(k)=
\int_{\BbbR^p}\widehat{f}(k-(k_1+\cdots+k_p))\widehat{f}(k_1)\cdots
\widehat{f}(k_{p-1})ik_p\widehat{f}(k_p)dk_1\cdots dk_p
\endgathered
$$
and
$$
\frac{d}{dk}\widehat{f^p f_x}(k)=\int_{\BbbR^p}\widehat{f}'(k-
(k_1+\cdots+k_p))\widehat{f}(k_1)\cdots\widehat{f}(k_{p-1})ik_p
\widehat{f}(k_p)dk_1\cdots dk_p.
$$
\pf
\enddemo

A bound on the kernel of the linear propagator will also prove
to be useful.

\proclaim
{Lemma 4}  There are constants $C_0$ and $C_1$ such that for 
any $\eta\in[0,1]$, $t\geq 1$, and for all $k$,
$$
\align
\left|\int_0^{t-1} e^{-s(|k|^{2\beta}+i\eta k^3)}ds\right|
&\leq\frac{C_0 t}{1+|k|^{2\beta}}, \tag3.5a \\
|k|\int_0^{t-1}\left|\frac{d}{dk}e^{-s(|k|^{2\beta}+i\eta k^3)}
\right|ds&\leq C_1 t(1+k^2). \tag3.5b
\endalign
$$
\endproclaim

\demo
{Proof}  These two inequalities follow by computing the integrals
in question and making elementary estimates.  \pf
\enddemo
\vskip .3in

A little later in the analysis, Duhamel's principle will be used
and it will then be helpful to have estimates on the functional
$N=N_{\eta}$ defined for $u\in C(1,T;\calb)$ by
$$
N(u)(x,t)=\int_1^t S(t-s+1)u^p(x,s)u_x(x,s)ds,
\tag3.6
$$
where $\{S(r)\}_{r\geq 1}=\{S_{\eta}(r)\}_{r\geq 1}$ is the semigroup
defined in (1.4) associated to the initial-value problem for the
linear equation (1.3).  By using the estimates in Lemmas 3 and 4,
one deduces that if $u\in C(1,T;\calb)$, then for $1\leq t\leq T$,
$$
\aligned
\supk(1&+|k|^3)|\widehat{N(u)}(k,t)|\leq
\supk(1+|k|^3)\left|\int_1^t e^{-(t-s)\theta(k)}
\widehat{u^p u_x}(k,s)ds\right|  \\
&\leq\supk\left\{(1+|k|^3)\underset{1\leq s\leq T}\to\sup
\left(\frac{C}{1+k^2}\|u(\cdot,s)\|^{p+1}\right)\int_1^t
e^{-(t-s)Re\theta(k)}ds\right\} \\
&\leq\|u\|_T^{p+1}\supk\left\{\frac{C(1+|k|^3)}{1+k^2}
\int_0^{t-1}e^{-r|k|^{2\beta}}dr\right\} \\
&\leq\|u\|_T^{p+1}\supk\left\{CC_0\frac{(1+|k|^3)}{(1+k^2)(1+|k|^{2\beta})}\right\} \\
&\leq C\|u\|_T^{p+1},
\endaligned
$$
provided that $\beta\geq \frac12$ and $t\leq T$.  Here $\theta(k)=
|k|^{2\beta}+i\eta k^3$ and we have introduced the space-time norm
$$
\|u\|_T=\|u\|_{C(1,T;\calb)}=\underset{1\leq t\leq T}\to\sup
\|u(\cdot,t)\|. 
\tag3.7
$$
In a similar vein, it transpires that for $u\in C(1,T;\calb)$,
$$
\gathered
	\supk\left|\widehat{N(u)}'(k,t)\right|=\supk\left|
	\frac{d}{dk}\int_1^t e^{-(t-s)\theta(k)}\widehat{u^p u_x}
	(k,s)ds\right| \\
%
	\leq\supk\frac{1}{p+1}\left|\int_1^t\frac{d}{dk}
	\left(e^{-(t-s)\theta(k)}\right)
	ik\widehat{u^{p+1}}(k,s)ds\right|  \\
%
	+\supk\left|\int_1^t e^{-(t-s)\theta(k)}\frac{d}{dk}
	\widehat{u^p u_x}(k,s)ds\right|  \\
%
	\leq\|u\|_T^{p+1}\supk\frac{C}{1+|k|^3}\int_0^{t-1}
	|k|\left|\frac{d}{dk} e^{-r\theta(k)}\right|dr \\
%
	+C\|u\|_T^{p+1}\supk\int_1^t e^{-(t-s)Re\theta(k)}ds \\
%
	\leq Ct\|u\|_T^{p+1}
\endgathered
\tag3.8
$$
for $1\leq t\leq T$, where $C$ is independent of both $T$ and $u$.

These estimates lead to the following result, which will find use
immediately in the attack on the main result.
\vskip .3in

\proclaim
{Proposition 2}  Suppose $\frac12<\beta\leq 1$ and that $u_1,u_2\in
C(1,T;\calb)$ where $T=L^{2\beta}$ with $L>1$ given.  Then the 
following inequalities hold:
$$
\align
\|N(u_1)\|_T&\leq CT\|u_1\|_T^{p+1}  \tag3.9  \\
\|N(u_1)-N(u_2)\|_T&\leq CT(\|u_1\|_T^p+
\|u_2\|_T^p)\|u_1-u_2\|_T. \tag3.10
\endalign
$$
\endproclaim

\demo
{Proof}  Inequality (3.8) shows that $\|N(u_1)(\cdot,t)\|\leq
Ct\|u_1\|_T^{p+1}$ for $1\leq t\leq T$.  Taking the supremum over
$t\in[1,T]$ thus gives (3.9).  Using the elementary relation
$$
|u^{p+1}-v^{p+1}|=|u-v\|\;u^p+u^{p-1}v+\cdot+v^p|\leq C|u-v|
(|u^p|+|v^p|)
$$
and the same sort of estimates that appear in (3.8), the
inequality (3.10) is likewise verified.  \pf
\enddemo
\vskip .3in

With these preliminaries in hand, we turn to the
primary task of determining the asymptotic behavior of solutions of
(1.1)--(1.2).  Let $f_0$ in $\calb$ be given.  Much as in the linear
case, we consider the sequence
$$
f_n=R_{L^n,1,1}f_0,\qquad\quad n=1,2,\dots,
$$
where $L>0$ will be specified later.  Because of the semigroup
property,
$$
f_n=R_{L,\alpha^{n-1},\gamma^{n-1}}\circ\cdots\circ R_{L,\alpha,\gamma}
\circ R_{L,1,1}f_0,
\tag3.11
$$
or what is the same,
$$
f_n=R_{L,\alpha^{n-1},\gamma^{n-1}}f_{n-1},
$$
for $n=1,2,\cdots$, where $\alpha=L^{2\beta-3}$ and
$\gamma=L^{2\beta-p-1}$ as above.

To analyse the sequence $\{f_n\}_{n=0}^{\infty}$, it is
convenient to consider the initial-value problem
$$
\gathered
\partial_t u_n=Mu_n-\alpha^{n-1}\partial_x^3 n_n-\gamma^{n-1}u_n^p
\partial_x u_n, \\
u_n(\cdot,1)=f.
\endgathered
\tag3.12
$$
Duhamel's principle is applied to the initial-value problem
(3.12) to obtain the formulation
$$
\aligned
u_n(\cdot,t)&=S_n(t)f+\gamma^{n-1}\int_1^t S_n(t-s)u^p(\cdot,s)
u_x(\cdot,s)ds \\
&=u_n^0(\cdot,t)+\gamma^{n-1}N_n(u_n)(\cdot,t),
\endaligned
\tag3.13
$$
where $S_n(t)=S_{\alpha^{n-1}}(t)$ is the semigroup defined in
(1.4) associated with the linear initial-value problem (1.3) with
$\eta=\alpha^{n-1}$.  Because of the previously derived results,
the terms on the right-hand side of (3.13) can be esimated thusly:
$$
\|u_n^0\|_T\leq C_2\|f\|
$$
\line{{and \hfil}{\hfil(3.14)}}
$$
\gamma^{n-1}\|N_n(u_n)\|_T\leq C_3 T\|u_n\|_T^{p+1}\gamma^{n-1},
$$
for $T>0$, where $C_2$ and $C_3$ are independent of $n,T,f,\alpha$
and $\gamma$. 

Define a mapping $T_n$ on functions $v\in C(1,T;\calb)$ by
$$
T_n(v)=u_n^0+\gamma^{n-1}N_n(v),
$$ 
where $u_n^0$ is as in (3.13).  Let $B_n^f=\{u\in C(1,T;\calb):
\|u-u_n^0\|_T\leq\|f\|\}$ and presume that $f$ is drawn from
$B_R=\{g\in\calb:\|g\|\leq R\}$.

It will now be shown that if $R$ is chosen small enough and
$L$ is sufficiently large, with $T=L^{2\beta}$, then $T_n$ defines a
contraction mapping of $B_n^f$, for all $n$.  First, because 
of the second inequality in (3.14), $T_n$ will map $B_n^f$ into
itself provided
$$
C_3 T\|u\|_T^{p+1}\gamma^{n-1}\leq \|f\|
\tag3.15
$$ 
for $u\in B_n^f$, and the latter holds if
$$
C_3 T\gamma^{n-1}(\|f\|+\|u_n^0\|_T)^{p+1}\leq \|f\|.
$$
Because of the first inequality in (3.14), this holds 
provided
$$
C_3 T\gamma^{n-1}(1+C_2)^{p+1}\|f\|^p\leq 1.
$$
Since $\gamma=L^{2\beta-p-1}$, $T=L^{2\beta}$ and $\|f\|\leq R$, (3.15)
will hold if
$$
C_3(1+C_2)^{p+1}R^p L^{2n\beta-(p+1)(n-1)}\leq 1.
$$
Since $p\geq 2$ and $\beta\leq 1$, the exponent is negative for
$n\geq 3$.  Moreover, the inequality (3.10) in 
Proposition 2 shows that $T_n$ is a contraction on $B_f$,
if $\|f\|$ is small enough, and thus (3.12) has 
a unique solution there.
%\enddemo

Taking $f=f_{n-1}$ in (3.12), then using (3.11) and (3.13) leads
to the formula
$$
\align
f_n(\cdot)&=L\,u_{f_{n-1}}(\cdot L,L^{2\beta})+\gamma^{n-1}L\,N(u_n)
(\cdot L,L^{2\beta}) \\
&=R_{L,\alpha^{n-1}}f_{n-1}(\cdot)+w_n(\cdot) \tag3.16
\endalign
$$
for $f_n$, where 
$$
\align
\|w_n\|&\leq L \gamma^{n-1}\|N(u_n)(\cdot L,L^{2\beta})\| \\
&\leq \gamma^{n-1}\|N(u_n)\|_L\leq C\,L^{2\beta}\gamma^{n-1}
\|u_n\|_L^{p+1}  \\
&\leq C\,L^{2\beta}\gamma^{n-1}\|f_{n-1}\|^{p+1}. \tag3.17
\endalign
$$
Writing $f_n=A_n f^*+g_n$ with a constant $A_n$ chosen so that
$\wg_n(0)=0$, we see from (3.14) that
$$
\aligned
f_n&=A_{n-1}R_{L,\alpha^{n-1}} f^*+R_{L,\alpha^{n-1}} g_{n-1}+w_n \\
&=A_{n-1}f^*+A_{n-1}(R_{L,\alpha^{n-1}} f^*-f^*)+R_{L,\alpha^{n-1}}
g_{n-1}+w_n.
\endaligned
$$
So $A_n=A_{n-1}+\ww_n(0)$ and
$$
g_n=A_{n-1}(R_{L,\alpha^{n-1}}^0 f^*-f^*)+R_{L,\alpha^{n-1}}^0 g_{n-1}+
w_n-\ww_n(0)f^*. \tag3.18
$$
As observed for the linear case, $A_n=A_{n-1}=A$, say, for all $n$.
Furthermore, we see that
$$
\align
\|g_n\|&\leq|A|\frac{C}{L^n}+C\frac{1}{L}\|g_{n-1}\|+C\|w_n\|  \\%\tag3.19 \\
&\leq|A|\frac{C}{L^n}+C\frac{1}{L}\|g_{n-1}\|+C\,L^{2\beta}
\gamma^{n-1}\|f_{n-1}\|^{p+1}, \tag3.19%\tag3.20
\endalign
$$
where we used Lemma's 1 and 2 from the linear case and the bound on
$w_n$ from (3.17).

Assume inductively that for all $m\leq n-1$, we have
\hskip .5in{\roster
\item"{(i)}"  $\|f_m\|\leq \|f_0\|\;,\;\|f_0\|\leq\frac{1}{L}$, and
\item"{(ii)}"  $\|g_m\|\leq C\,L^{-m(1-\epsilon)}\|f_0\|$.
\endroster}
Note that $\|g_0\|\leq C\|f_0\|$, and that (3.19) implies
$$
\|g_n\|\leq C\,L^{-n}\|f_0\|^{p+1}+C\,L^{-\epsilon}L^{-n(1-\epsilon)}
\|f_0\|+C\,L^{(2\beta-p-1)(n-1)}\|f_0\|^{p+1}.
$$
Since $p\geq 2$ and $\frac12\leq\beta\leq  1$, if follows that
$2\beta-p-1\leq-1$.  Hence, for $L$ large enough, we have 
$$
\|g_n\|\leq C\,L^{-n(1-\epsilon)}\|f_0\|
$$
for any $\epsilon>0$, where $C$ depends on $\epsilon$ but not on
$n$.

It follows that
$$
\|f_n-Af^*\|
\leq C\,L^{-n(1-\epsilon)}\|f_0\|. 
\tag3.20
$$
Setting $t=L^{2\beta n}$, we see from (3.11) that
$$
\|t^{\frac{1}{2\beta}}u(t^{\frac{1}{2\beta}}\cdot,t)-A\,f^*(\cdot)\|
\leq C\,
\|f_0\|(t^{-\frac{1}{2\beta}})^{1-\epsilon}
\tag3.21
$$
for $t$ large enough and $\|f_0\|$ small enough.

As we remarked in section II, this result can be extended to any 
$t\geq
L^{2\beta}$.

\proclaim
{Theorem 1$'$} Let $\frac12<\beta\leq 1$ and $p\geq 2$, then for
$\|f_0\|$ small enough and $t_0$ large enough, there exist constants
$A$ and $C>0$ such that for all $t\geq t_0$
$$
\|u(\cdot t^{\frac{1}{2\beta}},t)
-\frac{A}{t^{\frac{1}{2\beta}}}f^*(\cdot)\|
\leq C\,\|f_0\|(t^{-\frac{1}{\beta}})^{1-\epsilon}.
\tag3.22
$$
\endproclaim

\specialhead
{IV.\;\;$L_2$- and $L_{\infty}$-bounds}
\endspecialhead

The error estimates in (3.24) also imply estimates in the $L_2$- and
$L_{\infty}$-norms on $\BbbR$.  For any function $f\in L^2(\BbbR)$, 
Plancherel's Theorem states that
$\|f\|_2=\|\wf\|_{L^2}\leq C\,\|f\|$.
Observing that 
$$
\|f(\gamma\cdot)\|_{L^2}=\gamma^{-1/2}\|f(\cdot)\|_{L^2},
$$
(3.22) implies
$$
t^{\frac{1}{4\beta}}\|u(\cdot,t)
-\frac{A}{t^{\frac{1}{2 \beta}}}f^*(\cdot/t^{\frac{1}{2\beta}})\|_2\leq C
\|f_0\|(t^{-\frac{1}{2\beta}})^{1-\epsilon}.
\tag4.1
$$
Similarly $\|f\|_{\infty}\leq \|\wf\|_{L^1}\leq C\|f\|$,  
and so we have
$$
t^{\frac{1}{2\beta}}\|u(\cdot,t)-\frac{A}{t^{\frac{1}{2\beta}}}
f^*(\cdot/t^{\frac{1}{2\beta}})\|_{\infty}\leq C
\|f_0\|(t^{-\frac{1}{2\beta}})^{1-\epsilon}.
\tag4.2
$$
These are precisely the estimates claimed in Theorem 1.

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\by{Naumkin, P. I.}
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\endRefs
\enddocument