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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 65, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/65\hfil Energy decay for solutions]
{Energy decay for solutions to semilinear systems of
elastic waves in  exterior domains}
\author[M. V. Ferreira, G. P. Menzala\hfil EJDE-2006/65\hfilneg]
{Marcio V. Ferreira, Gustavo P. Menzala}  % in alphabetical order

\address{Marcio V. Ferreira \newline
Centro Universit\'ario Franciscano, Rua dos Andradas 1614,
Santa Maria, CEP 97010-032, RS, Brazil}
\email{ferreira@unifra.br}

\address{Gustavo Perla Menzala \newline
National Laboratory of Scientific Computation LNCC/MCT,
Av. Getulio Vargas 333, Petropolis, CEP 25651-070, RJ, Brasil \newline
and IM-UFRJ, P.O. Box 68530, RJ, Brazil}
\email{perla@lncc.br}

\date{}
\thanks{Submitted March 20, 2006. Published May 22, 2006.}
\subjclass[2000]{35Q99, 35L99}
\keywords{Uniform stabilization; exterior domain; system of elastic waves;
\hfill\break\indent  semilinear problem}

\begin{abstract}
 We consider the dynamical system of elasticity in the exterior
 of a bounded open domain in 3-D with smooth boundary.
 We prove that under the effect of ``weak" dissipation, the total
 energy decays at a uniform rate as $t \to +\infty$, provided the
 initial data is ``small" at infinity. No assumptions on the
 geometry of the obstacle are required. The results are then
 applied to a semilinear problem proving global existence and
 decay for small initial data.
\end{abstract}

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

\section{Introduction}

We study the uniform stabilization of the
solutions of a hyperbolic system of equations in an exterior
domain,  as $t \to +\infty$. A classical  example of this class
is the system of elastic waves. Let us describe the model:
Let $\mathcal{O}$ be an
open bounded region of $\mathbb{R}^3$ with smooth boundary and
$\Omega = \mathbb{R}^3\setminus\overline{\mathcal{O}}$. We consider the system
\begin{equation}
\begin{gathered}
 u_{tt} - \sum_{i,j=1}^3 \frac{\partial}{\partial x_i}
 \big(A_{ij} \frac{\partial u}{\partial x_j}\big) + u_t = f(u_t) \quad
 \text{in } \Omega \times \mathbb{R}\\
 u(x,0) = u_0(x), \quad  u_t(x,0) = u_1(x) \quad \text{in } \Omega\\
 u=0 \quad\text{on }\partial\Omega \times \mathbb{R}
\end{gathered}  \label{e1.1}
\end{equation}
Here $x = (x_1,x_2,x_3) \in \Omega$, $t$ is the time variable,
$u(x,t) = \big(u_1(x,t),u_2(x,t),u_3(x,t)\big)$ denotes the displacement
vector, $A_{ij} = [C_{kh}^{ij}]$ are $3\times3$ symmetric matrices
and $f = (f_1,f_2,f_3)$ is a nonhomogeneous vector-valued function.
Both $A_{ij}$ and $f$ will satisfy suitable assumptions.
Associated to the initial boundary valued problem \eqref{e1.1}
we have the total energy
\begin{equation}
E(t) = \frac 12 \int_\Omega \big\{|u_t|^2 + \sum_{i,j=1}^3 A_{ij}
\frac{\partial u}{\partial x_j} \cdot \frac{\partial u}{\partial x_i}\big\} dx
\label{e1.2}
\end{equation}
where $|u_t|^2 = u_t \cdot u_t = \sum_{j=1}^3
|\frac{\partial}{\partial t} u_j|^2$ and the dot  $\cdot$  denotes
the usual inner product in $\mathbb{R}^3$. Let $u$ be the solution of
problem \eqref{e1.1} in a suitable function space and assume for a moment
that $f \equiv 0$. Then, a formal calculation give us that the derivative
of $E(t)$:
\begin{equation}
\frac{d}{dt} E(t) = -\int_\Omega |u_t|^2\,dx \le 0. \label{e1.3}
\end{equation}
Thus, we may ask: Does $E(t)$ decays at a uniform rate as $t \to
+\infty$? Furthermore, in case the answer is  affirmative then we
can ask if the same result would still hold for a class of
functions $f$ and initial data $(u_0,u_1)$ satisfying suitable
assumptions. Both questions above are by now not very difficult to
answer in case $\Omega$ is a bounded domain (see for instance
Racke \cite{r1} and the references therein). In our case, since
$\Omega$ is an exterior domain, the uniform stabilization requires
a more detailed discussion which is our main objective in this
article. There is a large literature concerning the decay of
solutions of hyperbolic problems in exterior domains. In a
pioneering work, Morawetz \cite{m1,m2} studied the asymptotic
behavior of the local energy for the scalar wave equation in
exterior domains. Assuming geometric conditions on the obstacle
and initial data with compact support she obtained uniform rates
of decay. B. Kapitonov got similar results for the system of
elastic waves and the Maxwell equations, Zuazua \cite{z1}, Nakao
\cite{n2} and Ikehata \cite{i3} obtained also stabilization
results for scalar wave equations with localized damping (being
effective only near ``infinity"). As far as we know the results we
present in this article for system \eqref{e1.1} are the first of
the kind for the system of elasticity. We do not assume geometric
conditions on the obstacle nor special restrictions on the
Lam\'{e}'s coefficients in the isotropic case. Our strategy relies
on recent work due to  Ikehata \cite{i1} for the scalar wave
equation adapted conveniently to system \eqref{e1.1}.

 Let us make precise our assumptions on the matrices $A_{ij}$ and
 the nonlinearity $f$ in \eqref{e1.1}:
\begin{itemize}

\item[(H1)]   (a) Given a set of real numbers $\{a_{ijkh}\}$ with
$i,j,k,h \in \{1,2,3\}$ satisfying the symmetric properties
$a_{ijkh} = a_{jikh} = a_{khij}$,
we consider
$$
C_{kh}^{ij} = (1-\delta_{ih} \delta_{jk})a_{ikjh} + \delta_{ik} \delta_{jh} a_{ihjk}
$$
with $\delta_{\ell k} = \begin{cases} 1 \text{ if } \ell=k\\ 0
\text{ if } \ell \ne k\end{cases}$ and ``build" the $3\times3$
matrices $A_{ij} = [C_{kh}^{ij}]$.
\\
(b) We assume that there exist a constant $C_0 > 0$ such that
\begin{equation}
\sum_{i,j=1}^3 A_{ij} v_j \cdot v_i \ge C_0 \sum_{i=1}^3 |v_i|^2 \label{e1.4}
\end{equation}
for any vector $v_i = (v_i^1, v_i^2, v_i^3) \in \mathbb{R}^3$ where
$|v_i|^2 = v_i \cdot v_i$.


\item[(H2)]   Let $f = (f_1,f_2,f_3)$ with
$f_j\colon \mathbb{R}^3 \to \mathbb{R}$ satisfying the following assumptions:
Each $f_j \in C^2(\mathbb{R}^3)$ and
\\
(a) $|f(y)| \le C_1|y|^p$\quad for every $ y \in \mathbb{R}^3$
\\
(b) $|\nabla f(y)| \le C_2|y|^{p-1}$\quad for every $ y \in
\mathbb{R}^3$
\\
(c) $\sum_{i,j=1}^3 |\nabla \frac{\partial f_i(y)}{\partial
y_j}| \le C_3|y|^{p-2}$\quad for every $ y \in \mathbb{R}^3$
\\
where $C_j$ are positive constants $(1 \le j \le 3)$,
$\frac 73 < p \le 3$ and $|\nabla f(y)|^2 = \sum_{i=1}^3 |\nabla f_i(y)|^2$.

\end{itemize}

\begin{remark} \label{rmk1} \rm
In the simplest case, that is, when the medium is isotropic, the constants
$a_{ijkh}$ are
$$
a_{ijkh} = \lambda \delta_{ij} \delta_{kh} + \mu\big(\delta_{ik} \delta_{jh}
 + \delta_{ih} \delta_{jk}\big)
$$
where $\lambda$ and $\mu$ are Lam\'{e}'s constants $(\mu > 0,
\lambda+\mu > 0)$. Furthermore, \eqref{e1.4} holds with $C_0 =
\mu> 0$ and $\sum_{i,j=1}^3 \frac{\partial}{\partial x_i}
\big(A_{ij} \frac{\partial u}{\partial x_j}\big)$ reduces to $\mu
\Delta u + (\lambda+\mu)\nabla \mathop{\rm div} u$.
\end{remark}


\begin{remark} \label{rmk2} \rm
Due to the symmetry conditions on the numbers $a_{ijkh}$ it follows that
 $A_{ij}^* = A_{ji}$ .
\end{remark}


\section{The linear case}


In this section we consider the linear problem
\begin{equation}
\begin{gathered}
 u_{tt} - \sum_{i,j=1}^3 \frac{\partial}{\partial x_i}
 \Big(A_{ij} \frac{\partial u}{\partial x_j}\Big) + u_t = 0 \quad
 \text{in } \Omega \times \mathbb{R}\\
 u(x,0) = u_0(x), \quad  u_t(x,0) = u_1(x) \quad\text{in } \Omega\\
 u=0 \quad\text{on } \partial\Omega \times \mathbb{R}
\end{gathered}  \label{e2.1}
\end{equation}
Using standard semigroup theory we can easily prove the following result.


\begin{theorem} \label{thm2.1}
 Let $(u_0,u_1) \in [H_0^1(\Omega)]^3 \times [L^2(\Omega)]^3$ and $A_{ij}$
satisfy assumption (H1). Then, there exist a unique (weak)
solution $u$ of problem \eqref{e2.1} such that $u \in C\big(\mathbb{R};
[H_0^1(\Omega)]^3\big) \cap C^1\big(\mathbb{R};
[L^2(\Omega)]^3\big)$. If $(u_0,u_1) \in [H^2(\Omega) \cap
H_0^1(\Omega)]^3 \times [H_0^1(\Omega)]^3$, then, there
exist a unique (strong) solution $u$ of problem \eqref{e2.1} such that
$$
u \in C\big(\mathbb{R}; [H^2(\Omega) \cap H_0^1(\Omega)]^3\big)
\cap C^1\big(\mathbb{R}; [H_0^1(\Omega)]^3\big) \cap C^2
\big(\mathbb{R}; [L^2(\Omega)]^3\big).
$$
\end{theorem}

Here $H^m(\Omega)$ denotes the usual Sobolev space of order $m$ in
$\Omega$ and $H_0^1(\Omega) =\linebreak \big\{u \in H^1(\Omega),
u\big|_{\partial\Omega} = 0\big\}$. Now, we want to devote our attention
to the asymptotic behavior of the total energy $E(t)$ given by
\eqref{e1.2}. Our result in this case is as follows.


\begin{theorem} \label{thm2.2}
 Let $(u_0,u_1) \in [H_0^1(\Omega)]^3 \times [L^2(\Omega)]^3$ and
assume that the initial data satisfy the condition
\begin{equation}
\int_\Omega |x|^2 |u_0+u_1|^2\,dx < +\infty. \label{e2.2}
\end{equation}
Then, there exist a positive constant $C$ such that
\begin{gather*}
E(t) \le C I_0 \big(1+|t|\big)^2 \quad \text{for every }  t \in \mathbb{R}, \\
\int_\Omega |u(x,t)|^2\,dx \le C I_0\big(1+|t|\big)^{-1} \quad
\text{for every } t\in\mathbb{R}
\end{gather*}
where $I_0 = \|u_0\|^2_{[H^1(\Omega)]^3} + \|u_1\|^2 + \| | \cdot
| (u_0+u_1)\|^2$ and $\|g\|^2 = \sum_{j=1}^3 \int_\Omega
|g_j|^2\,dx$ whenever $g = (g_1,g_2,g_3) \in [L^2(\Omega)]^3$.
\end{theorem}

As far as we know,
results of this type for exterior domains are known only for
scalar wave equations and most of them require geometrical
conditions on the obstacle (like star-shaped condition). We need
some preliminary lemmas. Obviously, is sufficient to prove
Theorem \ref{thm2.2} for $t \ge 0$.


\begin{lemma} \label{lem2.3}
 Let $(u_0,u_1) \in [H^2(\Omega) \cap H_0^1(\Omega)]^3\times [H_0^1(\Omega)]^3$.
  Then, the solution of\eqref{e2.1} satisfies, for any $t \ge 0$,
\begin{gather}
E(t) + \int_0^t \int_\Omega |u_s(x,s)|^2\,dx\,ds = E(0),
\label{e2.3a} \\
\int_0^t \int_\Omega (1+s)|u_s(x,s)|^2\,dx\,ds + (1+t)E(t) = E(0)
+ \int_0^t E(s)\,ds ,\label{e2.3b} \\
\begin{aligned}
&\int_0^t \int_\Omega \sum_{i,j=1}^3 A_{ij} \frac{\partial u}{\partial x_j}
\cdot \frac{\partial u}{\partial x_i}\,dx\,ds
+ \frac 12 \int_\Omega |u(x,t)|^2\,ds \\
&= \frac 12 \|u_0\|^2 + \int_\Omega u_1 \cdot u_0\,dx
- \int_\Omega u_t \cdot u\,dx +
\int_0^t \int_\Omega |u_s|^2\,dx\,ds,
\end{aligned} \label{e2.3c}
\\
\begin{aligned}
&\int_0^t \int_\Omega (1+s) \sum_{i,j=1}^3 A_{ij} \frac{\partial u}{\partial x_j}
\cdot \frac{\partial u}{\partial x_i}\,dx\,ds + (1+t)
\int_\Omega |u|^2\,dx \\
&\le C + \frac 12 \int_0^t \int_\Omega |u|^2\,dx\,ds,
\end{aligned}\label{e2.3d}
\end{gather}
where $C$ is a positive constant which depends only on $E(0)$ and
$\|u_0\|$.
\end{lemma}


\begin{proof} Equality \eqref{e2.3a} follows directly from \eqref{e1.3} by
integration over $[0,t]$. Also, from \eqref{e1.3} it follows that
$$
(1+t) \frac{dE}{dt} = -\int_\Omega (1+t)|u_t|^2\,dx
$$
that is,
\begin{equation}
\int_\Omega (1+t)|u_t|^2\,dx = -\frac{d}{dt} \big\{(1+t)E(t)\big\} + E(t).
\end{equation}
Integration of this equality over $[0,t]$ proves \eqref{e2.3b}. Next,
we take the inner product in $[L^2(\Omega)]^3$ of system
\eqref{e2.1} with $u$ to obtain
\begin{equation}
\frac{d}{dt} \int_\Omega u_t \cdot u\,dx - \int_\Omega \sum_{i,j=1}^3
 \frac{\partial}{\partial x_i} \Big(A_{ij} \frac{\partial u}{\partial x_j}\Big)
 \cdot u\,dx + \frac 12 \frac{d}{dt} \int_\Omega |u|^2\,dx
 = \int_\Omega |u_t|^2\,dx. \label{e2.4}
\end{equation}
Using the divergence theorem and the boundary conditions we know that
$$
\int_\Omega \sum_{i,j=1}^3 \frac{\partial}{\partial x_i} \Big(A_{ij}
\frac{\partial u}{\partial x_j}\Big)\cdot u\,dx
= -\int_\Omega \sum_{i,j=1}^3 A_{ij} \frac{\partial u}{\partial x_i}
\cdot \frac{\partial u}{\partial x_j}\,dx.
$$
Substitution of the above identity into \eqref{e2.4} and integration over
$[0,t]$ proves \eqref{e2.3c}. To prove \eqref{e2.3d},  we proceed as above: Let us
take the inner product in $[L^2(\Omega)]^3$ of system \eqref{e2.1}
with $(1+t)u$ and use the divergence theorem to obtain
\begin{align*}
 &\frac 12 \frac{d}{dt} \int_\Omega t|u|^2\,dx
 + (1+t) \int_\Omega \sum_{i,j=1}^3 A_{ij} \frac{\partial u}{\partial x_j}
  \cdot \frac{\partial u}{\partial x_i} \,dx \\
 &= (1+t) \int_\Omega |u_t|^2\,dx + \frac 12 \int_\Omega |u|^2\,dx
 - \frac{d}{dt} \int_\Omega (1+t)u_t \cdot u\,dx.
\end{align*}
Integration of this equality over $[0,t]$ and using Holder's inequality
implies
\begin{equation}
\begin{aligned}
&\int_0^t\int_\Omega (1+s) \sum_{i,j=1}^3 A_{ij}
\frac{\partial u}{\partial x_j} \cdot \frac{\partial u}{\partial x_i}\,dx
\,ds + \frac t2 \int_\Omega |u|^2\,dx \\
& \le \int_\Omega u_1 \cdot u_0\,dx + \int_0^t \int_\Omega (1+s)|u_s|^2\,
dx\,ds + \frac 12 \int_0^t \int_\Omega |u|^2\,dx\,ds \\
&\quad + \frac{1+t}{4} \int_\Omega |u|^2\,dx + (1+t) \int_\Omega |u_t|^2
\,dx\,.
\end{aligned} \label{e2.6}
\end{equation}
From \eqref{e2.3b} and \eqref{e2.3c} in Lemma \ref{lem2.3},  we know that
\begin{equation}
\int_0^t \int_\Omega (1+s)|u_s|^2\,dx\,ds \le E(0) + \int_0^t E(s)\,ds \label{e2.7}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\int_0^t \int_\Omega \sum_{i,j=1}^3 A_{ij} \frac{\partial u}{\partial x_j}
\cdot \frac{\partial u}{\partial x_i}\,dx\,ds
+ \frac 14 \int_\Omega |u(x,t)|^2\,dx \\
& \le \frac 12 \|u_0\|^2 + \int_\Omega u_1 \cdot u_0\,dx
+ \int_\Omega |u_t|^2\,dx + E(0) - E(t).
\end{aligned} \label{e2.8}
\end{equation}
 From the above inequality, and using again \eqref{e2.3a}, we deduce that
\begin{equation}
2 \int_0^t E(s)\,ds + \frac 14 \int_\Omega |u|^2\,dx \le 2E(0)
+ \frac 12 \|u_0\|^2 + \int_\Omega u_1 \cdot u_0\,dx. \label{e2.9}
\end{equation}
Using the estimates \eqref{e2.7}, \eqref{e2.8} and \eqref{e2.9} we obtain
 from \eqref{e2.6} the inequality
 \begin{equation}
\begin{aligned}
&\int_0^t \int_\Omega (1+s) \sum_{i,j=1}^3 A_{ij} \frac{\partial u}{\partial x_j}
\cdot \frac{\partial u}{\partial x_i}\,dx\,ds
+ \frac{(1+t)}{4} \int_\Omega |u|^2\,dx \\
&\le 3 \int_\Omega u_1 \cdot u_0\,dx + 5E(0) + \|u_0\|^2
+ \frac 12 \int_0^t \int_\Omega |u|^2\,dx\,ds + 2(1+t)E(t).
\end{aligned} \label{e2.10}
\end{equation}
It remains to estimate $2(1+t)E(t)$. Observing that
$$
\frac{d}{dt} \big\{(1+t)E(t)\big\} = E(t)+(1+t) \frac{dE}{dt} \le E(t).
$$
Consequently
\[
2(1+t)E(t) \le 2E(0) + 2 \int_0^t E(s)\,ds \le 4E(0)
+ \frac 12 \|u_0\|^2 + \int_\Omega u_1 \cdot u_0\,dx.
\]
Substitution of this inequalit into \eqref{e2.10} completes
the proof
\end{proof}


\begin{lemma} \label{lem2.4}
 Let $(u_0,u_1) \in [H^2(\Omega) \cap H_0^1(\Omega)]^3 \times [H_0^1(\Omega)]^3$
 and
$(u_0,u_1)$ satisfy \eqref{e2.2}. Then the solution $u$ of problem \eqref{e2.1}
satisfies
$$
\int_\Omega |u|^2\,dx + \int_0^t \int_\Omega |u|^2\,dx\,ds \le \|u_0\|^2 + \frac{4}{C_0} \int_\Omega |x|^2 |u_0+u_1|^2\,dx
$$
where $C_0$ is the positive constant which appears in \eqref{e1.4}.
\end{lemma}


\begin{proof} First, let us observe that whenever $u_j \in H_0^1(\Omega)$
then Hardy's inequality states that
$$
\int_\Omega \frac{|u_j|^2}{|x|^2}\,dx \le 4 \int_\Omega |\nabla u_j|^2\,dx.
$$
Therefore, $u = (u_1,u_2,u_3)$ satisfies
\begin{equation}
\int_\Omega \frac{|u|^2}{|x|^2}\,dx \le 4 \int_\Omega \sum_{i,j=1}^3 |
\frac{\partial u_j}{\partial x_i}|^2 dx \le \frac{4}{C_0}
\int_\Omega \sum_{i,j=1}^3 A_{ij} \frac{\partial u}{\partial x_j}
\cdot \frac{\partial u}{\partial x_i}\,dx \label{e2.12}
\end{equation}
due to \eqref{e1.4}. Let $w(x,t) = \int_0^t u(x,s)\,ds$. It follows that
$w(x,t)$ satisfies the equation
\begin{equation}
\begin{gathered}
 w_{tt} - \sum_{i,j=1}^3 \frac{\partial}{\partial x_i} \Big(A_{ij}
 \frac{\partial w}{\partial x_j}\Big) + w_t = u_0+u_1\quad \text{in }
 \Omega \times \mathbb{R}^+\\
 w(x,0) = 0,  \quad w_t(x,0) = u_0(x) \quad \text{in } \Omega\\
 w=0 \quad\text{on } \partial\Omega \times \mathbb{R}^+
\end{gathered}  \label{e2.13}
\end{equation}
Let us consider the inner product in $[L^2(\Omega)]^3$ of the above
equation with $w_t$ and use the divergence theorem to obtain
\[
\frac 12 \frac{d}{dt} \int_\Omega \big\{|w_t|^2 + \sum_{i,j=1}^3 A_{ij}
\frac{\partial w}{\partial x_j} \cdot \frac{\partial w}{\partial x_i}\big\}
 dx + \int_\Omega |w_t|^2\,dx = \frac{d}{dt} \int_\Omega (u_0+u_1) \cdot w\,dx.
\]
Integrating this equality over $[0,t]$, using H\"older's
inequality and \eqref{e2.12} implies that
\begin{align*}
&\frac 12 \int_\Omega \big\{|w_t|^2
+ \sum_{i,j=1}^3 A_{ij} \frac{\partial w}{\partial x_j} \cdot
\frac{\partial w}{\partial x_i}\big\} dx + \int_0^t \int_\Omega |w_s|^2\,dx\,ds \\
&= \int_\Omega (u_0+u_1) \cdot w\,dx + \frac 12 \|u_0\|^2 \\
&\le \Big(\int_\Omega |x|^2 |u_0+u_1|^2\,dx\Big)^{1/2}
\Big(\int_\Omega \frac{|w|^2}{|x|^2}\,dx\Big)^{1/2} + \frac 12 \|u_0\|^2  \\
&\le \big(\frac{4}{C_0}\big)^{1/2}
\Big(\int_\Omega \sum_{i,j=1}^3 A_{ij} \frac{\partial w}{\partial x_j}
\cdot \frac{\partial w}{\partial x_i}\,dx\Big)^{1/2}
\Big(\int_\Omega |x|^2 |u_0+u_1|^2\,dx\Big)^{1/2} + \frac 12 \|u_0\|^2 \\
&\le \frac 14 \int_\Omega \sum_{i,j=1}^3 A_{ij} \frac{\partial w}{\partial x_j}
\cdot \frac{\partial w}{\partial x_i}\,dx + \frac{4}{C_0}
\int_\Omega |x|^2 |u_0+u_1|^2\,dx + \frac 12 \|u_0\|^2.
\end{align*}
This inequality proves Lemma \ref{lem2.4} because $w_t = u$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm2.2}]
It follows from Lemmas \ref{lem2.3} and \ref{lem2.4} that
\begin{equation}
\int_0^t \int_\Omega (1+s) \sum^3 A_{ij} \frac{\partial u}{\partial x_j}
\cdot \frac{\partial u}{\partial x_i}\,dx\,ds
+ (1+t) \int_\Omega |u|^2\,dx \le C I_0 \label{e2.16}
\end{equation}
for any $t \ge 0$. Observing that
$$
\frac{d}{dt} \big\{(1+t)^2 E(t)\big\} = 2(t+1)E(t) + (1+t)^2 \frac{dE}{dt} \le 2(1+t)E(t)
$$
it follows that
\begin{align*}
(1+t)^2 E(t) &\le E(0) + 2 \int_0^t (1+s)E(s)\,ds \\
&\le E(0) + C I_0 + \int_0^T \int_\Omega (1+s)|u_s|^2\,dx\,ds \\
&\le 2 E(0) + C I_0 + \int_0^t E(s)\,ds \le \widetilde C I_0\,.
\end{align*}
Here we used Lemma \ref{lem2.3} and \eqref{e2.9}, with  $\widetilde C$  a positive
constant. This completes the proof of Theorem \ref{thm2.2}.
\end{proof}

\begin{remark} \label{rmk3} \rm
 It is quite interesting to mention here
that a similar procedure to the one presented above was done by
the first author (M.F) in \cite{f1} for the Maxwell equations in
exterior domains and the requirement \eqref{e2.2} was not needed in order
to obtain uniform decay rates.
\end{remark}

\begin{remark} \label{rmk4} \rm
The above procedure could be extended to include the anisotropic case,
that is, when the coefficients $a_{ijkh}$ do depend on each $x \in \Omega$.
In that case $A_{ij} = A_{ij}(x)$ and assumptions (a) and (b) would
be required to be valid for each $x \in \Omega$ with $C_0 > 0$
independent of $x \in \Omega$. As it is clear in the proof of
Lemma \ref{lem2.3}
additional assumptions on the behavior of partial derivatives
$\frac{\partial}{\partial x_i} A_{ij}(x)$ would be required to arrive
to the conclusion of Theorem \ref{thm2.2}.
\end{remark}



\section{The semilinear problem}

This section, we apply the results obtained in Section 2 to study the
asymptotic behavior of the solutions of the semilinear model.
 We will sketch the proof that for small enough initial data the solution
 of problem \eqref{e1.1} exists globally and enjoys the same rate of decay
as $t \to +\infty$ as the solution of the linear model \eqref{e2.1}.
We will assume that $f$ satisfies all conditions given in (H2).
Local existence will be done via contraction arguments and the
global existence as well as the asymptotic behavior using the
decay rates for the linear part obtained in Section 2. Due to the character
of the nonlinearity in problem \eqref{e1.1} we will require more regular
solutions. First, let us rewrite problem \eqref{e1.1} as a first order
evolution system:
\begin{equation}
\frac{dU}{dt} = AU + F(U), \quad U(0) = U_0 \label{e3.1}
\end{equation}
where $U = (u,u_t)$, $U_0 = (u_0,u_1)$, $F(U) = (0,f(u_t)+u)$ and $A$ with
domain  $\mathcal{D}(A) = [H^2(\Omega) \cap H_0^1(\Omega)]^3
 \times [H_0^1(\Omega)]^3$ given by
$$
A(u,v) = \Big(v, \sum_{i,j=1}^3 \frac{\partial}{\partial x_i} \Big(A_{ij}
 \frac{\partial u}{\partial x_j}\Big) - u - v\Big)
$$
for every $  (u,v) \in \mathcal{D}(A)$. The operator $A$ is the
infinitesimal generator of a $C_0$ group of operators
$\{T(t)\}_{t\in\mathbb{R}}$ in the Hilbert space $X
=[H_0^1(\Omega)]^3 \times [L^2(\Omega)]^3$. The main result of
this section for the solution of problem \eqref{e1.1} will be
present with initial data with compact support. However, it seems
to us that using recent work due to Todorova and  Yordanov
\cite{t1} and Ikehata and Matsuyana \cite{i2} for the scalar wave
equation then our result may be improved for initial data
satisfying only \eqref{e2.2}. We want to prove the following
result.



\begin{theorem} \label{thm3.1}
 Assume condition (H1) and (H2). Let $(u_0,u_1) \in \mathcal{D}(A^2)$ with
 compact support. Then, there exist $\delta > 0$ such that if
$\widetilde I < \delta$ then problem \eqref{e1.1} has a unique global
solution $(u,u_t)$ such that
$$
(u,u_t) \in C(\mathbb{R}; \mathcal{D}(A^2)) \cap C^1(\mathbb{R};
\mathcal{D}(A)) \cap C^2(\mathbb{R};X)
$$
and satisfies
\begin{gather*}
\int_\Omega |u|^2\,dx \le C \widetilde I \big(1+|t|\big)^{-1}
\quad \forall t \in \mathbb{R}\\
E(t)+E_1(t)+E_2(t) \le C \widetilde I \big(1+|t|\big)^{-2} \quad
\forall t \in \mathbb{R},
\end{gather*}
where $E(t)$ is given by \eqref{e1.2} and $E_1$ and $E_2$ will be given
by \eqref{e3.7} and \eqref{e3.9} and $C > 0$ is a positive constant.
Here $\widetilde I$ depends only on the Sobolev norms (up to order three)
of the initial data.
\end{theorem}

First, we sketch the proof of existence of a local solution.
Let $T > 0$ and consider the space
$$
Y(T) = C\big([0,T]; \mathcal{D}(A^2)\big) \cap C^1\big([0,T];
\mathcal{D}(A)\big) \cap C^2\big([0,T]; X\big)
$$
with norm
\begin{equation}
\|U\|_{Y(T)} = \sup_{[0,T]} \|U(t)\|_{\mathcal{D}(A^2)}
+ \sup_{[0,T]} \|U_t(t)\|_{\mathcal{D}(A)} + \sup_{[0,T]}
\|U_{tt}(t)\|_X . \label{e3.2}
\end{equation}
Clearly $Y(T)$ is a Banach space. Let $U = (u,v) \in Y(T)$. Using our
assumptions (H2) on $f$ and the embedding
$H_0^1(\Omega) \hookrightarrow L^q(\Omega)$ for $2 \le q \le 6$ and
$H^2(\Omega) \hookrightarrow L^\infty(\Omega)$ we obtain the estimates
\begin{gather*}
\|f(v)\| \le C\|v\|_{[L^{2p}(\Omega]^3}^p
\le C\|v\|_{[H_0^1(\Omega)]^3}^p,\\
\|\nabla f(v)\| \le C\|v\|_{[H^2(\Omega)]^3}^{p-1}
\|v\|_{[H_0^1(\Omega)]^3},
\\
\| \frac{\partial^2 f(v)}{\partial x_i\partial x_j}\|
\le C\|v\|_{[H^2(\Omega)]^3}^p, \quad i,j=1,2,3.
\end{gather*}
We recall that $\|g\|^2 = \sum_{j=1}^3 \int_\Omega |g_j|^2\,dx$ whenever
$g = (g_1,g_2,g_3) \in [L^2(\Omega)]^3$. The above estimates imply
$$
f(v) \in C\big([0,T]; [H^2(\Omega) \cap H_0^1(\Omega)]^3\big).
$$
Now, we claim that $f(v) \in C^1\big([0,T]. [H_0^1(\Omega)]^3\big)
\cap C^2\big([0,T]; [L^2(\Omega)]^3\big)$.
 In fact,
$$
\frac{d}{dt} f(v) = \big(\nabla f_1(v) \cdot v_t,
 \nabla f_2(v) \cdot v_t,    \nabla f_3(v) \cdot v_t\big)\,.
$$
Therefore, using assumption (H2) and H\"older's inequality we obtain
\begin{align*}
\| \frac{d}{dt} f(v)\|^2
&\le \int_\Omega \sum_{j=1}^3 |\nabla f_j(v) \cdot v_t|^2\,dx \\
&\le C \int_\Omega |v|^{2(p-1)} |v_t|^2\,dx \\
&\le C\|v_t\|_{[L^{2p}]^3}^2 \|v\|_{[L^{2p}]^3}^{2(p-1)} \\
&\le C\|v_t\|_{[H_0^1]^3}^2  \|v\|_{[H_0^1]^3}^{2(p-1)}  .
\end{align*}
Similarly, we can estimate
$$
| \frac{\partial}{\partial x_j} \Big(\frac{d}{dt} f(v)\Big)|
\le C|v_t| |\frac{\partial v}{\partial x_j}| |v|^{p-2} + C|
\frac{\partial v_t}{\partial x_j}| |v|^{p-1}
$$
for some positive constant $C$. Consequently,
$$
\|\frac{\partial}{\partial x_j} \Big(\frac{d}{dt} f(v)\Big)\|
\le C\|v\|_{[H^2]^3}^{p-1} \|v_t\|_{[H_0^1]^3}
$$
for $j=1,2,3$. It follows from the above discussion that
$$
f(v) \in C^1\big([0,T]. [H_0^1(\Omega)]^3\big).
$$
By a similar procedure we can prove that
$f(v) \in C^2\big([0,T]; [L^2(\Omega)]^3\big)$ which proves our claim.
Thus, whenever we consider an element
$\widetilde U = (\tilde u, \tilde v) \in Y(T)$ then, the nonlinearity
$F(\widetilde U) = \big(0, f(\tilde v)+\tilde u\big)$ belongs to
$$
C^1\big([0,T]; \mathcal{D}(A)\big) \cap C^2\big([0,T];X\big).
$$
It follows by semigroup theory that the nonhomogeneous problem
\begin{equation}
\frac{dU}{dt} = AU + F(\widetilde U), \quad U(0) = U_0 = (u_0,u_1) \label{e3.3}
\end{equation}
has a unique (local) solution $U = (u,v) \in Y(T)$ provided
$U_0 \in \mathcal{D}(A^2)$.



\begin{lemma} \label{lem3.2}
Assume (H1) and (H2). Let $U_0 = (u_0,u_1) \in \mathcal{D}(A^2)$. Then,
there exist $T_0 > 0$ such that problem \eqref{e1.1} has a unique solution
$U = (u,u_t)$ belonging to the space
$$
C\big([0,T_0]; \mathcal{D}(A^2)\big) \cap C^1\big([0,T_0]; \mathcal{D}(A)\big)
\cap C^2\big([0,T_0]; X\big).
$$
\end{lemma}


\begin{proof}[Sketch of proof] We consider the map
 $\Phi\colon Y(T) \mapsto Y(T)$ given by $\Phi(\widetilde U) = U$ where
$U$ is the solution of \eqref{e3.3} and we will prove that $\Phi$ has a
unique fixed point in $Y(T)$ as long as we choose $T$ sufficiently small.
We achieve this in the following way: Using the formula of variation of
parameters and our assumptions of $f$ we can prove that the solution $U$
of \eqref{e3.3} satisfies
\begin{equation}
\|U\|_{Y(T)} \le C(U_0) + CT\left\{\|\widetilde U\|_{Y(T)}^p
+ \|\widetilde U\|_{Y(T)}\right\} \label{e3.4}
\end{equation}
where $C(U_0)$ depends only on the norm $\|A^2U_0\|_X$ and the Sobolev
norms (up to order three) of $U(0)$ and $U_t(0)$. Next, we choose $K \ge 1$
and consider the set
$$
B_K = \big\{\widetilde U \in Y(t); \widetilde U(0) = U_0,
\widetilde U_t(0) = U_1,   \|\widetilde U\|_{Y(T)} \le K\big\}
$$
where
$$
U_1 = \Big(u_1, \sum_{i,j=1}^3 \frac{\partial}{\partial x_i}
\big(A_{ij} \frac{\partial u_0}{\partial x_j}\big) - u_0-u_1\Big).
$$
We claim that $\Phi(B_K) \subseteq B_K$, if we choose $T$ small and $K$ large.
In fact, let $\widetilde U \in B_K$ then, from \eqref{e3.4} we obtain
$$
\|U\|_{Y(T)} \le C(U_0) + CT\big\{K^p + K\big\}.
$$
Now, we choose $K$ such that $C(U_0) \le K/2$ and $T > 0$ such that
$T < [2C(K^{p-1}+1)]^{-1}$. Thus $\|U\|_{Y(T)} \le K$. Obviously
$U(0) = U_0$ and $U_t(0) = U_1$ . Using the semigroup properties and the
formula of variation of parameters we can prove that $\Phi$ is a
contraction map, that is for any $\widetilde U$ and $\widetilde W$ belonging
 to $B_K$ we have
$$
\|\Phi(\widetilde U)-\Phi(\widetilde W)\|_{Y(T)} \le \alpha \|\widetilde U
- \widetilde W\|_{Y(T)}
$$
where $0 < \alpha = \alpha(K,T) < 1$ as long as we choose $K$ large
and $T > 0$ sufficiently small. This proves Lemma \ref{lem3.2}.
\end{proof}


Next we prove Theorem \ref{thm3.1}. First, we extend the local solution we found
in Lemma \ref{lem3.2} to the maximal interval of  existence $[0,T_{\rm max})$.
Technically it will be more convenient to rewrite problem \eqref{e1.1} as
\begin{equation}
\frac{dU}{dt} = \widetilde A U + \widetilde F(U), \quad
U(0) = U_0 = (u_0,u_1) \label{e3.5}
\end{equation}
with
\[
\widetilde A(u,v) = \Big(v, \sum_{i,j=1}^3 \frac{\partial}{\partial x_j}
\big(A_{ij} \frac{\partial u}{\partial x_i}\big)-v\Big)
\]
and $\widetilde F(U) = (0,f(u_t))$ where $U = (u,v)$, $v = u_t$.
Let $\{S(t)\}$ be the semigroup associated to the generator $\widetilde A$.
Then Theorem \ref{thm2.1} tell us that the solution of the linear equation satisfies
\begin{equation}
E(t) \le C I_0 \big(1+t\big)^{-2} \quad \forall  t \ge 0. \label{e3.6}
\end{equation}
 In this article, we denote by $C$ various positive constants
 which may vary from line to line.
 Let $v = u_t$ . Taking the derivative in time of equation \eqref{e2.1}
  we deduce that $v$ satisfies
\begin{gather*}
v_{tt} - \sum^3 \frac{\partial}{\partial x_i} \Big(A_{ij}
\frac{\partial v}{\partial x_i}\Big) + v_t = 0 \quad \text{in }
 \Omega \times [0,\infty)\\
 v(x,0) = u_1(x), \quad   v_t(x,0) = \sum^3 \frac{\partial}{\partial x_j}
  \Big(A_{ij} \frac{\partial u_0}{\partial x_i}\Big) - u_1(x)\\
 v=0 \quad\text{on } \partial\Omega \times [0,+\infty)
\end{gather*}
Applying the same reasoning as in the proof of Theorem \ref{thm2.2},
\begin{equation}
E_1(t) = \frac 12 \int_\Omega \big\{|v_1|^2 + \sum_{i,j=1}^3 A_{ij}
\frac{\partial v}{\partial x_j} \cdot \frac{\partial v}{\partial x_i}\big\}dx
 \le C I_1(1+t)^{-2} \label{e3.7}
\end{equation}
with $v = u_t$, where $I_1$ depends on the Sobolev norms (up to order two)
of the initial data and the quantity
$\int_\Omega |x|^2 \big| \sum_{i,j=1}^3 \frac{\partial}{\partial x_j}
\Big(A_{ij} \frac{\partial u_0}{\partial x_i}\Big)\big|^2 dx$.
Thus, from the equation \eqref{e2.1} we also obtain
\begin{equation}
\big\| \sum_{i,j=1}^3 \frac{\partial}{\partial x_j} \Big(A_{ij}
\frac{\partial u}{\partial x_i}\Big)\big\|^2 \le C(I_0+I_1)(1+t)^{-2} \label{e3.8}
\end{equation}
Similarly, if $w = v_t = u_{tt}$ we obtain
\begin{equation}
E_2(t) = \frac 12 \int_\Omega \big\{|w_t|^2 + \sum_{i,j=1}^3
\sum_{i,j=1}^3 A_{ij} \frac{\partial w}{\partial x_j} \cdot
\frac{\partial w}{\partial x_i}\big\}dx \le C I_2(1+t)^{-2} \label{e3.9}
\end{equation}
where $I_2$ depends on the Sobolev norm (up to order three) of the initial
data and the quantity
$\int_\Omega |x|^2 \big| \sum_{i,j=1}^3 \frac{\partial}{\partial x_j}
\big(A_{ij} \frac{\partial u_1}{\partial x_i}\big)\big|^2 dx$.
Let $\widetilde I = I_0+I_1+I_2$ and $K > 1$ such that
\begin{gather}
\|u_0\|^2 < K \widetilde I, \label{e3.10} \\
E(0) + E_1(0) + E_2(0) + \|Lu_0\|^2 + \|Lu_1\|^2 < K \widetilde I, \label{e3.11}
\end{gather}
where $L = \sum_{i,j=1}^3 \frac{\partial}{\partial x_j}
\big(A_{ij} \frac{\partial}{\partial x_i}\big)$.

We proceed to prove Theorem \ref{thm3.1}: Let $(u,u_t)$ be the local solution
for the semilinear model \eqref{e1.1} obtained in Lemma \ref{lem3.2}. Clearly, by
continuity of the quantities on the left hand side of \eqref{e3.6}, \eqref{e3.7}
and \eqref{e3.9} then in an small interval $[0,t)$ we will have that
\begin{gather}
(1+t)\|u(\cdot,t)\|^2 < K \widetilde I, \label{e3.12} \\
(1+t)^2 \big\{E(t) + E_1(t) + E_2(t) + \|Lu(\cdot,t)\|^2
+ \|Lu_t\|^2\big\} < K \widetilde I \label{e3.13}
\end{gather}
are valid. We want to prove that \eqref{e3.12} and \eqref{e3.13} hold for
 any $t \ge 0$. To do this we will choose $K$ large and after $\widetilde I$
 small. Suppose that \eqref{e3.12} and \eqref{e3.13} are not valid for any
 $\widetilde T$ ``near" $T_{\rm max}$ .
Therefore, there must exist $T \in [0,\widetilde T]$ such that
 \eqref{e3.12} and \eqref{e3.13} hold in $[0,T)$ but
\begin{equation}
(1+T)\|u(\cdot,T)\|^2 = K \widetilde I \label{e3.14}
\end{equation}
and/or
\begin{equation}
(1+T)^2 \big\{E(T) + E_1(T) + E_2(T) + \|Lu(\cdot,T)\|^2
+ \|Lu_t(\cdot,T)\|^2\big\} = K \widetilde I \label{e3.15}
\end{equation}
 From \eqref{e3.5} it follows that
$$
U(t) = S(t)U_0 + \int_0^t S(t-r)\widetilde F(r)\,dr.
$$
Consequently, from Theorem \ref{thm2.2} we deduce
\begin{equation}
E(t) \le C \widetilde{I}(1+t)^{-1} + C \int_0^t (1+t+r)^{-1} J(r)\,dr \label{e3.16}
\end{equation}
where $J(r) = \|f(u_r)\| + \| |\cdot| f(u_r)\|$.
Using assumptions (H2) and Gagliardo-Nirenberg's inequality we
obtain
$$
\|f(u_r)\| \le C\|u_r\|_{L^{2p}}^p \le C\|u_r\|^{(1-\theta)p}
\Big(\int_\Omega \sum_{i,j=1}^3 A_{ij} \frac{\partial u_r}{\partial x_j}
\cdot \frac{\partial u_r}{\partial x_i}\,dx\Big)^{\theta p/2}
$$
where $0 < \theta = \frac{3(p-1)}{2p} \le 1$ because $\frac 73 < p \le 3$.
 Due to \eqref{e3.12}-\eqref{e3.15} it follows that
\begin{equation}
\|f(u_r)\| \le C\big\{K \widetilde I(1+r)^{-1}\big\}^{(1-\theta)p}
\big\{K \widetilde I(1+r)^{-1}\big\}^{\theta p} = C K^p \widetilde I^p (1+r)^{-p}
\label{e3.17}
\end{equation}
for any $r \in [0,T]$. Now we use finite propagation speed valid
for the solution of problem \eqref{e1.1}: If
$\mathop{\rm supp}u_0 \cup \mathop{\rm supp} u_1
\subseteq \{x \in \mathbb{R}^3, |x| \le R\}$ then in the interval of
existence
$(u,u_t)=(0,0)$, if $|x| \ge C_1t + R$ where
$C_1 = \|A\|\big/\sqrt{C_0}$, $\|A\|^2 = \sum_{i,j=1}^3
\|A_{ij}\|^2$ and $C_0$ is as in \eqref{e1.4}. We estimate
\begin{align*}
\| |\cdot| f(u_r)\|^2
&\le C \int_\Omega |x|^2 |u_r(x,r)|^{2p}\,dx \\
&= C \int_{\Omega\cap\{|x|\le C_1r+R\}} |x|^2 |u_r(x,r)|^{2p}\,dx \\
&\le (C_1r+R)^2  C\|u_r(\cdot,r)\|_{L^{2p}}^{2p}
\end{align*}
and by Gagliardo-Nirenberg it follows that
\begin{equation}
\| |\cdot| f(u_r)\| \le C(C_1r+R)K^p \widetilde I^p (1+r)^{-p} \label{e3.18}
\end{equation}
 From \eqref{e3.16}, \eqref{e3.17} and \eqref{e3.18} we deduce
\begin{equation}
\begin{aligned}
E(t) &\le C \widetilde I(1+t)^{-1} + C K^p \widetilde I^p
\int_0^t (1+t-r)^{-1}(1+r)^{-p+1}\,dr\\
&\le (C \widetilde I + C K^p  \widetilde I^p))(1+t)^{-1}
\end{aligned} \label{e3.19}
\end{equation}
for any $t \in [0,T]$. Here we used a calculus type lemma
(see \cite[Lemma 7.4]{r1}). Using the formula of variation of parameters
we also obtain
$$
\|u(\cdot,t)\| \le C I_0(1+t)^{-1/2} + C \int_0^t (1+t-r)^{-1/2} J(r)\,dr
$$
where $J(r)$ is as in \eqref{e3.16}. Due to our above calculation we get
\begin{align*}
\|u(\cdot,t)\| &\le C I_0(1+t)^{-1/2} + C K^p \widetilde I^p
\int_0^t (1+t-r)^{-1/2} (1+r)^{-p+1}\,dr\\
&\le (C I_0 + C K^p \widetilde I^p)(1+t)^{-1/2}.
\end{align*}
Next, we differentiate in time equation \eqref{e1.1} and use the same
sequence of ideas given above to obtain that $v = u_t$ satisfies
\[
E_1(t) \le C(\widetilde I+ \widetilde I^p + K^p \widetilde I^p)(1+t)^{-2} %\label{e3.21}
\]
where $E_1(t)$ is given as in \eqref{e3.7}. Using the equation it follows that
\[
\|Lu(\cdot,t)\| \le C(\widetilde I + \widetilde I^p + K^p \widetilde
I^p)(1+t)^{-2} %\label{e3.22}
\]
for any $t \in [0,T]$. Finally, we differentiate twice in time
equation \eqref{e1.1} and repeat the above reasoning to obtain that $w =
v_t = u_{tt}$ satisfies
\begin{gather}
E_2(t) \le C(\widetilde I + \widetilde I^p + \widetilde I^{2p-1)}
+ K^p \widetilde I^p)(1+t)^{-2}, \label{e3.23} \\
 \|Lu(\cdot,t)\| \le C(\widetilde I + \widetilde I^p + \widetilde I^{2p-1)}
 + K^p \widetilde I^p)(1+t)^{-2}\,. \label{e3.24}
\end{gather}
Collecting information from \eqref{e3.19} up to \eqref{e3.24}, we have
\begin{equation}
(1+t)\|u(\cdot,t)\|^2 \le C(1+K^p \widetilde I^{p-1})\widetilde I,
\label{e3.25}
\end{equation}
and
\begin{equation}
\begin{aligned}
&(1+t)^2 \big\{E(t) + E_1(t) + E_2(t) + \|Lu(\cdot,t)\|^2
+ \|Lu_t\|^2\big\} \\
&\le C\big(1+\widetilde I^{p-1} + \widetilde I^{2p-2}
+ K^p \widetilde I^{p-1}\big)\widetilde I
\end{aligned}\label{e3.26}
\end{equation}
for any $t \in [0,T]$ and some positive constant $C$. Now we choose
$K$ large so that $K > C$ and
$$
\widetilde I < \min\Big\{\big(\frac{K-C}{3C}\big)^{1/p-1},
\big(\frac{K-C}{3C}\big)^{1/2p-2},
\big(\frac{K-C}{3CK^p}\big)^{1/p-1}\Big\}\,.
$$
With this choice, we clearly have that
$$
C\big(1+\widetilde I^{p-1} + \widetilde I^{2p-2}
+ K^p \widetilde I^{p-1}\big) < K.
$$
Consequently from \eqref{e3.25} and \eqref{e3.26}, we deduce that
\begin{gather*}
(1+t)\|u(\cdot,t)\|^2 < K \widetilde I, \\
(1+t)^2 \big\{E(t) + E_1(t) + E_2(t) + \|Lu(\cdot,t)\|^2
+ \|Lu_t\|^2\big\} < K \widetilde I
\end{gather*}
for any $t \in [0,T]$ which is a contradiction with \eqref{e3.14}
and \eqref{e3.15}. It follows that \eqref{e3.12} and \eqref{e3.13}
should be valid for any $t \in [0,T_{\rm max})$;
therefore, the solution  of \eqref{e1.1} exists globally and decays
at the desired rate.


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