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\begin{document}
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\begin{titlepage}
\begin{center}
{\Large\bf Long-Time Asymptotics for a Classical Particle
\medskip\\
Interacting with a Scalar Wave Field }\\
\vspace{2cm}
{\large Alexander Komech}\medskip
\footnote{Supported partly by
French-Russian A.M.Liapunov Center of Moscow State University, and by
research grants of RFBR and of Volkswagen-Stiftung.}\\
Department of Mechanics and Mathematics of\\
Moscow State University, Moscow 119899, Russia\\
email: komech@mech.math.msu.su\bigskip\\
{\large Herbert Spohn}\medskip\\
Theoretische Physik, Ludwig-Maximilians-Universit\"at\\
Theresienstrasse 37, D-80333 M\"unchen, Germany\\
email: spohn@stat.physik.uni-muenchen.de\bigskip\\
{\large Markus Kunze}\medskip\\
Mathematisches Institut der Universit\"at K\"oln\\
Weyertal 86, D-50931 K\"oln, Germany\\
email: mkunze@mi.uni-koeln.de\\
\end{center}
\date{April 11, 1996}
\vspace{4cm}
{\bf Abstract} We consider the Hamiltonian system consisting
of scalar wave field
and a single particle coupled in a translation
invariant manner. The point particle
is subject to a confining
external potential. The stationary solutions of the system are
a Coulomb type wave field centered at those particle positions for which
the external force vanishes.
We prove that solutions of
finite energy converge, in suitable local energy seminorms, to the set of
stationary solutions in
the long time limit $t\to\pm\infty$.
The rate of relaxation
to a stable stationary solution
is determined by spatial decay of initial data.
\end{titlepage}
\section{Introduction and Main Results}
We consider the Hamiltonian system consisting of a real scalar field
$\phi(x),~~x\in\R^3$, and a point particle with position $q\in\R^3$.
The field is governed by the standard linear wave equation. The point particle
is subject to an external potential, $V$,
which is confining in the sense that $V(q)\to\infty$ as $|q|\to\infty$.
The interaction between the particle and the scalar field is local,
translation invariant, and linear in the field. We would like to
understand the long-time behavior of the coupled system.
On a physical level one argues that the force due to the potential $V$
accelerates the particle. Thereby energy is transferred to the wave field
a part of which is eventually transported to infinity.
Thus the particle feels a sort
of friction and we expect that as $t\to\infty$
it will come to rest at some critical point $q^*$ of $V$, where
$\nabla V(q^*)=0$. Our main achievement here is to give this argument a
precise mathematical setting.
The by far most important physical realization of our system is the
electromagnetic field governed by Maxwell's equations and coupled to
charges by the Lorentz force. The physical mechanism just described
is then known as radiation damping, an ubiquituous phenomenon.
There is a huge literature on this subject
\ci{{Ba},{Dir},{Lor},{Ro},{FW},{Ya}}.
Somewhat surprisingly, there is however little mathematical work,
notable exceptions being \ci{{B},{BG}}.
In our paper we simplify somewhat by ignoring the vector character
of the electromagnetic field and hope to come back to the full
coupled Maxwell-Lorentz equations at some later time.
Let $\pi(x)$ be the canonically conjugate field to $\phi(x)$ and let
$p$ be the momentum of the particle.
The Hamiltonian (energy functional) reads then
\beqn\la{1}
H(\phi,q,\pi,p)\equiv
(1+p^2)^{1/2}+V(q)+
\fr 12\int d^3x\,(|\pi(x)|^2+|\nabla\phi(x)|^2)+\int d^3x\,\phi(x)\rho(x-q).
\eeqn
The mass of the particle and the propagation speed for $\phi$ have been set
equal to $1$. A relativistic kinetic energy has been chosen only to
ensure that $|\dot q|<1$.
In spirit the interaction term is simply $\phi(q)$. This would result however
in an energy which is not bounded from below.
Therefore we smoothen out the coupling by
the function $\rho(x)$,
which is assumed to be radial and to have compact support.
In analogy to the Maxwell-Lorentz equations we call $\rho(x)$ the
``charge distribution''.
Taking formally variational derivatives in (\re{1}), the coupled dynamics
becomes
\beqn\la{2}
\ba{ll}
\dot \phi(x,t)=\pi(x,t),&\dot\pi(x,t)=\Delta\phi(x,t)-\rho(x-q(t)),\\
~\\
\dot q(t)= p(t)/(1+p^2(t))^{1/2},&\dot p(t)=-\nabla V(q(t))+
\int d^3x \phi(x,t)\nabla\rho(x-q(t)).
\ea
\eeqn
The stationary solutions for (\re{2}) are easily determined. We define
for $q\in\R^3$
$$
\phi_{q}(x) =
-\int\fr{d^3y}{4\pi|y-x|}\rho(y-q).
$$
Let $S=\{q^*\in\R^3: \nabla V(q^*)=0\}$ be the set of critical points for
$V$. Then the set of stationary solutions, ${\cal S}$, is given by
\be\la{ss}
{\cal S}=\{(\phi, q, \pi, p)=(\phi_{q^*}, q^*, 0, 0)=:Y_{q^*}|~~q^*\in S\}.
\ee
One natural goal is
to investigate the domain of attraction for ${\cal S}$ and in particular
to prove that a solution of (\re{2}) converges to some
stationary state $Y_{q^\ast}=(\phi_{q^{\ast}}, q^\ast, 0, 0)\in {\cal S}$ in
the long time limit $t\to\infty$. Since the total energy is
conserved, the only meaningful notion of convergence is a local
comparison, i.e. a comparison between the true time-dependent solution and the
asymptotic stationary solution in suitable local norms.
More ambituously one would like to estimate the rate of convergence
to $Y_{q^\ast}$. As a preliminary step
one linearizes (\re{2}) at some stationary state $Y_{q^*},~~q^*\in S$.
One observes that the stability of $Y_{q^*}$ is in correspondence
with the ``stability'' of the potential $V(q)$ at the point $q^*$.
In fact, if $d^2V(q^*)>0$ as a quadratic form,
then on the linearized level the relaxation to
$Y_{q^*}$ is exponentially fast.
For small deviation from $Y_{q^*}$ the linearized part should
dominate the nonlinear part of (\re{2}) and one expects
a full neighborhood of $Y_{q^*}$
to be contracted
to $Y_{q^*}$ at an exponential rate in time. This should still hold
if the initial data
have an exponential decay in space.
For a power
decay of initial data in space
one cannot hope for more than
a power rate of
contraction.
On the other hand if $d^2V(q^*)$ has some negative eigenvalues
then $Y_{q^*}$ is linearly unstable.
An interesting case is when $d^2V(q^*)\geq 0$ including a
zero eigenvalue, which however
will not be discussed here.
To state our main results we need some assumptions on $V$ and $\rho$.
The potential is in fact fairly arbitrary. We only need
$$
~~~~~~~~~~~~~~~~
V\in C^2(\R^3),
~~~~~~\lim_{|q|\to\infty} V(q)=\infty.
~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~(P)
$$
For the charge distribution $\rho$ we assume that
$$
~~~~~~~~~~~~~~~~~\rho\in C_0^\infty(\R^3),
~~~~~\rho(x)=0{\rm ~~for~~}|x|\geq R_\rho,~~~~\rho(x)=\rho_r(|x|).
~~~~~~~~~~~~~~~~~~~(C)
$$
However, as to be explained, we also need that all ``modes'' of the wave field
couple to the particle. This is formalized by the Wiener
condition
$$
~~~~~~~~~~~~~~~~~~~~~
\hat\rho(k)= \int d^3x\, e^{ik x}\rho(x)\not=0\mbox{ ~~~~~~for~~ }k\in\R^3.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(W)
$$
In particular the total charge $\hat\rho(0)$ does not vanish.
If $(W)$ is violated, then on the linearized
level one can construct periodic solutions provided the coupling
strength is adjusted to the zeros of $\hat\rho$. If such a
periodic solution would persist for the full nonlinear equations
(\re{2}), then global convergence to ${\cal S}$ would fail.
In the Appendix we will construct examples of charge distributions
satisfying both $(C)$ and $(W)$.
Next we must have a little closer look at (\re{2}). This means we have
to introduce a suitable phase space and have to establish
existence and uniqueness of solutions.
A point in phase space is referred to as state.
Let $L^2$ be the real
Hilbert space $L^2(\R^3)$
with norm $\br\cdot\br$ and scalar product $(\cdot,\cdot)$, and let
$D^{1,2}$ be the completion of real space $C_0^\infty(\R^3)$
with norm $\Vert\phi(x)\Vert=\br\nabla\phi(x)\br$.
Equivalently, using Sobolev's embedding theorem,
$D^{1,2}=\{\phi(x)\in L^6(\R^3):~~|\nabla\phi(x)|\in L^2\}$ (see \ci{Li}).
Let $\br\phi\br_R$ denote the norm in $L^2(B_R)$ for $R>0$,
where $B_R=\{x\in\R^3: |x|\le R\}$. Then the seminorms
$\Vert\phi\Vert_R=\br\nabla\phi\br_R+\br\phi\br_R$ are
continuous on $D^{1,2}$.
We denote by ${\cal E}$ the Hilbert space
$D^{1,2}\oplus\R^3\oplus L^2\oplus\R^3$ with finite norm
$$
{\Vert\,Y\Vert}_{\cal E}=\Vert\phi\Vert + |q| + \br\pi\br +|p|
\quad\mbox{for}\quad Y=(\phi, q, \pi, p)\,.
$$
For smooth $\phi(x)$ vanishing at infinity we have
\begin{eqnarray}\la{HB}
-\fr 1 {8\pi}\int\int~d^3x~d^3y~\fr{\rho(x)\rho(y)}{|x-y|}
=
\fr 12(\rho,\Delta^{-1}\rho)
&&\leq
\fr {1}{2}\br\nabla\phi\br^2+
(\phi(x),\rho(x-q))\nonumber\\
&&\leq
~~\br\nabla\phi\br^2-
\fr 12(\rho,\Delta^{-1}\rho).
\end{eqnarray}
Therefore ${\cal E}$ is the space of finite energy states
and
in particular $\Vert Y_q\Vert_{\cal E}<\infty$.
Let us note that $D^{1,2}$ is not contained in $L^2$
and for instance $\br\phi_q\br=\infty$.
The lower bound in (\re{HB}) implies that the energy (\re{1})
is bounded from below.
In the point charge limit
this lower bound tends to $-\infty$.
We define the local energy seminorms by
\be\la{6}
{\Vert Y\Vert}_R={\Vert\phi\Vert}_R + |q|
+ {\br\pi\br}_R + |p|
\quad\mbox{for}\quad Y=(\phi, q, \pi, p)
\ee
for every $R>0$, and denote by ${\cal E}_F$ the phase
space ${\cal E}$ equipped with the Fr\'echet topology
induced by these local energy seminorms.
Let ${\rm dist}_R$ denote the distance in the seminorm (\re{6}).
Note that the spaces ${\cal E}_F$ and ${\cal E}$ are metrisable.
\begin{pro}\la{Ex}
For every $Y^0=(\phi^0, q^0, \pi^0, p^0)\in {\cal E}$
the Hamiltonian system (\re{2}) has a unique solution $Y(t)
= (\phi(t), q(t), \pi(t), p(t))\in C(\R,{\cal E})$
with $Y(0) = Y^0$.
\end{pro}
We refer to Section 2 where also the precise notion of solution
is explained.
>From physical intuition one is tempted to conjecture that
every solution $Y(t)$ of finite energy will converge to some
stationary state $Y_{q^*}$ as $t\to\infty$. We do not achieve such a
global result. First of all, the decay of initial fields at
infinity should be as required by finite energy but with some additional
smoothness. Secondly the set $S$ need not be discrete.
In this case $Y(t)$ may never settle to a definite $Y_{q^*}$ but
wander around to approach ${\cal S}$ only as a set.
\begin{theorem}\la{A}
Let $(P)$, $(C)$, $(W)$ hold. Let the
initial state $Y^0=(\phi^0, q^0, \pi^0, p^0)\in {\cal E}$
have the following
decay at infinity: For some $R_0>0$ the functions
$\phi^0(x)$,$\pi^0(x)$
are $C^2$,$C^1$-differentiable respectively outside the ball
$B_{R_0}$ and for $|x|\to\infty$
\be\la{8}
DY^0(x)=|\phi^0(x)|+
|x|(|\nabla\phi^0(x)|+|\pi^0(x)|)+
|x|^2(|\nabla\nabla\phi^0(x)|+|\nabla\pi^0(x)|)={\cal O}(|x|^{-\sigma})
\ee
with some $\sigma>1/2$. Then for the solution $Y(t)\in C(\R,{\cal E})$
to (\re{2}) with $Y(0)=Y^0$
\\
(i) $Y(t)$ converges as $t\to\infty$ in Fr\'echet topology of
the space ${\cal E}_F$ to the set ${\cal S}$, i.e.
for every $R>0$
\be\la{9}
\lim_{t\to\infty}{\rm dist}_R (Y(t), {\cal S})=0\,.
\ee
(ii) If the set $S$ is discrete, then
there exists a point $q^*\in S$ such that
\be\la{10}
Y(t)\longrightarrow Y_{q^*}
{\rm ~~in~~} {\cal E}_F{\rm ~~~as~~}t\longrightarrow\infty.
\ee
\end{theorem}
{\it Remarks.}
(i) Since the Hamiltonian system (\re{2}) is invariant under
time-reversal, our results also hold for $t\to-\,\infty$.\\
(ii)
The assumption $(C)$ can be weakened to finite differentiability and
some decay of $\rho(x)$ at infinity.
\medskip
To prove Theorem \re{A} we will estimate the energy dissipation by decomposing
$\phi$ into a near and far field. Energy is dissipated in the far
field. Since energy is bounded from below, such dissipation
cannot go on forever
and a certain energy dissipation functional has to be bounded.
This dissipation functional can be written as a convolution. By a
Tauberian theorem of Wiener, using $(W)$, we conclude that
$\lim_{t\to\infty}\ddot q(t)=0$, and also $\lim_{t\to\infty}\dot q(t)=0$
since $|q(t)|$ is bounded by some $q_0<\infty$ due to $(P)$.
This implies that ${\cal A} = \{(\phi_q, q, 0, 0): |q|\le q_0)\}$
is a compact attracting set.
Relaxation and compactness reduce ${\cal A}$ to ${\cal S}$
as a minimal attractor.
To establish the rate of convergence in Theorem \re{A} the point $q^*\in S$
must be stable in the following sense.
\begin{definition}\la{1.1}
A point $q^*\in S$ is said to be stable if
$d^2V({q^\ast})>0$ as a quadratic form.
\end{definition}
Even for a stable $q^*\in S$, the slow decay of the initial fields
in space will transform into a slow decay in time.
\begin{theorem}\la{C}
Let all assumptions of Theorem \re{A} hold,
$V\in C^3(\R^3)$
and let $Y(t)\in C(\R,{\cal E})$ be a solution of the system
(\re{2}) converging
to
$Y_{q^*}$ as in (\re{10}) with the stable point $q^*\in S$. Then\\
i) for every $R,\ve>0$
\be\la{EE}
~ {\|Y(t)-Y_{q^*}\|}_R={\cal O}(t^{-\sigma+\ve})~~~as~~~t\to\infty.
\ee
ii) Let additionally
\be\la{8'}
~~~~~~~~~DY^0(x)~={\cal O}(e^{-\alpha|x|})~~~as~~~|x|\to\infty
\ee
with some $\alpha>0$. Then
there exists a $\gamma^*=\gamma(q^*)>0$ such that
for every $R>0$
\be\la{EE'}
~~~ {\|Y(t)-Y_{q^*}\|}_R={\cal O}(e^{-\beta t})~~~~as~~~~t\to\infty~~
\ee
with
$\beta=\alpha$ if
$\alpha<\gamma^*$
and with arbitrary
$\beta<\gamma^*$ if $\alpha\geq\gamma^*$.
\end{theorem}
We will prove Theorem \re{C} in Sections 6 to 9 by controlling the
nonlinear part of (\re{2}) by
the linearized equation.
For the linearized equation
exponential convergence can be
established by Paley-Wiener technics for complex Fourier transforms \ci{PW}.
As a byproduct in Theorem \re{B} we will also establish
exponential convergence for
initial states not covered by Theorem \re{C}.
Before entering into the proofs it may be useful to
put our results in
the context of related works.
We establish here that solutions of a Hamiltonian system
converge to an attractor, possibly consisting of an infinite
number of points, in the long time limit. Such a behavior
is familiar from dissipative systems.
The mechanism is however completely different.
For a dissipative system there is a local loss of ``energy'',
whereas here energy is propagated to infinity.
If the wave field in (\re{2}) would be enclosed in some finite volume,
then Theorem \re{A} would not be valid.
Propagation of energy to infinity is also the
essence of scattering theory for Hamiltonian linear wave equations
\ci{{LMP},{LP}, {M}}, \ci{V68}--\ci{V89} and for Hamiltonian
nonlinear wave equations
either with a
unique ``zero'' stationary solution
\ci{{Ch},{GV},{GS1},{GS2}, {MS},{Se},{St}}
or with small initial data \ci{{H},{K}}.
Note that the attractor consists then only of the zero solution
in contrast to the case considered here.
Somewhat closer to our investigation are \ci{KUs}--\ci{KMo} where
a one dimensional version of (\re{2}) is studied: the particle
is coupled to an infinite string and moves only transversally
subject to some confining external potential.
The interaction with the string generates then a linear friction
term for the dynamics of the particle and
the attraction to stationary states
can be studied by ordinary differential equation methods.
\ci{KMa} considers several such oscillators coupled to a string.
In this case the effects of retarded interaction have to be controlled.
For wave equations with local nonlinear terms a result
similar to Theorem \re{C}
is proved in \ci{KV}.
%%%%%%%%%%%%%%%%%% Section 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Existence of dynamics, a priori estimates}
We consider the Cauchy problem for the Hamiltonian system (\re{2}),
which we write as
\be\la{2.1}
\dot Y(t)=F_0(Y(t))+F_1(Y(t)),\quad t\in\R,\quad Y(0)=Y^0.
\ee
All derivatives are understood in the sense of distributions.
Here $Y(t)=(\phi(t), q(t), \pi(t), p(t))$, $Y^0=(\phi^0, q^0, \pi^0, p^0)
\in {\cal E}$, and $F_0: Y\mapsto (\pi, 0, \Delta\phi, 0)$.
One is interested also in situations where the particle is
allowed to travel to infinity, e.g. when the external potential $V(q)$
vanishes identically. The existence of dynamics and the relaxation
of the
acceleration $\ddot q(t)$ are in fact true under such
more general conditions.
We state then as a weaker form of $(P)$,
$$
~~~~~~~~V\in C^2(\R^3), ~~~~~~
V_{0}:=\inf_{q\in \R^3}
V(q)>-\infty.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~(P_{\min})
$$
\begin{lemma}\label{existence} Let $(C)$ and $(P_{\min})$ hold.
Then\\
(i) For every $Y^0\in {\cal E}$ the Cauchy problem (\re{2.1}) has a unique
solution $Y(t)\in C(\R, {\cal E})$.\\
(ii) For every $t\in\R$ the map $W_t: Y^0\mapsto Y(t)$ is continuous both on
${\cal E}$ and on ${\cal E}_F$.\\
(iii) The energy is conserved, i.e.
\[ H(Y(t))=H(Y^0)~~~for~~t\in\R. \]
(iv) The energy is bounded from below, and
\be\la{Hm}
\inf_{Y\in {\cal E}} H(Y)=1+V_{0}
-\fr 1{8\pi(2\pi)^3}\int~d^3k~
\fr{|\hat\rho(k)|^2}{|k|^2}\,.
\ee
(v) The speed is bounded,
\be\la{2.1'}
|\dot q(t)|\leq q_1<1~~for~~t\in\R.
\ee
(vi) If $(P)$ holds, then the time derivatives $q^{(k)}(t)$, $k=0,2,3$,
also are uniformly bounded,
i.e.~there are constants $q_k>0$, depending only on the initial data,
such that
\be\la{2.4} |q^{(k)}(t)|\le q_k~~~for~~t\in\R.
\ee
\end{lemma}
{\bf Proof} Let us fix an arbitrary $b>0$ and prove (i)--(iii) for
${\Vert Y^0\Vert}_{\cal E}\le b$ and $|t|\le\varepsilon=\varepsilon(b)$ for
some sufficiently small $\varepsilon(b)>0$.\\
{\it ad (i)} Fourier transform provides the existence and uniqueness of
solution $Y_0(t)\in C(\R,{\cal E})$ to the linear problem (\re{2.1})
with $F_1=0$. Let $W^0_t: Y^0\mapsto Y_0(t)$ be the corresponding
strongly continuous group of
bounded linear operators on ${\cal E}$. Then
uniqueness of solution to the (inhomogeneous)
linear problem implies that (\re{2.1}) for
$Y(t)\in C(\R,{\cal E})$ is equivalent to
\be\la{2.5}
Y(t)=W^0_tY^0+\int_0^t\,ds\, W^0_{t-s}F_1(Y(s)),
\ee
because $F_1(Y(\cdot))\in C(\R,{\cal E})$ in this case. The latter follows
from Lipschitz continuity of the map
$F_1$ in ${\cal E}$: for each $b>0$ there exist
a $\kappa=\kappa (b)>0$ such that for all $Y, Z\in {\cal E}$
with ${\Vert Y\Vert}_{\cal E}, {\Vert Z\Vert}_{\cal E}\le b$
\be\la{Lip}
{\Vert F_1(Y)-F_1(Z)\Vert}_{\cal E}\le\kappa
{\Vert Y-Z\Vert}_{\cal E}\,.
\ee
For example, we have
\[ \bigg|\int d^3x\, (\phi_1(x)-\phi_2(x))\nabla\rho(x-q)\bigg|
\le\br\nabla(\phi_1-\phi_2)\br\br\rho\br. \]
Moreover, by the contraction mapping principle, (\re{2.5}) has a unique local
solution $Y(\cdot)\in C([-\varepsilon,\varepsilon], {\cal E})$ with
$\varepsilon>0$ depending only on $b$. \\
{\it ad (ii)} The map $W_t: Y^0\mapsto Y(t)$ is continuous in the norm
${\Vert\cdot\Vert}_{\cal E}$ for $|t|\le\varepsilon$ and
${\Vert Y^0\Vert}\le b$. To prove continuity of $W_t$ in ${\cal E}_F$, let us
consider Picard's successive approximation scheme
\[ Y^N(t)=W^0_tY^0+\int_0^t ds\,W^0_{t-s}F_1(Y^{N-1}(s)),~~~N=1,2,\dots \]
The third equation in this system implies $|\dot q^N(t)|<1$ and therefore
$|q(t)|<|q^0|+|t|$. Now we fix $t\in\R$ and choose $R>|q^0|+|t|+R_\rho$ with
$R_\rho$ from $(C)$. From the explicit solution of the free wave equation
$W^0_t Y^0$ we conclude that every Picard's approximation $Y^N(t)$ and
hence the solution
$Y(t)=(\phi(x,t), q(t), \pi(x,t), p(t))$ for $|x||q(t)|+R_\rho$
the energy $E_R(t)$ in the ball $B_R$ at time $t>0$
is defined by
\begin{eqnarray}\la{ER}
E_R(t) & = & \fr 12\int_{B_R}d^3x (|\pi(x,t)|^2+|\nabla\phi(x,t)|^2)
\nonumber\\
& & + (1+p^2(t))^{1/2}+V(q(t))+\int d^3x \phi(x,t)\rho(x-q(t))\,.
\end{eqnarray}
Let us denote $\omega(x)=x/|x|$, $d^2x$ the surface
area element of $\pa B_R$ and
$\ov R_0=\max (R_0, |q^0|+R_\rho)$.
Let $\Delta_R=[\ov R_0+R,(R-\ov R_0)/q_1]$. Since $0R_0+|x|$. Moreover, the estimate (\re{2.1'})
insures that $|q(t)|<|q^0|+q_1t$ for $t>0$. Hence, the
definition (\re{ER}) leads to
\be\la{3.3}
\fr d{dt}E_R(t)
=\int_{\pa B_R}d^2 x ~~\omega (x)\cdot\nabla\phi(x,t)~\pi(x,t)
\mbox{~~~~for~~~} t\in\Delta_R.
\ee
The differentiability follows from the integral representation
of the solution. Namely,
\be\la{Dec}
\phi(x,t)=\phi_r(x,t)+\phi_0(x,t)
{\rm ~~~~for~~}x\in\R^3,~~t>0,
\ee
where $\phi_r(x,t)$
is the retarded potential and $\phi_0(x,t)$ is the Kirchhoff integral,
\beqn
\phi_r(x,t)&= -&\fr 1{4\pi}\int
\fr{d^3y~\theta(t-|x-y|)}{|x-y|}\rho(y-q(t-|x-y|))
\la{retp},\\
\phi_0(x,t)&= &\fr 1{4\pi t}\int_{S_t(x)}~d^2y~\pi^0(y)+
\fr \pa {\pa t}~\Bigg(~\fr 1{4\pi t}\int_{S_t(x)}~d^2y~\phi^0(y)~\Bigg).
\la{K}
\eeqn
$S_t(x)$ denotes here the sphere $\{y:~|y-x|=t\}$.
Step 2. We have, $\nabla\phi(x,t)=\nabla\phi_r(x,t)+\nabla\phi_0(x,t)$
and
$\pi(x,t)=\pi_r(x,t)+\pi_0(x,t)$, where
$\pi_r(x,t)=\dot\phi_r(x,t)$ and
$\pi_0(x,t)=\dot\phi_0(x,t)$.
Hence (\re{3.3}) reads
$$
\fr d{dt}E_R(t)
=\int_{\pa B_R}d^2 x ~~\omega(x) \cdot
(\nabla\phi_r~\pi_r+\nabla\phi_r~\pi_0+\nabla\phi_0~\pi_r+
\nabla\phi_0~\pi_0)
\mbox{~~for~~}
t\in \Delta_R.
$$
We separate the first term which turns out to be negatively
definite from the remainder which is controlled simply by Cauchy-Schwarz.
We have
$$
\fr d{dt}E_R(t)\leq
\int_{\pa B_R}d^2 x
\Big(
\omega(x) \cdot\nabla\phi_r~\pi_r+ \fr 14(|\nabla\phi_r|^2+|\pi_r|^2)
+ 2(|\nabla\phi_0|^2+|\pi_0|^2)
\Big)\nonumber\\
\mbox{ for }t\in\Delta_R.\nonumber
$$
Integrating in $t$ we obtain for
$\ov R_00\nonumber
$$
with a constant $I<\infty$ not depending on $R,T$.
Therefore, (\re{3.7}) implies
\beqn\la{3.7'}
&& - \int_{\ov R_0+R}^{T+R}dt
\int_{\pa B_R}d^2 x ~~
\Big(
(\omega(x) \cdot\nabla\phi_r~\pi_r+
\fr 14
(|\nabla\phi_r|^2+|\pi_r|^2)
\Big)
\nonumber\\
&&\leq I+
2
\int_{\ov R_0+R}^{T+R}dt
\int_{\pa B_R}d^2x~
(|\nabla\phi_0|^2+|\pi_0|^2){\rm ~~~for~~}\ov R_0\ov R_0$
\be\la{kr}
\nabla \phi_r(x,t)=
-\pi_r(x,t)~\omega(x)+{\cal O}(|x|^{-2})
\mbox{ in the region }
\ov R_00.
\ee
\end{lemma}
Using Lemma \re{KR} and \re{I0} in (\re{3.7'}), we obtain
for every fixed $T>0$ and sufficiently large
$R>R_T\sim Tq_1/(1-q_1)$
\beqn
\int_{\ov R_0+R}^{T+R}dt
\int_{\pa B_R}d^2 x ~~|\pi_r(x,t)|^2
\leq C( I+I_0)+T{\cal O}(R^{-2}).\nonumber
\eeqn
Hence, (\re{retp}) implies
\be\la{db}
\int_{\ov R_0+R}^{T+R}dt \int_{\pa B_R}d^2x~~
|\int_{Q_T} d^3y\fr 1{|x-y|}\fr \pa{\pa t}\rho(y-q(t-|x-y|))|^2\leq
C(I+I_0)+T{\cal O}(R^{-2}),
\ee
where $Q_T=\{y\in\R^3:~|y|\leq \max_{[0,T]}|q(t)|+R_\rho\}$.
Furthermore,
uniformly in
$x\in\pa B_R$ and $y\in Q_T$
\[ |x-y|\sim R \quad\mbox{and}\quad
t+R-|x-y |=t+\omega \cdot y+{\cal O}(R^{-1}), ~~~~\omega=x/|x|.\]
Thus taking the limit $R\to\infty$ in (\re{db})
we obtain
$$
\int_{\ov R_0}^Tdt\int_{S^2} d^2\omega
{\bigg|\int d^3y \fr{\pa}{\pa t}
\rho(y-q(t+\omega\cdot y))\bigg|}^2 \le C(I+I_0).
\nonumber
$$
Since this bound also holds in the limit $T\to\infty$, we are left with
rewriting the integrand as in (\re{3.1}). For this purpose we note that
\[ \int d^3y \fr{\pa}{\pa t}\rho(y-q(t+\omega\cdot y))=
-\int d^3y\,\dot q(t+\omega\cdot y)\cdot
\nabla\rho(y-q(t+\omega\cdot y)) \]
and
\[ \dot q(t+\omega\cdot y)\cdot\nabla_y(\rho(y-q(t+\omega\cdot y)))=
(1-\omega\cdot\dot q(t+\omega\cdot y))\dot q(t+\omega\cdot y)
\cdot\nabla\rho(y-q(t+\omega\cdot y)). \]
Thus, by partial integration, and because of $|\dot q(t)|<1$,
we finally obtain
\[ \int_Q d^3y \fr{\pa}{\pa t}\rho(y-q(t+\omega\cdot y))=
\int_Q d^3y \,\rho(y-q(t+\omega\cdot y))\nabla_y\cdot
\big(\dot q(t+\omega\cdot y)(1-\omega\cdot\dot
q(t+\omega\cdot y))^{-1}\big), \]
which agrees with the integrand in (\re{3.1}). {\hfill$\Box$}
\medskip
We still have to prove Lemma \re{KR}, \re{I0}.
\medskip\\
{\bf Proof of Lemma \re{KR}}
The representation (\re{retp}) implies for $\ov R_0<|t|-|x|\ov R_0+|x|$,
\be\la{dt}
\nabla\phi_0(x,t)=\sum_{|\alpha|\leq 1} t^{|\alpha|-2}
\int_{S_t(x)} d^2y~a_\alpha(x-y)\pa^\alpha\pi^0(y)+
\sum_{|\alpha|\leq 2} t^{|\alpha|-3}
\int_{S_t(x)} d^2y~b_\alpha(x-y)\pa^\alpha\phi^0(y).
\ee
Here all derivatives are understood in a classical sense,
and the coefficients $a_\alpha(\cdot)$ and $b_\alpha(\cdot)$ are bounded.
A similar representation holds for $\pi_0(x,t)$.
The coefficients are homogeneous functions of order
zero and smooth outside the origin.
Hence, taking into account our assumption (\re{8}),
we obtain from (\re{dt}) for $t>\ov R_0+|x|$
\be\la{bdt}
|\nabla\phi_0(x,t)|\leq
C \sum_{0\leq k\leq 1} t^{k-2}~\int_{S_t(x)} d^2y~|y|^{-\sigma-1-k}
+
C \sum_{0\leq k\leq 2} t^{k-3}~\int_{S_t(x)} d^2y~|y|^{-\sigma-k}.
\ee
We can always adjust $\sigma$ such that $\sigma+k\not= 2$. Then
by explicit computation a typical term reads,
$$
I^k(x,t):=~ \int_{S_t(x)} d^2y~|y|^{-\sigma-k}
=\fr{2\pi t}{|x|(2-\sigma-k)}
\Bigg( (t+|x|)^{2-\sigma-k}~-~(t-|x|)^{2-\sigma-k} \Bigg).
$$
Hence the contribution of the corresponding term from (\re{dt})
in the left hand side of (\re{ub})
can be majorized by
\beqn\la{ba}
B^\alpha_{R,T}:=&&C_1\int_{\ov R_0+R}^{T+R}dt \int_{\pa B_R}d^2x
~\Bigg| t^{|\alpha|-3} I^{|\alpha|}(x,t)\Bigg|^2\nonumber\\
=&& C_2\int_{\ov R_0}^Tdt~(R+t)^{2(|\alpha|-2)}
\Bigg| (2R+t)^{2-\sigma-|\alpha|}~-~t^{2-\sigma-|\alpha|}\Bigg|^2.
\eeqn
We may adjust $\sigma$ slightly larger than $1/2$.
Then for
$|\alpha|\leq 1$, we have $\sigma+|\alpha|\leq 2$ and (\re{ba}) implies
$$
B^\alpha_{R,T}\leq C\int_{\ov R_0}^Tdt~(R+t)^{-2\sigma}\leq B^\alpha<
\infty\mbox{ for }R,T\geq 0.
\nonumber
$$
For $|\alpha|=2$ the bound
(\re{ba}) implies
$$
B^\alpha_{R,T}\leq C\int_{\ov R_0}^Tdt~t^{-2\sigma}\leq B^\alpha<
\infty\mbox{ for }R,T\geq 0.
\nonumber
$$
All contributions of other terms from (\re{dt}) can be majorized
in a similar fashion, correspondingly for $\pi_0(x,t)$
on the left hand side of
(\re{ub}).{\hfill$\Box$}
\medskip\\
{\it Remark}
The representations (\re{K}), (\re{dt}) and the corresponding
representation for
$\pi_0(x,t)$ together with (\re{8})
imply for every $R>0$
\be\la{Rdec}
\max_{|x|\leq R} \Big(
|\phi_0(x,t)|+t|\pi_0(x,t)|+t|\nabla\phi_0(x,t)|
\Big)={\cal O}(t^{-\sigma}) \mbox{~~as~~}t\to\infty,
\ee
where the derivatives are understood in a classical sense.
%%%%%%%%%%%%%%%%%%%%%%%%% Section 4 %%%%%%%%%%%%%%%%%%%%%
\section{Relaxation of the particle velocity}
In this section
we will deduce from Proposition \ref{aussen}
that $\dot q(t)$,
$\ddot q(t)\to 0$ as $t\to\infty$ provided we add
the assumptions $(W)$ and $(P)$.
Assumption $(P)$ implies the bounds (\re{2.1'}) and (\re{2.4})
with $k=2,3$
due to Lemma \re{existence} (vi). Hence, the function
\be\la{4.2}
R_{\omega}(t)= \int d^3y\,\rho(y-q(t+\omega\cdot y))
\frac{\omega\cdot\ddot q(t+\omega\cdot y)}
{{(1-\omega\cdot\dot q(t+\omega\cdot y))}^2}
\ee
is Lipschitz continuous in $\omega$ and $t$. Thus by Proposition \ref{aussen}
\be\la{4.3}
\lim_{t\to\infty} R_{\omega}(t)=0
\ee
uniformly in $\omega\in S^2$.
Let $r(t)=\omega\cdot q(t)\in\R$, $s=\omega\cdot y$, and
$\rho_a(q_3)=\int dq_1dq_2\rho(q_1,q_2,q_3)$. In (\re{4.2}) we
decompose the $y$-integration along and transversal to $\omega$. Then
\begin{eqnarray}\la{4.4}
R_{\omega}(t) & = & \int ds \,\rho_a(s-r(t+s))\,\frac{\ddot r(t+s)}
{{(1-\dot r(t+s))}^2}\nonumber\\ & = &
\int d\tau\, \rho_a(t-(\tau-r(\tau)))\,
\frac{\ddot r(\tau)}{{(1-\dot r(\tau))}^2}\nonumber\\
& = & \int d\theta\, \rho_a(t-\theta) g_\omega(\theta)=\rho_a*g_\omega(t).
\end{eqnarray}
Here we substituted $\theta=\theta(\tau)=\tau-r(\tau)$,
which is a nondegenerate diffeomorphism
since $|\dot r|\leq q_1<1$ due to (\re{2.1'}), and we set
$$
g_{\omega}(\theta)=(1-\dot r(\tau(\theta)))^{-3}\ddot r(\tau(\theta)).
$$
Let us extend $q(t)$ smoothly to zero for $t<0$.
Then $\rho_a*g_\omega\,(t)$ is
defined for all $t$ and agrees with $R_\omega(t)$ for sufficiently
large $t$. Hence (\re{4.3}) reads as a convolution limit
\be\la{4.6}
\lim_{t\to\infty} \rho_a*g_{\omega}(t)=0.
\ee
Now note that (\re{2.1'}) and (\re{2.4}) with $k=2,3$ imply that
$g^\prime_\omega(\theta)$
is bounded.
Hence (\re{4.6}) and $(W)$ imply by
Pitt's extension to Wiener's Tauberian Theorem,
cf.~\cite[Thm.~9.7(b)]{Ru},
\be\la{4.7}
\lim_{\theta\to\infty} g_{\omega}(\theta)=0.
\ee
Because $\omega\in S^2$ is arbitrary
and $\theta(t)\to\infty$ as $t\to\infty$, we have proved
\begin{lemma}\la{Wi}
Let
all assumptions of Proposition \re{aussen} hold, and the potential $V$
satisfy $(P)$.
If $(W)$ holds, then
\be\la{4.7'}
\lim_{t\to\infty} \ddot q(t)=0.
\ee
\end{lemma}
{\it Remarks.} (i) For a point charge $\rho(x)=\delta(x)$, (\re{4.6}) implies
(\re{4.7}) directly.\\
(ii) Parseval's identity, (\re{4.4}) and (\re{3.1}) give
\[ \int_{S^2} d^2\omega\int d\xi\,
|\hat\rho_a(\xi) \hat g_{\omega}(\xi)|^2<\infty. \]
If $|\hat\rho_a(\xi)|\ge C>0$, then
$\int d^2\omega\int dt\,|g_{\omega}(t)|^2<\infty$, and (\re{4.7}) would follow
from the Lipschitz continuity of $g_\omega$. Thus, the main difficulty results
from the rapid decay of the Fourier transform (``symbol") $\hat\rho_a$, due
to the smoothness of the kernel $\rho_a$.\\
(iii) Condition $(W)$ is necessary. Indeed, if $(W)$ is violated, then
$\hat\rho_a(\xi)=0$ for some $\xi\in\R$, and with the choice
$g(t)=\exp(i\xi t)$ we have $\rho_a*g(t)=0$ whereas $g$ does not decay to
zero.
\begin{cor}
Let all assumptions of Theorem \re{A} hold. Then
\be\la{rel}
\lim_{t\to\infty}\dot q(t)=0.
\ee
\end{cor}
{\bf Proof} Since $|q(t)|\le q_0$ due to (\re{2.4}) with $k=0$,
(\re{4.7'}) implies (\re{rel}).
{\hfill$\Box$}
\medskip\\
{\it Remark} Lemma \re{Wi}
holds even under an
assumption on the potential, weaker than $(P)$,
$$
~~~V\in C^2(\R^3),~~\inf_{q\in\R^3}V(q)>-\infty~~~~
{\rm ~~and~~~~~~}
\sup_{q\in\R^3} |\pa^\alpha V(q)|<\infty{\rm ~~for~}|\alpha|= 1
{\rm ~and~}2.
~~~(P_w)
$$
The proof of Lemma \re{Wi} with $(P_w)$ instead of $(P)$
remains unchanged. Firstly,
$(P_w)$ includes $(P_{\min})$, hence it provides (\re{3.1}) and
(\re{2.1'}).
Secondly, $(P_w)$ implies the bounds (\re{2.4}) with $k=2,3$
similarly to $(P)$ in Lemma \re{existence} {\it (vi)}.
%%%%%%%%%%%%%%%%%%%%%%%%% Section 5 %%%%%%%%%%%%%%%%%%
\section{A compact attracting set, proof of Theorem \re{A}}
\begin{definition}\la{CA}
Let ${\cal A}=
\{ Y_q:q\in\R^3,~~ |q|\le q_0\}$, where $Y_q$ is defined in
(\re{ss}).
\end{definition}
Since ${\cal A}$ is homeomorphic to a closed ball
in $\R^3$, ${\cal A}$ is compact in ${\cal E}_F$.
\begin{lemma}\la{CAT}
Let all assumptions of Theorem \re{A} hold. Then
$Y(t)\to{\cal A}$ in
${\cal E}_F$ as $t\to\infty$.
\end{lemma}
{\bf Proof.} For every $R>0$
\begin{eqnarray}\la{5.1}
&&{\rm dist}_R (Y(t),{\cal A})=
\inf_{Y_q\in {\cal A}}{\Vert Y(t)- Y_q \Vert}_R \nonumber\\
&& =
|p(t)|+{\br\pi(t)\br}_R
+\,\inf_{|q|\le q_0}\big({\br\nabla(\phi(t)- \phi_q)\br}_R
+ {\br\phi(t)-\phi_q\br}_R+|q(t)-q|\big).
\end{eqnarray}
Let us
estimate each term separately.\\
i) (\re{rel}) implies $|p(t)|\to 0$ as $t\to\infty$.\\
ii) Let us denote $R_\infty=q_0+R_\rho$.
Then (\re{3.5}) implies
for $t>R+R_\infty$ and $|x|R+R_\infty$ and $x\in B_R$, and therefore
(\re{rel}) implies ${\br\pi_r(t)\br}_R\to 0$ as $t\to\infty$.
Then also ${\br\pi(t)\br}_R\to 0$ due to (\re{Dec}) and (\re{Rdec}).\\
iii)
To estimate the infimum over $q$ in
(\re{5.1}), we may substitute $q(t)$ for $q$. Then the
last term vanishes, and (\re{retp}) implies
for $t>R+R_\infty$ and $|x|0$
for some $R,\varepsilon>0$
and a sequence $t_k\to\infty$.
Then Lemma \re{CAT} and the compactness of ${\cal A}$ imply that,
for a suitable subsequence,
$Y(t_{k^\prime})\to\ov Y$ in ${\cal E}_F$,
where $\ov Y\in{\cal A}$.
Then $\ov Y\in\Omega$ by definition. Since
${\rm dist}_R(\ov Y,{\cal S})\geq\varepsilon>0$
we obtain a contradiction to
$\Omega\subset{\cal S}$. \\
{\it ad (ii)} Lemma \re{CAT} together with (\re{9}) imply that
$Y(t)\to {\cal A}\cap{\cal S}$ in ${\cal E}_F$ as $t\to\infty$.
If the set $S$ of critical points of $V$ is
discrete, then the set ${\cal S}$ is discrete
in ${\cal E}_F$. Moreover, the set ${\cal S}$ is closed in ${\cal E}_F$.
Hence, the intersection
${\cal A}\cap{\cal S}$ is a finite set,
as the intersection of a compact set and of a closed discrete set.
In (\re{9}) the states $Y(t)$
approach the finite subset ${\cal A}\cap{\cal S}$
in
the metrisable space
${\cal E}_F$. Therefore
the continuity of $Y(t)$
implies (\ref{10}).
{\hfill$\Box$}
%%%%%%%%%%%%%%%%%%%%% Section 6 %%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Linearization around a stationary state}
If the particle is close to a stable minimum of $V$, we expect the nonlinear
evolution to be dominated by the linearized dynamics.
As to be explained in Section 7 the linearized dynamics
has exponentially
fast convergence provided the initial fields have compact support.
For the nonlinear dynamics this corresponds to an initial state
of the form $Y_{q^*}$ plus perturbation of compact support.
In such a situation we expect then exponential convergence to $Y_{q^*}$.
The precise estimate is given in Theorem \re{B} below. In Theorem \re{C}
the initial fields (and not the perturbation) have compact support.
Thus it still requires some effort to prove Theorem \re{C}, which
is deferred to Section 9.
In this section we
establish the required estimate.
\begin{theorem}\la{B}
Let $(C)$, $(W)$ hold, and $V\in C^3(\R^3)$.
Let $q^*\in S$ be a stable point and let
the initial state $Y^0\in {\cal E}$ be such that
\be\la{L}
\phi^0(x)=\phi_{q^*}(x),~~~\pi^0(x)=0 ~~for~~ |x|\ge M
\ee
with some $M>0$.
Let $Y(t)\in C(\R,{\cal E})$ be corresponding solution to
(\re{2}) with $Y(0)=Y^0$.
Then
there exist a $\delta=\delta (M)>0$
and a $\gamma^*=\gamma(q^*)>0$, such that
if ${\|Y^0-Y_{q^*}\|}_{\cal E}\le\delta$,
the bounds hold
for every $R>0$ and every $\gamma<\gamma^*$
\be\la{E}
{\|Y(t)-Y_{q^*}\|}_R\le C_R\,e^{-\gamma t},\,\,t\geq 0
\ee
with some suitable constant $C_R=C_R(M,\gamma)>0$.
\end{theorem}
For notational simplicity we also assume isotropy in the sense that
\begin{equation}\label{hesse}
\partial_i\partial_j V(q^*)=\omega^2_0\delta_{ij},\
i,j=1,2,3,\ \omega_0>0\,.
\end{equation}
Without loss of
generality we take $q^\ast=0$.
Let $Y_{q^{\ast}} = Y_0 = (\phi_0, 0, 0, 0)$ be the
stationary state of (\ref{2}) corresponding to
$q^\ast=0$, and
$Y^0=(\phi^0, q^0, \pi^0, p^0)\in {\cal E}$ be an arbitrary
state satisfying (\re{L}). We denote by $Y(t)=(\phi(x,t), q(t), \pi(x,t), p(t))
\in{\cal E}$ the solution to (\re{2}) with $Y(0)=Y^0$.
To linearize (\ref{2}) at $Y_0$, we
set $\psi(x,t)=\phi(x,t)-\phi_0(x)$. Then (\ref{2}) becomes
\beqn\label{nonlin}
\ba{lll}
\dot{\psi}(x, t)=\pi (x, t),
&\dot{\pi}(x, t) = &\Delta \psi(x, t)+\rho(x)-\rho(x-q(t)),
\\
\dot{q}(t) = p(t)/{(1+p^2(t))}^{1/2},
&\dot{p}(t) = & -\nabla V(q(t))+
\int d^{\,3}x\,\,\psi(x, t)\,\nabla\rho(x-q(t)) \\
& & +\,\int d^{\,3}x\,\,\phi_0(x)[\nabla\rho (x-q(t))-\nabla\rho(x)]\,.
\ea
\eeqn
We define the linearization by
\beqn\label{linear}
\ba{lll}
&\dot{\Psi}(x, t) = \Pi (x, t),\quad
&\dot{\Pi}(x, t) = \Delta \Psi(x,t) + \nabla\rho(x) \cdot Q(t),\quad
\\
&\dot{Q}(t) = P(t),\quad\quad\quad
&\dot P(t) = -(\omega^2_0+\omega^2_1) Q(t)+
\int d^3x\,\Psi(x,t)\nabla\rho(x)\,.
\ea
\eeqn
Here
\be\la{O1}
\omega^2_1\delta_{ij}=\frac{1}{3}\,{\br{\rho}\br}^2\delta_{ij}=-\,
\int d^{\,3}x\,\pa_i\phi_0(x)\pa_j\rho(x) \,,
\ee
where the factor
$1/3$ is due to a spherical symmetry of $\rho(x)$ (see $(C)$).
We rewrite (\ref{linear}) as
\be\la{lin}
\dot Z(t)=AZ(t),~~t\in\R.
\ee
Here $Z(t)=(\Psi(\cdot,t),Q(t),\Pi(\cdot,t),P(t))$ and $A$ is
the linear operator defined by
$$
A:\,Z=(\Psi,Q,\Pi,P)
\mapsto (\Pi,P,\Delta\Psi+\nabla\rho\cdot Q,-\omega^2 Q+
\int d^3x\,\Psi(x)\nabla\rho(x)),
$$
where $\omega^2=\omega_0^2+\omega_1^2$.
(\ref{lin}) is a formal Hamiltonian system with the quadratic
Hamiltonian
\be\la{H0}
H_0(Z)=
\fr 12 \Big( {P^2} +\omega^2 Q^2+
\int d^3x\,(|\Pi(x)|^2+|\nabla\Psi(x)|^2-2\Psi(x)\nabla\rho(x)\cdot Q) \Big),
\ee
which is the formal
Taylor expansion of $H(Y_0+Z)$ up to second order at $Z=0$.
Introducing $X(t)=Y(t)-Y_0=(\psi(t),q(t),\pi(t),p(t))\in C(\R, {\cal E})$,
we rewrite the nonlinear system (\ref{nonlin}) in the form
\be\la{AB}
\dot X(t)=AX(t)+B(X(t)).
\ee
Due to (\ref{O1})
the nonlinear part is given by
\beqn\la{BB}
B(X)& = &\Big(0,\,p/(1+p^2)^{1/2}-p,\,
\rho(x)-\rho(x-q)-\nabla\rho(x)\cdot q, \nonumber\\ & &
\,\,\,-\nabla V(q)+\omega_0^2 q
+\,\int d^{\,3}x\,\psi(x) [\nabla\rho(x-q)-\nabla\rho (x)]
\nonumber\\ & & \,\,\,+\,\int d^{\,3}x \,\nabla\phi_0(x)
[\rho(x)-\rho(x-q)-\nabla\rho (x)\cdot q]\,\Big)
=:(\psi^1(x),q^1,\pi^1(x),p^1)
\eeqn
for
$X=(\psi,q,\pi,p)\in {\cal E}$.
This definition immediately implies
\begin{lemma}\label{Best}
Let $(C)$ hold,
$V\in C^3(\R^3)$,
and $b>0$ be some
fixed number. Then for
$|q|\leq b$ \\
(i) with notations (\ref{BB}),
\be\la{Bloc}
\psi^1(x)=\pi^1(x)=0~~for~~|x|\geq R_\rho+b~;
\ee
(ii) for every $R>0$
\be\la{BXest}
{\|B (X)\|}_R \le
C_b\,{\|X\|}^2_{R_\rho + b}.
\ee
\end{lemma}
Let us consider the Cauchy problem for the linear equation (\ref{lin}) with
initial condition
\be\la{ic}
Z|_{t=0}=Z^0.
\ee
\begin{lemma}\label{exlin}
Let $(C)$ hold. Then\\
(i) For every $Z^0\in{\cal E}$ the Cauchy problem (\ref{lin}), (\ref{ic})
has a unique solution $Z(\cdot)\in C(\R,{\cal E})$.\\
(ii) For every $t\in\R$ the map $U(t):Z^0\mapsto Z(t)$ is continuous
both on ${\cal E}$ and on ${\cal E}_F$.\\
(iii) For $Z^0\in{\cal E}$ the energy $H_0$ is finite and conserved, i.e.
\be\la{ec}
H_0(Z(t))=H_0(Z^0)~~for~~t\in\R.
\ee
iv) For $Z^0\in{\cal E}$
\be\la{linb}
\Vert Z(t)\Vert_{\cal E} \leq B~~~for~~t\in\R
\ee
with $B$ depending only on the norm $\Vert Z^0\Vert_{\cal E}$
of the initial state.
\end{lemma}
The proof of this lemma is almost identical with the proof of the Lemma 2.1.
In fact, for the linearized system the Hamiltonian
is nonnegative, since (\ref{H0}) with the definition (\ref{O1}) implies
$$
2H_0(Z)=
{P^2} + \omega_0^2 Q^2+
\int d^3x\,(|\Pi(x)|^2+
||\nabla|\Psi(x)-|\nabla|^{-1}(\nabla\rho(x)\cdot Q)|^2)\geq 0.
$$
Thus (\ref{linb}) follows from (\ref{ec}) because of $\omega_0>0$.
{\hfill$\Box$}
%%%%%%%%%%%%%%%%% Section 7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section
{ Decay estimates for the linearized system}
We prove the local decay of solutions $Z(t)$ to the linearized
system (\ref{lin}).
\begin{pro}\label{lElin}
Let $(C)$ and $(W)$ hold, and $\omega_0>0$.
Let
$Z(t)\in C(\R,{\cal E})$ be a solution to the Cauchy problem
(\ref{lin}), (\ref{ic}) and
the initial state $Z^0=(\Psi^0,Q^0,\Pi^0,P^0)\in{\cal E}$ have
compact support,
\be\la{CS}
\Psi^0(x)=\Pi^0(x)=0~~for~~|x|\geq M
\ee
with some $M>0$.
Then there exist a $\gamma^*>0$
such that for every $R>0$ and every $\gamma<\gamma^*$
\be\la{Elin}
\Vert Z(t)\Vert_R\leq C_Re^{-\gamma t}\Vert Z^0\Vert_{\cal E}~~~~~for~~t\geq 0
\ee
with suitable $C_R=C_R(M,\gamma)>0$.
\end{pro}
To prove this proposition we solve the Cauchy problem
(\ref{lin}), (\ref{ic}) explicitly through Laplace transform
$$
\ti Z(\lambda)=\int_0^\infty dt~e^{-\lambda t} Z(t),~~~\Re \lambda >0.
$$
Note that, by the bound (\ref{linb}), $\ti Z(\lambda )$ is an analytic
function in complex right half plane $\C_+=\{\lambda \in\C:~\Re \lambda >0\}$
with values in the Hilbert space ${\cal E}$.
We have
\beqn\la{Law}
\ba{lll}
\ti \Psi(\lambda ) & = & (-\Delta+\lambda ^2)^{-1}\,\big(\lambda
\Psi^0 + \Pi^0\big) +
(-\Delta+\lambda ^2)^{-1}\big(\nabla\rho (x)\cdot\ti Q (\lambda )\big), \\
\ti Q(\lambda ) & = & (a(\lambda ))^{-1}\Big(
\lambda Q^0 + P^0 + \int d^3y \,\Big[(-\Delta+\lambda ^2)^{-1}\,(\lambda
\Psi^0 + \Pi^0)\Big](y)
\cdot\nabla\rho(y)\,
\Big)
\ea
\eeqn
provided $\Re \lambda >0$. Here $a(\lambda )$ is a matrix
$a_{ij}(\lambda )=\alpha(\lambda )\delta_{ij}$ with
$i,j=1,2,3$,
\beqn\label{alphadef}
a_{ij}(\lambda )&&=
(\lambda ^2+ \omega^2_0+\omega^2_1)\delta_{ij}
+((-\Delta+\lambda ^2)^{-1}\pa_i\rho,\pa_j\rho)\nonumber\\
&&=\Big(\lambda ^2\Big[1+
\fr 13((-\Delta+\lambda ^2)^{-1}\rho,\rho)\Big]
+ \omega^2_0\Big)\delta_{ij}=\alpha(\lambda )\delta_{ij}
{\rm ~~~~~~for~~}\Re \lambda >0
\eeqn
due to (\re{O1}).
In order to estimate the decay of $Z(t)$ we first have to investigate the zeros of
$\alpha(\lambda )$.
Denote
$ \C_{\beta} = \{\lambda \in\C: \Re \lambda >\beta\}$
for $\beta\in\R$.
\begin{lemma}\label{alphaeig} Let $\alpha$ be defined by (\ref{alphadef})
for $\lambda \in\C$ with $\Re \lambda >0$ and let $(C)$ hold.
Then
\begin{itemize}\item[(i)] $\alpha$ has an analytic continuation to $\C$.
\item[(ii)] For every $\beta<0$ there exists $d_\beta>0$ such
that $\displaystyle |\alpha (\lambda )| \ge {|\lambda |}^2/2$
for $\lambda \in \C_{\beta}$
with $|\lambda | \ge d_\beta$\,.
\item[(iii)]
If the Wiener condition $(W)$ holds, then
there exists $\gamma >0$ such that $\alpha(\lambda)\not= 0$
%%$\displaystyle|\alpha (\lambda )|\ge \alpha_0 >0$
for $\lambda \in {\C_{-\gamma}}$.
\end{itemize}
\end{lemma}
{\bf Proof} {\it ad (i)} (\re{alphadef}) and
$(-\Delta+\lambda ^2)\Big(e^{-\lambda |x|}/(4\pi |x|)\Big)=\delta(x)$ imply
\be\label{analyt}
\alpha (\lambda ) = \lambda ^2
\left(1+{1\over 3}\int\int
d^{\,3}x\,d^{\,3}x'\,
\frac{\rho(x)\rho(x')}{4\pi|x-x'|}\,e^{- \lambda |x-x'|}
\right)+\omega^2_0.
\ee
The right-hand side of this expression is defined and analytic in all of
$\C$ and is thus an analytic continuation of $\alpha$.\\
{\it ad (ii)} The assertion follows from (\ref{analyt}),
because
\be\label{beta}
I(\lambda )=
\int\int d^{\,3}x\,d^{\,3}x'\,
\frac{\rho(x)\rho(x')}{4\pi|x-x'|}\,e^{- \lambda |x-x'|}
\to 0 {\rm ~~~~as~~}|\lambda |\to\infty ~~\mbox{with}~~\lambda \in\C_{\beta}.
\ee
{\it ad (iii)}
Since $\alpha(\lambda)\not= 0$ for $\Re\lambda>0$ due to
(\ref{Law}) and (\ref{linb}),
we only have to exclude that
$\alpha(\lambda )$ has zeros on the imaginary axis.
For this the Wiener condition
$(W)$ will be needed. For $\lambda =iy,~~y\in\R$,
we obtain from (\ref{alphadef})
\be\la{iy}
\alpha(iy)=-y^2
\Big( 1+\fr 1{3(2\pi)^3}
\int d^3k\fr{|\hat\rho(k)|^2}{k^2-(y-i0)^2} \Big)+\omega_0^2.
\ee
Clearly, $\alpha(0)=\omega_0^2>0$.
Let $y\not=0$ and define
$$
D(y)=\int d^3k \fr{|\hat\rho(k)|^2}{k^2-(y-i0)^2}=
\int_0^\infty d\nu\fr{g(\nu)}{\nu^2-(y-i0)^2}=
\int_0^\infty d\nu\fr{g(\nu)}{(\nu-y+i0)(\nu+y-i0)}
$$
Because of $(W)$
$$
g(\nu)=\int_{|k|=\nu} d^2k \,|\hat\rho(k)|^2
>0{\rm ~~~~for~~}\nu>0.
$$
Furthermore, because of $(C)$,
$g(\nu)\in C^\infty ([0,\infty))$ and
$\max_{0<\nu<\infty}|g(\nu)|(1+\nu^p)<\infty$ for every $p>0$.
Therefore by the Plemelj formula \ci{Gel}
$$
D(y)=
-i\pi \fr{g(|y|)}{2y}+\mbox{pv}\int_0^\infty d\nu \fr {g(\nu)}{(\nu-y)(\nu+y)}
$$
and $\Im D(y)\not= 0$.
Then (\ref{iy}) implies that $\Im\alpha(iy)\not=0$. {\hfill$\Box$} \medskip\\
{\it Remarks}. (i) For $\rho=0$ the zeros of $\alpha$ are at
$\pm i\omega_0$. If $\rho=\varepsilon\rho_0$ with
some fixed $\rho_0$ satisfying $(C)$, we can follow
perturbatively how these zeros move to the left of the imaginary axis as the
coupling strength $\varepsilon$ is turned on.\\
(ii) If $(W)$ does not hold, then $g$ and thus the imaginary part of $D$
vanish at isolated points. To make also the real part of $D$ vanish at some
of these points we need to vary at most ``one parameter'' in $\rho$. In this
sense, for
$\alpha(\lambda )$ to have some zeros on the imaginary axis is a codimension one
property in the space of $\rho$'s.\\
\begin{definition}\la{Dg*}
For a stable $q^*\in S$ denote by $\alpha(\lambda)$ the function
defined in (\ref{analyt}) and (\ref{hesse}),
by $\Lambda(q^*)$ the set of zeros $\{\lambda\in\C: \alpha(\lambda)=0\}$,
and by
\be\la{g*}
\gamma^*=\gamma(q^*)=-\max\limits_{\lambda\in\Lambda(q^*)} \Re \lambda.
\ee
\end{definition}
To prove the exponential decay of $Y(t)$, we need the following
lemma about the inverse Laplace transform of $1/\alpha(\lambda )$ given
for arbitrary $\gamma<\gamma^*$
by
$$
h(t) = \fr 1{2\pi}
\int_{\Re \lambda =-\gamma}d\lambda~\frac{e^{\lambda t}}{\alpha (\lambda )}
{\rm ~~~for~~}t>0.
$$
\begin{lemma}\label{habfall} For
$j=0,1,2,\dots$ and every $\gamma<\gamma^*$
\be\la{hj}
|h^{(j)}(t)|
={\cal O}(e^{-\gamma t})
\quad\mbox{as}\quad t\to\infty~.
\ee
\end{lemma}
{\bf Proof} By Lemma \ref{alphaeig} (ii) and (iii), the bound on
$h$ follows. To prove the same bound for the derivatives $h^{(j)}(t)$,
we consider corresponding Laplace transforms $\lambda ^j/\alpha(\lambda )$.
For large $N=1,2,3,\dots$ we are going to establish that
\be\la{Lahj}
|(\fr d{d\lambda })^N\fr{\lambda ^j}{\alpha(\lambda )}|
\leq \fr {C_N(\gamma)}{(1+|\lambda |)^{\ov N}}
{\rm ~~~for~~}\Re\lambda=-\gamma,
\ee
where $\ov N\to\infty$ as $N\to\infty$. This implies then
$$
\max_{0\leq t<\infty}|t^N e^{\gamma t}h^{(j)}(t)|<\infty
$$
for large $N$ and (\ref{hj}) follows.
To prove (\ref{Lahj}), we note that
$\alpha(\lambda )=\lambda ^2(1+I(\lambda )/3+\omega_0^2/\lambda ^2)$.
Hence $1/(\lambda ^2\alpha(\lambda ))$ can be expanded due to (\ref{beta}) as a power
series in $I(\lambda )$ and
$1/\lambda $ for
$\Re\lambda=-\gamma$
with large $|\lambda |$. Therefore
it suffices to establish the bounds of type
(\ref{Lahj})
for $I(\lambda )$,
$$
|(\fr d{d\lambda })^N I(\lambda )|
\leq \fr {C_N}{(1+|\lambda |)^{\ov N}}
{\rm ~~~for~~}\Re\lambda=-\gamma,
$$
where $\ov N\to\infty$ as $N\to\infty$.
These estimates follow from the representation
(\ref{beta}) of the function $I(\lambda )$.
{\hfill$\Box$} \medskip\\
{\bf Proof of Proposition 7.1}
Using Lemma 7.3 we estimate the decay of $Z(t)$ in the
local energy seminorms.
From (\ref{Law}) we obtain
\begin{eqnarray}\la{dec}
\Psi(x, t)& = &\Psi_1(x, t) + \int^t_0 ds\,h(t-s) \Psi_2(x, s)\nonumber\\
&&+ \int^t_0 ds\,h(t-s)\int^{s}_0 ds^\prime\,(\nabla\rho, \Psi_1(s-s^\prime))
\cdot \Psi_3 (x, s^\prime),\nonumber\\
Q(t)& = &\dot{h}(t)Q^0 + h(t)P^0 + \int^t_0 ds\,h(t-s)
(\Psi_1(s),\nabla\rho).
\end{eqnarray}
Here $\Psi_1$ is the solution of the homogeneous wave equation with initial
data $(\Psi^0, \Pi^0)$, $\Psi_2$ the solution with initial data
$(Q^0\cdot\nabla\rho, P^0\cdot\nabla\rho)$, and $\Psi_3$ the solution
with initial data $(0, \nabla\rho)$. Due to the strong Huygens principle
${\br \Psi_1(\cdot, t)\br}_R$, ${\br \Psi_2 (\cdot, t)\br}_R$,
${\br \Psi_3(\cdot, t)\br}_R$ and $(\nabla\rho, \Psi_1(t))$
vanish for sufficiently large $t$.
Therefore,
the claimed
estimate (\ref{Elin}) follows from energy estimate
(\ref{linb}) for the solutions
$\Psi_{1,2,3}$ and from bounds (\ref{hj}) with $j=0,1,2$.
{\hfill$\Box$} \medskip\\
%%%%%%%%%%%%%%%%%%%%%%%%% Section 8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem \re{B}}
In essence we follow \cite{KV}. We assume (\ref{hesse}) and a stable
$q^*=0$.
Let us note that (\ref{L}) implies that $X(0)=Y(0)-Y_0$ has a compact support,
i.e. the property (\ref{CS}) holds
for $Z^0=X(0)$. So
we have to prove for the solution $X(t)=Y(t)-Y_0\in C(\R,{\cal E})$
of (\ref{AB}) with initial values of compact support and with small
$\Vert X(0)\Vert_{\cal E}$,
\be\la{EX}
\Vert X(t)\Vert_R\leq C_R e^{-\gamma t}{\rm ~~~for~~}t\geq 0.
\ee
Let us note that
$B(X(t))\in C(\R,{\cal E})$ for such $X(t)$ because of (\ref{BXest}).
Therefore the integrated version of (\ref{AB}) holds
\begin{equation}\label{kolb}
X(t)=U(t)X(0)+\int^t_0 ds\,U(t-s) B(X(s)).
\end{equation}
Let us fix an arbitrary number $b>0$.
We restrict ourselves to $0\leq t0$
for small $\Vert X(0)\Vert_{\cal E}$.
Then
Lemma \ref{Best} (ii) implies for all $R>0$
\[ {\|B (X(t))\|}_R \le
C_b\,{\|X(t)\|}^2_{R_\rho + b}
{\rm ~~~ for~~}t< t_b.
\]
Hence, Proposition \ref{lElin},
(\ref{Bloc}), and (\ref{kolb}) imply the integral inequality
\be\la{iin}
{\|X(t)\|}_R\le C_R(M,b)\,\bigg(
e^{-\gamma t}{\|X(0)\|}_{\cal E} + \int^t_0 ds \,e^{-\gamma (t-s)}
{\|X(s)\|}^2_{R_\rho + b}\bigg)
{\rm ~~~ for~~}t< t_b.
\ee
Therefore, it suffices to prove that for sufficiently small
$\delta=\Vert X(0)\Vert_{\cal E}$
(a) $t_b=\infty$, i.e. $|q(t)|\leq b$ for all $t\geq 0$, and
(b) (\ref{EX}) with $R=R(b)=R_\rho + b$.
For this purpose we denote
$n (t)=e^{\gamma t}\|X(t)\|_{R(b)}$. Then (\ref{iin})
with $R=R(b)$ implies that
\be\la{iin1}
n (t) \le C(M,b)\,\bigg(\delta +
\int^t_0 ds\, e^{-\gamma s}\,n^2 (s)\bigg)
{\rm ~~~ for~~}t< t_b.
\ee
Further we denote $m(t)=\max_{0\leq s\leq t}n(s)$ for $t\geq 0$. Then
(\ref{iin1}) implies the quadratic inequality
\be\la{qin}
m(t)\leq
C(M,b,\gamma)\,\bigg(\delta + m^2 (t)\bigg)
{\rm ~~~ for~~}t< t_b.
\ee
Let us choose $\delta >0$ so small
that the quadratic equation
$$
m=C(M,b,\gamma)\,\bigg(\delta + m^2\bigg)
$$
has two positive roots $m_13R_\rho +1$.
\begin{lemma}\label{l9.1}
For arbitrary $\delta>0$
there exist $t_{*}>0$ and a solution
$$
Y_{*}(t)=
(\phi_{*}(x,t),q_{*}(t),\pi_{*}(x,t),p_{*}(t))
\-\in
C([t_{*},\infty),~{\cal E})
$$
to the system (\ref{2}) such that \\
(i) $Y_{*}(t)$ coincides with $Y (t)$ in some future cone,
\beqn\la{9.1}
\ba{rll}
\phi_{*}(x,t)&=\phi(x,t)& ~~~for~~|x|t_{*}.
\ea
\eeqn
(ii) $Y_{*}(t_{*})$ admits a decomposition
$Y_{*}(t_{*})=Y_0+W^0+Z^0$, where $Z^0=(\Psi^0~,Q^0~,\Pi^0~,P^0)$
satisfies
\beqn
\Psi^0(x)=\Pi^0(x)=0~~for~~|x|\geq M_*~, \la{CS'}\\
\Vert Z^0\Vert_{\cal E}\leq \delta.~~~~~~~~~~~~~~~\la{9.3}
\eeqn
$W^0$ satisfies for every $R>0$ and every $\gamma<\gamma(q^*)$
\be\la{A1}
\Vert U(\tau)W^0\Vert_R\leq C_R
\Big(
(|t_*+\tau|+1)^{-\sigma}+e^{-\gamma \tau}(|t_*|+1)^{-\sigma}
\Big)
~~~for~~\tau>0,
\ee
where $C_R=C_R(\gamma)$ does not depend on $\de$.
\end{lemma}
This lemma leads to
\medskip\\
{\bf Proof of Theorem \re{C}}
(\ref{EE}) follows from (\ref{9.1}) provided we
establish that for every $R,\ve>0$
\be\la{A2}
\Vert Y_*(t)-Y_0\Vert_R={\cal O}(t^{-\sigma+\ve})
{\rm ~~~as~~}t\to\infty.
\ee
We generalise the integral
inequality method used in the proof of Theorem \re{B}
of the previous Section.
We set $X(\tau)=Y_*(t_*+\tau)-Y_0$. Then $X(0)=W^0+Z^0$ and
(\ref{kolb}) reads
\begin{equation}\label{kolb'}
X(\tau)=U(\tau)W^0+U(\tau)Z^0+\int^\tau_0 ds\,U(\tau-s) B(X(s)),
\end{equation}
since $U(\tau)$ is a linear operator.
(\ref{A1}) implies an integral inequality similar to (\ref{iin}),
\beqn\la{iin'}
{\|X(\tau)\|}_R& \le &C_R\,\bigg((|t_*+\tau|+1)^{-\sigma}
+e^{- \gamma \tau}(|t_*|+1)^{-\sigma}\nonumber\\
&& +e^{-\gamma \tau}{\|Z^0\|}_{\cal E} + \int^\tau_0 ds \,e^{-\gamma (\tau-s)}
{\|X(s)\|}^2_{R_\rho + b}\bigg){\rm ~~~~for~~}\tau< \tau_b.
\eeqn
Here
$C_R=C_R(M_*,b)$,
$b>0$ is an arbitrary fixed number and
$\tau_b=\sup\{\tau\geq0:~|q(t_*+\tau)\leq b\}>0$
for sufficiently large $t_*$ due to (\ref{10}).
Denoting $\mu(\tau)=(|\tau|+1)^{-\sigma+\ve}$ and
$n(\tau)=\Vert X(\tau)\Vert_{R(b)}/\mu(\tau)$
with $R(b)=R_\rho +b$, we rewrite (\ref{iin'}) as
\be\la{iin1'}
n(\tau)\le C^1_R\,\bigg(
\fr{(|t_*+\tau|+1)^{-\sigma}} {(|\tau|+1)^{-\sigma+\ve}}+
\fr{ e^{- \gamma \tau}(|t_*|+1)^{-\sigma}} {(|\tau|+1)^{-\sigma+\ve}}+
\de + \int^\tau_0 ds \,\fr{\mu(\tau-s)\mu^2(s)}{\mu(\tau)} n^2(s)\bigg)
,~~\tau< \tau_b.
\ee
We use now an extension of the basic inequality in \ci{KV}
\be\la{CI}
\sup_{\tau>0}\int_0^\tau ds\fr{\mu(\tau-s)\mu^2(s)}{\mu(\tau)}\leq B(\sigma-\ve)<\infty
{\rm ~~~for~~}\sigma-\ve>\fr 12.
\ee
A further remark is that for every $\ve, \gamma>0$
\be\la{t*}
\sup_{\tau>0}
\left(
{
\fr{(|t_*+\tau|+1)^{-\sigma}}{(|\tau|+1)^{-\sigma+\ve}}+
\fr{ e^{- \gamma \tau}(|t_*|+1)^{-\sigma}} {(|\tau|+1)^{-\sigma+\ve}}
}\right)
\to 0 {\rm ~~~as~~}t_*\to\infty.
\ee
Finally, let us
denote by $m(\tau)=\max_{0\leq s\leq \tau} n(s)$ and let us choose
$0<\ve<\sigma-1/2$, $0< \gamma<\gamma(q^*)$, and $t_*$
sufficiently large. Then
(\ref{iin1'})-(\ref{t*}) lead to a quadratic
inequality of
the form (\ref{qin}) and the proof can be continued as in Section 8.
{\hfill $\Box$}
\medskip\\
{\bf Proof of Lemma 9.1}
The convergence (\ref{10}) with $q^*=0$
implies that for every $\varepsilon>0$ there exist $t_{\varepsilon}$
such that
\be\la{9.6}
|q(t)|+|\dot q(t)|<\varepsilon {\rm ~~~for~~} t>t_{\varepsilon}.
\ee
Let us denote
\be\la{9.6'}
t_{0,\varepsilon}=t_{\varepsilon}+R_\rho,
~~t_{1,\varepsilon}=t_{0,\varepsilon}+1,
~~t_{2,\varepsilon}=t_{1,\varepsilon}+\varepsilon+R_\rho,
~~t_{3,\varepsilon}=t_{2,\varepsilon}+\varepsilon+R_\rho.
\ee
Then there exist a function $q_\varepsilon(\cdot)\in C^1(\R)$ such that
\beqn\la{9.7}
q_\varepsilon(t)=
\left\{
\ba{rl}
q(t),&t>t_{1,\varepsilon},\\
0,&t0,
\ee
where
\be\la{5.12}
\phi_{r,\ve}(x,t)=
-\int~\fr{d^3y\theta(t-|x-y|)}{4\pi|x-y|}\rho(y-q_\ve(t-|x-y|))~.
\ee
Then $\phi_\ve(x,t)$ is a solution to the wave equation
\be\la{5.13}
\ddot \phi_\ve(x,t)=\Delta \phi_\ve(x,t)-
\rho(x-q_\ve(t)){\rm ~~for~~}t>0~.
\ee
By (\re{5.12}), (\re{retp}), and (\re{9.7}) we have
\beqn
\phi_{r,\ve}(x,t)=
\phi_r(x,t)&~~{\rm for~~}|x|t-t_\varepsilon.~~~\la{9.12}
\eeqn
Moreover, $\phi_{r,\ve}(\cdot,\cdot)\in C^1(\R^4)$ and (\ref{9.7})
implies
\be\la{9.13}
\sup_{x\in\R^3,t\in\R}
(|\dot\phi_{r,\ve}(x,t)|+
|\nabla\phi_{r,\ve}(x,t)-\nabla\phi_0(x)|+
|\phi_{r,\ve}(x,t)-\phi_0(x)|)
={\cal O}(\ve).
\ee
Let us define
\beqn\la{9.14}
Y_{*}(t)=(\phi_\varepsilon(\cdot,t),q(t),
\dot\phi_\varepsilon(\cdot,t), p(t))
~~~\mbox{for}~~t>t_{3,\varepsilon}=t_{*},~~~~~~~~~\nonumber\\
W^0=(\phi_0(\cdot ,t_*),0,\dot\phi_0(\cdot ,t_*),0),~~~
Z^0=(\phi_{r,\ve}(\cdot ,t_*)-\phi_0(x), q(t_*),
\dot\phi_{r,\ve}(\cdot ,t_*), p(t_*)).
\eeqn
It is easy to check that $t_{*}$ and
$Y_{*}(t)$, $W^0$, $Z^0$
satisfy all requirements of Lemma 9.1, provided $\varepsilon>0$
be sufficiently small.
Firstly, $Y_{*}(t)$ is a solution of
the system (\ref{2}) for $t>t_*$.
Indeed, (\ref{5.14}) and (\ref{9.6'}) imply
$$
\phi_\varepsilon(x,t)= \phi(x,t){\rm ~~for~~}|x|<\varepsilon+R_\rho
{\rm ~~and~~}t>t_{3,\varepsilon}.
$$
Hence, $Y_{*}(t)$ together with $Y(t)$
is a solution to the system
(\ref{2})
in the region $|x|<\varepsilon+R_\rho$.
On the other hand, (\ref{9.6}) implies
$$
\rho(x-q(t))= 0{\rm ~~for~~}|x|>\varepsilon+R_\rho
{\rm ~~and~~}t>t_\varepsilon.
$$
Hence, $Y_{*}(t)$ satisfies the equations (\ref{2})
in the region $|x|>\varepsilon+R_\rho$
by (\ref{5.13}). In addition,\\
{\it ad (i)} (\ref{9.1}) follows from (\ref{5.14}) and (\ref{9.14}).\\
{\it ad (ii)} (\ref{CS'}) for $M_*=3R_\rho+2\varepsilon+1$
follows from (\ref{9.12}).
(\ref{9.3}) follows from (\ref{CS'}) and (\ref{9.13}).
We deduce
(\ref{A1}) from the bounds (\ref{Rdec}) and the representation (\ref{dec})
for the linearized dynamics $U(\tau)$.
Denote by
$U(\tau)W^0=(\Psi(x,\tau),Q(\tau),\Pi(x,\tau),P(\tau))$ and let us prove the
bounds of the type (\ref{A1})
for
$(\Psi(\tau),\dot\Psi(\tau))$, for instance. Let us rewrite the representation
(\ref{dec}) for $\Psi(x,\tau)$ in the form
$$
\Psi(x, \tau) = \Psi_1(x, \tau) + \int^\tau_0 ds\,h(\tau-s) \Psi_2(x, s)
+\Psi_{1,3}(x,\tau)
$$
and consider every term separately.
At first,
$(\Psi_1(x,\tau), \dot\Psi_1(x,\tau))=
w^0_\tau (\phi_0(\cdot,t_*),\dot\phi_0(\cdot,t_*))$
where $w^0_\tau$ is a dynamical group of the free wave equation.
On the other hand
$(\phi_0(\cdot,t_*), \dot\phi_0(\cdot,t_*))=w^0_{t_*}(\phi^0,\pi^0)$
due to the Kirchhoff formula (\ref{K}).
Therefore
$(\Psi_1(x,\tau), \dot\Psi_1(x,\tau))=w^0_{t_*+\tau}(\phi^0,\pi^0)$
and the bound $C_R(|t_*+\tau|+1)^{-\sigma}$ for
$\Vert\Psi_1(x,\tau)\Vert_R + \br \dot\Psi_1(x,\tau)\br_R$
follows from (\ref{Rdec}).
Secondly, $(\Psi_2(x,\tau), \dot\Psi_2(x,\tau))=
w^0_\tau (0,0)=0$.
Finally, $(\Psi_3(x,\tau), \dot\Psi_3(x,\tau))=
w^0_\tau (0,\nabla\rho(x))$ and then
$\Vert \Psi_3(x,\tau)\Vert_R=0$ for $|\tau|>R+R_\rho$.
Therefore the representation (\ref{dec})
of the term $\Psi_{1,3}(x,\tau)$
with $\Psi_3(x,s^\prime)$,
together with (\ref{hj}) and bounds for
$\Psi_1$,
imply the bound
$$
\Vert\Psi_{1,3}(x,\tau)\Vert_R\leq
C_R\int_0^\tau ds e^{-\ti\gamma(\tau-s)}
\int_0^{R+R_\rho}ds^\prime(|t_*+s-s^\prime|+1)^{-\sigma}~
$$
with $\gamma<\ti\gamma<\gamma(q^*)$.
{\hfill$\Box$}
\medskip\\
\noindent {\bf Acknowledgement.} A.K. thanks Yu.E.Egorov for pointing out the
argument in the proof to Lemma \ref{attract}. H.S. thanks M.Kiessling
for comments on a previous version of this paper.
%%%%%%%%%%%%%%%%%%%%% APPENDIX %%%%%%%%%%%%%%%%%%
\section{Appendix: Densities of Wiener type}
It is not completely evident that $(C)$ and $(W)$ can be satisfied
simultaneously. To construct a generic example fix a real
$\varphi\in C_0^{\infty}(\R)$ with $\varphi(s)\not\equiv 0$. Then
$\hat\varphi$ may be extended as an analytic function to the
whole complex plane and there exists an $\alpha\in\R$ such that
$\hat\varphi(\xi+i\alpha)\not= 0$ for all $\xi\in\R$.
By replacing $\varphi(s)$ with $\varphi(s)\exp(\alpha s)$ we may assume
that $\alpha=0$. Clearly $\phi(x_1)\phi(x_2)\phi(x_3)$ satisfies $(W)$
and $(C)$ except for rotation invariance. Since by rotational averaging we
could pick up a zero, we first let $\rho_1=\varphi* \psi$ with
$\psi(s)=\varphi(-s)$. Then again $\rho_1\in C_0^{\infty}(\R)$ and
$\hat\rho_1(\xi)={|\hat\varphi(\xi)|}^2>0$ for all $\xi\in\R$.
Let $\rho$ be the rotational average of $\rho_1(x_1)\rho_1(x_2)\rho_1(x_3)$.
Then $\rho$ satisfies both $(C)$ and $(W)$.
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\end{document}