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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 48, 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}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/48\hfil Differentiability, analyticity and optimal rates]
{Differentiability, analyticity and optimal rates of decay for
 damped wave equations}

\author[L. H. Fatori, M. Z. Garay, J. E. M. Rivera\hfil EJDE-2012/48\hfilneg]
{Luci Harue Fatori, Maria Zegarra Garay, Jaime E. Mu\~noz Rivera}  % in alphabetical order

\address{Luci Harue Fatori \newline
Department of Mathematics,
 Universidade Estadual de Londrina, PR, Brazil}
\email{lucifatori@uel.br}

\address{Maria Zegarra Garay \newline
 Universidad Nacional Mayor de San Marcos,
 Facultad de Ciencias, Lima, Peru}
\email{mzgaray@hotmail.com}

\address{Jaime E. Mu\~noz Rivera \newline
 National Laboratory of Scientific Computations, LNCC/MCT,
 Institute of Mathematics, UFRJ, RJ, Brazil}
\email{rivera@lncc.br}

\thanks{Submitted December 22, 2011. Published March 27, 2012.}
\thanks{Luci Fatori was supported by grant 14423/2009 from the
Funda\c{c}\"ao Arauc\'aria}
\subjclass[2000]{35L10, 47D06}
\keywords{Dissipative systems; decay rate; analytic semigroups; \hfill\break\indent
  polynomial stability}

\begin{abstract}
 We give necessary and sufficient conditions on the damping term of a 
 wave equation for the corresponding  semigroup to be analytic. 
 We characterize damped  operators for which the corresponding semigroup 
 is analytic, differentiable, or exponentially stable. 
 Also when the damping operator is not strong enough to have the above 
 properties,  we show that the solution decays  polynomially, and 
 that the polynomial rate of decay is optimal.
\end{abstract}

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

\section{Introduction}

This article is concerned with analyticity, differentiability and
asymptotic stability of the $C_0$ semigroups associated with the
 initial-value problem
\begin{gather}
u_{tt}+Au+Bu_t=0\label{eq1.1} \\
u(0)=u_0,\quad u_t(0)=u_1\label{eq1.2}
\end{gather}
where $A$,  and $B$ are a self-adjoint positive definite operators
with domain $D(A^\alpha)= D(B)$ dense in a Hilbert space $H$.
We use the following hypotheses:
\begin{itemize}
\item[(H1)] There exists positive constants $C_1$ and $C_2$ such that
$$ C_1A^{\alpha}\leq B\leq C_2 A^{\alpha}.
$$
which means
$$
C_1(A^{\alpha}u,u)\leq (Bu,u)\leq C_2 (A^{\alpha}u,u)
$$
for any $u\in D(A^\alpha)$.

\item[ (H2)] The bilinear
form $b(u,w)=(B^{1/2}u,B^{1/2}w)$ is  continuous  on
$D(A^{\alpha/2})\times D(A^{\alpha/2}).$ By the Riesz
representation theorem, assumption (H2) implies that there exists
an operator $S\in \mathcal{L}(D( A^{\alpha/2}))$  such that
$$
(Bu,w)=(A^{\alpha/2}Su, A^{\alpha/2}w)
$$
 for any $u,w \in D(A^{\alpha/2})$.
\end{itemize}

There exists a large body of literature about the above problem dealing
with  asymptotic behaviour of the solutions to the damped wave
equation see for example \cite{ l3N191, p3AN88, chrussl, f7AN76,
c7J175, 96, 98} and the references therein. In contrast to this
results, there exists only a few publications dealing with
regularity properties of the damped wave equation, like
analyticity and differentiability of the corresponding  semigroup.
Here we mention two references. First, in \cite{ChenTri} the
authors proved that the  semigroup associated to the damped wave
equation is analytic if $1/2\leq \alpha\leq 1$. This result
established a fortiori the conjectures by  Chen and Russel on structural
damping for elastic systems, which referred
to the case $\alpha=1/2$. Second,  Liu and  Liu \cite{w5D197}
proved also the analyticity of the
corresponding semigroup when $\alpha\in [1/2,1]$ and the
differentiability  of the semigroup provides $\alpha\in ]0,1/2]$.
Their  proof is simpler than the proof in \cite{ChenTri}, the
method the authors  used is based on contradiction arguments.

In the two above cited papers there is no information
about the behaviour of the semigroup for $-1\leq\alpha\leq 1/2$, which frequently appears in applications.
We also cite the book by Liu and Zheng \cite{six}, for questions related questions
to this problem.


In this article we show a class of operators $A$ and $B$, for
which the above equation is analytic, differentiable and
exponentially stable.  Here we develop a  proof simpler than the one in
\cite{ChenTri,w5D197},
 without using contradiction arguments.  In addition, we show in case that the
semigroup is not exponentially stable, that the solution of
 \eqref{eq1.1} decays polynomially to zero as
 time appraoches infinity. We  show the our rate decay is  optimal. To do so, we show for any
 contraction semigroup, a necessary condition to get the polynomial rate of decay.
 That is to say,
the  main result of this paper is to get a fully characterization of the damping term
for $-1\leq \alpha\leq 1$.
 We show as in \cite{ChenTri, w5D197} that the semigroup is analytic if and
 only if $1/2\leq \alpha\leq 1$, it is differentiable when
$\alpha\in ]0,1[$ and that it is exponentially stable if
and only if $\alpha\in [0,1].$ Finally, in case of $\alpha
=-\gamma<0$ we show that the corresponding semigroup decays
polynomially to zero as $t^{-1/\gamma}$
 and we show that this rate of decay is optimal in $D(A)$ in the sense that is not possible to improve the rate $t^{-1/\gamma }$ with initial data over the domain of the operator $A$.

This paper is organized as follows. In sections 2 and 3 we show the
analyticity and differentiability  of the semigroup respectively.
In section 4 we show the polynomial rate of decay of the
semigroup when $\alpha<0$ and we prove the optimality of the rates
of decay. Finally, in section 5 we give some applications of  the
above results.


\section{Analyticity}\label{analytic}


Let us denote
$\mathcal{H} = \mathcal{D} (A^{1/2})\times H$.
Denoting by $U=(u,v)$ we define the norm in $\mathcal{H}$ as
$$
\|U\|_{\mathcal{H}}^2=\|A^{1/2}u\|^2+\|v\|^2.
$$
Putting $v=u_t$, \eqref{eq1.1} can be written as the
 initial-value problem
\begin{equation}
\begin{gathered}
\frac{dU}{dt}=A_BU\\
U(0)=U_0
\end{gathered}
\end{equation}
with $U=(u,v)^t,\ U_0=(u_0,u_1)^t$. Let us define
\begin{equation}
\mathcal{D}(A_B)=\left\{(u,v)\in \mathcal{D}(A)\times \mathcal{D}(A^{1/2}):
Au+  Bv\in H\right\}
\end{equation}
and
\begin{equation}
A_B=\begin{pmatrix}
0& I\\
-A&-B
\end{pmatrix}, \quad
A_BU=\begin{pmatrix}
v\\
-(Au +  Bv)
\end{pmatrix}.
\end{equation}
Clearly, for $U\in \mathcal{D}(A_B)$,
\[
(A_BU,U)= (A^{1/2}v,A^{1/2}u)-(Au+  B v,v)=-\|  B^{1/2}v\|\leq 0.
\]
Thus $A_B$ is a dissipative operator. Therefore we have the following result;
see Pazy \cite{z3LS83}.


\begin{theorem}\label{thm2.2}
Let us assume that $A$ and $B$ are self adjoint
operators  positive definite and also a bijection operator from
$D(A_B)$ to $\mathcal{H}$. Then the operator $A_B$ is the
infinitesimal generator of a $C_0$-semigroup $S_B(t)$ of
contraction in $\mathcal{H}$.
\end{theorem}

In this section we will show that the semigroup is analytic. Our
 main tool is the following theorem whose proof is found in \cite{six}.


\begin{theorem}\label{thm2.1}
Let $S(t)=e^{At}$ be a $C_0$-semigroup of contractions on Hilbert space.
Then $S(t)$ is analytic if and only if
\[
\rho (A) \supseteq \{i\beta :\beta \in \mathbb{R}\} \equiv i\mathbb{R}
\]
and
\[
\limsup_{|\beta |\to \infty} |\beta|\,\|(i\beta I-A)^{-1}\| < \infty,
\]
 where $\rho (A)$ is the resolvent set of $A$.
\end{theorem}

The main result of this section is to show that the semigroup is
analytic if and only if $1/2\leq \alpha\leq 1$.

\begin{theorem}\label{anali}
The semigroup $S_B(t)=e^{A_Bt}$ is analytic if and only if
 $1/2\leq \alpha\leq 1$.
\end{theorem}


\begin{proof}
For $1/2\leq \alpha\leq 1$, the domain of the operator $A_B$ is
\begin{equation}
\mathcal{D}(A_B)=\{(u,v)\in \mathcal{D}(A^{1/2})\times \mathcal{D}(A^{1/2}):
Au+  Bv\in H\}.
\end{equation}
Note that in general it is not possible  to conclude that $u\in D(A)$.
Using the spectral equation we obtain
\begin{gather}
i\beta u - v = f \quad \text{in } D(A^{1/2})\label{esp1}\\
i\beta v + Au+B v = g \quad \text{in } H.\label{esp2}
\end{gather}
 As in the above section we obtain
\begin{equation}\label{relA0}
\|A^{\alpha/2}v\|^2\leq C\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}}.
\end{equation}
Multiplying  \eqref{esp2} by
$A^{\gamma}\overline{u}$  and using \eqref{esp1} we obtain
$$
 \|A^{(1+\gamma)/2}u\|^2+(B v,A^\gamma u) =\| A^{\gamma/2} v\|^2
+( A^\gamma v,f)+(g,A^\gamma u);
$$
that is,
\begin{equation} \label{chave}
\begin{split}
& \|A^{(1+\gamma)/2}u\|^2+(A^{\alpha/2 } Sv, A^{\gamma+\alpha/2}u)\\
& =\| A^{\gamma/2} v\|^2 +( A^{\gamma-1/2}
v,A^{1/2}f)+(g,A^\gamma u).
\end{split}
\end{equation}
 Taking $\gamma= 1-\alpha$ in the above identity  we obtain
$$
\|A^{(2-\alpha)/2}u\|^2\leq \|A^{(1-\alpha)/2}v\|^2+C\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}}.
$$
From where we have that there exists a positive constants $C$ such that
\begin{equation}
\|A^{(2-\alpha)/2}u\|^2\leq C\|A^{\alpha/2}v\|^2+C\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}} \leq C_0\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}}\label{mm}.
\end{equation}
 Multiplying \eqref{esp1} by $A\overline{u}$  we obtain
$$
i\beta\|A^{1/2}u\|^2 =(
A^{1/2}f,A^{1/2}u)+(A^{\alpha/2}v,A^{1-\alpha/2} u).
$$
Then using  \eqref{relA0} and \eqref{mm} we obtain
\[
\beta\|A^{1/2}u\|^2 \leq C\|U\|_{\mathcal{H}}\|F\|_{\mathcal{H}}\label{chave2}.
\]
 Let us decompose $v$ as $v=v_1+v_2$ such that
\begin{gather}
i\beta v_1 +B v_1 = g \quad \text{in } H\label{deco1}\\
i\beta v_2 + Au+B v_2 = 0 \quad \text{in } H\label{deco2}.
\end{gather}
Multiplying  \eqref{deco1} by $\overline{v}_1$ and taking
imaginary and real part we obtain
\begin{equation}\label{betav}
|\beta|\|v_1\|\leq \|F\|_{\mathcal{H}},\quad \|B^{1/2}v_1\|\leq
\|F\|_{\mathcal{H}}.
\end{equation}
Note that $\|v_1\| \leq  \|v\|+\|v_2\|$
and
\begin{align*}
\|A^{\alpha/2} v_2\|&\leq  \|A^{\alpha/2} v\|+\|A^{-\alpha/2}B v_1\|\\
&\leq  c_1\|U\|^{1/2}_{\mathcal{H}}\|F\|^{1/2}_{\mathcal{H}}
 +c_2\|v_1\|^{1/2}\|F\|^{1/2}_{\mathcal{H}}\\
&\leq  c\|U\|^{1/2}_{\mathcal{H}}\|F\|^{1/2}_{\mathcal{H}}
 +c_2\|v_2\|^{1/2}\|F\|^{1/2}_{\mathcal{H}}.
\end{align*}
 From \eqref{deco2} and the above inequality, we obtain
\[
|\beta|\|A^{-\alpha/2}v_2\| \leq  \|A^{(2-\alpha)/2}u\|+\|A^{\alpha/2} v_2\|
\leq c\|U\|^{1/2}_{\mathcal{H}}\|F\|^{1/2}_{\mathcal{H}}+c_2\|v_2\|^{1/2}\|F\|^{1/2}_{\mathcal{H}}.
\]
 Using interpolation we obtain
\begin{align*}
\|v_2\|^2&\leq  c\|A^{-\alpha/2} v_2\|\|A^{\alpha/2} v_2\|\\
&\leq  \frac{c}{\beta}(\|U\|^{1/2}_{\mathcal{H}}\|F\|^{1/2}_{\mathcal{H}}
 +c_2\|v_2\|^{1/2}\|F\|^{1/2}_{\mathcal{H}})^2\\
&\leq  \frac{c}{\beta}(\|U\|_{\mathcal{H}}\|F\|_{\mathcal{H}}
 +\|v_2\|\|F\|_{\mathcal{H}}).
\end{align*}
From where we have
$$
\beta^2\|v_2\|^2\leq c\beta\|U\|_{\mathcal{H}}\|F\|_{\mathcal{H}}+c_0\|F\|^2_{\mathcal{H}}.
$$
 From the above inequality and \eqref{betav} we obtain
$$
\beta^2\|v\|^2\leq 2\beta^2(\|v_1\|^2+\|v_2\|^2)\leq
c\beta\|U\|_{\mathcal{H}}\|F\|_{\mathcal{H}}+c_0\|F\|^2_{\mathcal{H}}.
$$
 Using relation \eqref{chave2}, we obtain
$$
\beta^2(\|v\|^2+\|A^{1/2}u\|^2)
\leq  c\beta\|U\|_{\mathcal{H}}\|F\|_{\mathcal{H}}+c_0\|F\|^2_{\mathcal{H}}
$$
which is equivalent to
$$
\beta^2\|U\|^2_{\mathcal{H}}\leq  c\beta\|U\|_{\mathcal{H}}\|F\|_{\mathcal{H}}
+c_0\|F\|^2_{\mathcal{H}}
$$
which implies
$$
\beta^2\|U\|^2_{\mathcal{H}}\leq  c_1\|F\|^2_{\mathcal{H}}.
$$
From where  the analyticity follows.

Now we show that the corresponding semigroup is not analytic for
$0\leq \alpha<1/2$. Here, we consider that the operator $A$ and
$B$ have infinite eigenvector in common. Let us  construct a
sequence $F_\nu$ such that the solutions of
$$
i\beta_\nu U_\nu -\mathcal{A}U_\nu =F_\nu
$$
satisfies
$|\beta_\nu| \|U_\nu\|_{\mathcal{H}} \to \infty$,
which in particular implies
$$
\|\beta_\nu (i\beta_\nu I-A)^{-1}\|_{\mathcal{H}}  \to \infty
$$
which means that the corresponding semigroup is not analytic. To
see this, let us consider the spectral system
\begin{gather} \label{espec1}
i\beta u_\nu  - v_\nu =0 \\
i\beta v_\nu  +Au_\nu + B v_\nu =w_\nu \label{espec2}
\end{gather}
where $w_\nu$ is an unitary eigenvector of $A$ and $B$. Let us
denote by $\lambda_\nu$ and $\lambda_{B\nu}$ the
 eigenvalues of $A$ and $B$ respectively.
 So we have
$$
-\beta^2 u_\nu  +Au_\nu + i\beta B u_\nu =w_\nu.
$$
Therefore, we can assume that $u_\nu =Kw_\nu$, with $K\in\mathbb{C}$.
Substitution of $u_\nu$ yields
$$
(-\beta^2   +\lambda_\nu  + i\beta \lambda_{B\nu})K w_\nu =w_\nu.
$$
Taking $\beta^2  =\lambda_\nu$ we obtain that
$$
i\beta \lambda_{B\nu} K  =1\quad\Rightarrow\quad
K:=K_\nu=-i\lambda_{\nu}^{-1/2}\lambda_{B\nu}^{-1},
$$
since
$$
v_\nu = i\beta u_\nu =i\beta K_\nu
w_\nu=-i\lambda_{B\nu}^{-1}w_\nu.
$$
Therefore, 
\begin{equation}\label{espec3}
\|U_\nu \|^2_{\mathcal{H}}=\|A^{1/2}u_\nu \|^2+\|v_\nu
\|^2=2\lambda_{B\nu}^{-2} \quad\Rightarrow\quad \beta_\nu\|U_\nu
\|_{\mathcal{H}}=\sqrt{2}\lambda_\nu^{1/2}\lambda_{B\nu}^{-1}.
\end{equation}
From (H1) we conclude that
\begin{equation}\label{eqivalor}
C_0\lambda_\nu^\alpha\leq \lambda_{B\nu}\leq
C_1\lambda_\nu^\alpha.
\end{equation}
Therefore, if $\alpha<1/2$ we obtain
$$
\beta_\nu\|U_\nu
\|_{\mathcal{H}}\geq c_0\lambda_\nu^{1/2-\alpha}\quad\Rightarrow\quad
\beta_\nu\|U_\nu \|_{\mathcal{H}}\to  \infty
$$
From where our conclusion follows.
\end{proof}


\section{Differentiability}

Our main tool to show differentiability is the following theorem,
 Pazy \cite[Theorem 4.9]{z3LS83}.

\begin{theorem}\label{thm2.1b}
Let $S(t)=e^{At}$ be a
$C_0$-semigroup of contractions on Hilbert space. Then $S(t)$ is
 differentiable if $i\mathbb{R}\subset\rho (A)$ and
\[
\limsup_{|\beta|\to \infty} (\ln|\beta|)\|(i\beta I-A)^{-1}\| < \infty.
\]
\end{theorem}

We use the above result to show that the semigroup $S_B$ is
differentiable when $0<\alpha<1/2$. The differentiability for
$1/2\leq \alpha\leq 1$ is an immediate consequence of the
analyticity.

\begin{theorem}\label{diff}
Suppose that $0<\alpha<1/2$. Then the semigroup
$S_B(t)$ is differentiable.
\end{theorem}

\begin{proof}
To show the above relation, let us consider the spectral equation
$$
i\beta U -A_BU = F.
$$
In terms of the coefficients we have \eqref{esp1}--\eqref{esp2}.
Multiplying  \eqref{esp2} by $\overline{v}$ we obtain
$$
i\beta\|v\|^2+(A^{1/2}u,A^{1/2}v)+\|B^{1/2}v\|^2=(g,v).
$$
Multiplying  \eqref{esp1} by $A\overline{u}$ we obtain
$$
i\beta\|A^{1/2}u\|^2-(A^{1/2}v,A^{1/2}u)=(A^{1/2}f,A^{1/2}u).
$$
Adding  the above equations and taking the real part we obtain
\begin{equation}\label{b1/2v}
\|B^{1/2}v\|^2\leq C\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}}.
\end{equation}
From  (H1) we obtain
\begin{equation}\label{rel0}
\|A^{\alpha/2}v\|^2\leq C\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}}.
\end{equation}
In particular,
\begin{equation}\label{v}
\|v\|^2\leq C \|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}}.
\end{equation}
Multiplying  \eqref{esp2} by $\overline{u}$ we obtain
$$
(i\beta v,u)+\|A^{1/2}u\|^2+(Bv, u)=(g,u).
$$
Using  \eqref{esp1}, we obtain
$$
\|A^{1/2}u\|^2\leq \|v\|^2-(B^{1/2}v, B^{1/2}u)+C\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}}.
$$
Since $\alpha\leq 1/2$, using \eqref{b1/2v}, hypothesis (H1) and
\eqref{v}, we obtain
\begin{equation}\label{rel11}
\|A^{1/2}u\|^2\leq c\|v\|^2+C\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}}\leq C
\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}}.
\end{equation}
From \eqref{v} and \eqref{rel11} we conclude that
\begin{equation}\label{U}
\|U\|_{\mathcal{H}}\leq C \|F\|_{\mathcal{H}}.
\end{equation}
From  \eqref{esp1}, \eqref{rel0} and \eqref{U}
we obtain that
\begin{equation}\label{rel1}
|\beta|\|A^{\alpha/2}u\|\leq \|A^{\alpha/2}v\|+\|F\|_{\mathcal{H}}\leq
C\|F\|_{\mathcal{H}}.  \end{equation} This because $\alpha\leq 1/2$.
Multiplying  \eqref{esp2} by $A^{\gamma}\overline{u}$ we obtain
$$
(i\beta v,A^{\gamma}u)+\|A^{(\gamma+1)/2}u\|^2+(Bv,
A^{\gamma}u)=(g,A^{\gamma}u)
$$
or equivalent
$$
-( A^{\gamma}v,i\beta u)+\|A^{(\gamma+1)/2}u\|^2+(Bv,
A^{\gamma}u)=(g,A^{\gamma}u).
$$
From (H2) we obtain
$$
-( A^{\gamma}v,i\beta u)+\|A^{(\gamma+1)/2}u\|^2+(A^{\alpha/2}Sv,
A^{\gamma+\alpha/2}u)=(g,A^{\gamma}u).
$$
Using \eqref{esp1} we obtain
\begin{equation}
\begin{split}
\|A^{(\gamma+1)/2}u\|^2&\leq  \|A^{\gamma/2}v\|^2+(A^{\alpha/2}S
v,
A^{\gamma+\alpha/2}u)+C\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}} \\
&\leq  \|A^{\gamma/2}v\|^2+(A^{\alpha/2}S v,
A^{\gamma+\alpha/2}u)+C\|F\|^2_{\mathcal{H}}.
\end{split}\label{relA2}
\end{equation}
From the above relation we conclude that our best choice for
$\gamma$ is $\gamma=\alpha$, then we obtain
$$
\|A^{(\alpha+1)/2}u\|^2\leq \|A^{\alpha/2}v\|^2+(A^{\alpha/2}S v,
A^{3\alpha/2}u) +C\|F\|^2_{\mathcal{H}}.
$$
Since $\alpha\leq 1/2$ we obtain
$$
\|A^{3\alpha/2}u\|\leq \|A^{(\alpha+1)/2}u\|
$$
which implies
$$
\|A^{(\alpha+1)/2}u\|^2\leq c\|A^{\alpha/2}v\|^2+C\|F\|^2_{\mathcal{H}}.
$$
From \eqref{rel0} and \eqref{U} we obtain that there exists a
positive constant $C$ such that
\begin{equation}\label{fin1}
\|A^{(\alpha+1)/2}u\|^2\leq C\|F\|^2_{\mathcal{H}}.
\end{equation}
Now we  use interpolation
$$
1/2=\theta (\alpha+1)/2+(1-\theta)\alpha/2 \quad\Rightarrow\quad
\theta=1-\alpha.
$$
Therefore,
$$
\|A^{1/2}u\|\leq c\|A^{(\alpha+1)/2}u\|^{\theta}\|A^{\alpha/2}u\|^{1-\theta}.
$$
Then
$$
|\beta|^{1-\theta}\|A^{1/2}u\|\leq
c\|A^{(\alpha+1)/2}u\|^{\theta}(|\beta|\|A^{\alpha/2}u\|)^{1-\theta}.
$$
So from \eqref{rel1}--\eqref{fin1}  we have
\begin{equation}\label{rel3}
|\beta|^{\alpha}\|A^{1/2}u\|\leq C\|F\|_{\mathcal{H}}^{\theta}\|F\|_{\mathcal{H}}^{1-\theta}\leq C \|F\|_{\mathcal{H}}.
\end{equation}
Applying $A^{(\alpha-1)/2}$ in \eqref{esp2}  we obtain
$$
i\beta A^{(\alpha-1)/2}v+A^{(\alpha+1)/2}u + A^{(\alpha-1)/2}
Bv=A^{(\alpha-1)/2}g.
$$
From hypothesis (H2) the operator $B$ can be written as
$Bv=A^{\alpha}S v$, so we have
$$
|\beta|\| A^{(\alpha-1)/2}v\|\leq
\|A^{(\alpha+1)/2}u\|+\|A^{(3\alpha-1)/2} S v\|+c\|F\|_{\mathcal{H}}.
$$
Since $\alpha-1<0$ and $(3\alpha-1)/2\leq \alpha/2$, provided
$0<\alpha<1/2$,  we obtain that
$$
|\beta|\| A^{(\alpha-1)/2}v\|\leq
c\|A^{(\alpha+1)/2}u\|+c\|A^{\alpha/2}v\|+c\|F\|_{\mathcal{H}}
$$
for $\beta>1$.
From \eqref{rel0}, \eqref{U}  and \eqref{fin1} we obtain
\begin{equation}\label{inter1}
|\beta|\| A^{(\alpha-1)/2}v\|\leq C\|U\|_{\mathcal{H}}^{1/2}\|F\|_{\mathcal{H}}^{1/2}
+c\|F\|_{\mathcal{H}}\leq C\|F\|_{\mathcal{H}}.
\end{equation}
Using interpolation once more,
$$
0=\theta(\alpha-1)/2 +(1-\theta)\alpha/2 \quad\Rightarrow\quad \theta=\alpha,
$$
we obtain
$$
\|v\|\leq
c\|A^{(\alpha-1)/2}v\|^\theta\|A^{\alpha/2}v\|^{1-\theta}.
$$
So we have that
$$
|\beta|^\alpha\|v\|\leq
c(|\beta|\|A^{(\alpha-1)/2}v\|)^\alpha\|A^{\alpha/2}v\|^{1-\alpha}.
$$
From \eqref{rel0}, \eqref{U} and \eqref{inter1} it follows that
\begin{equation}\label{rel5}
|\beta|^\alpha\|v\|\leq c(\|F\|_{\mathcal{H}}
)^\alpha\|F\|_{\mathcal{H}}^{1-\alpha}=C\|F\|_{\mathcal{H}}.
\end{equation}
From relation \eqref{rel3} and \eqref{rel5} we obtain,for $\beta $
large,
$$
 |\beta|^{2\alpha}\|U\|_{\mathcal{H}}^2\leq C\|F\|_{\mathcal{H}}^2.
$$
Therefore our conclusion follows.
\end{proof}

\section{Polynomial rate of decay and optimality}

In this section we prove that the  solution of \eqref{eq1.1} for $\alpha=-\gamma<0$  
decays polynomially to zero as  time approaches infinity. 
We will show that the corresponding energy decays to zero as $t^{-1/\gamma}$. Moreover
we show that this rate of decay is optimal. This result improves
the rates established in \cite{M3z205}.
Our result is based on \cite{SeOptPo}. See also also \cite{Batty,Ammari}.

\begin{theorem}\label{Tomi}
 Let $S(t)$ be a bounded $C_0$-semigroup on a Hilbert space $\mathcal{H}$ with
generator $A$ such that $i\mathbb{R}\subset\varrho(A)$. Then
$$
\frac{1}{|\eta|^\alpha}\|(i\eta I-A)^1\|\leq C,\quad \forall\eta\in\mathbb{R}\quad 
\Leftrightarrow\quad \|S(t)A^{-1}\|\leq \frac{c}{t^{1/\alpha}}
$$
\end{theorem}

To prove polynomial rate of decay we should show that there
exist positive constant $C>0$ independent of $\beta$, $l$  or $f$
such that
\begin{align*}
\sup_{\|f\|\leq 1 }\frac{1}{\beta^l}\|U\|=\sup_{\|f\|\leq 1
}\frac{1}{\beta^l}\|(i \beta I -A)^{-1}f\| \leq C.
\end{align*}

\begin{remark} \label{rmk} \rm
Note that  we can improve the
polynomial rate of decay by improving the regularity of the
initial data, that is
$$
\|S(t)A^{-k}\|\leq \frac{c_k}{t^{k/\alpha}}
$$
for the proof see \cite{p3ME06}. In that sense it is important to remark what
optimality means. The optimality of course will depend on the
domain. So fixing the domain taking $k=1$,
 we prove that the rate $1/\gamma$ can not be improved.
\end{remark}

Under the above conditions we can establish the main result of this section.

\begin{theorem}\label{decpolim} 
Let  $\alpha=-\gamma<0$ be a negative real number where $0< \gamma \leq 1$.
Then  the semigroup $S_B(t)$ decays polynomially to zero as
$$
\|S_B(t)U_0\|\leq C ( \frac{1}{t})^{1/\gamma}\|U_0\|_{D(A)}.
$$
Moreover, when $B$ and $A^{-\gamma}$ have infinite common
eigenvectors the rate $1/\gamma$ can not be improved over $D(A)$.
\end{theorem}

\begin{proof} 
 We consider spectral \eqref{esp1} and  \eqref{esp2} when
$\alpha \in ]-1,0[$ or equivalently
\begin{gather}
i \beta u - v = f \in \mathcal{D}(A) \label{ep1}\\
i \beta v +Au+ B v = g \in H . \label{ep2}
\end{gather}
 Multiplying \eqref{ep2} by $\overline{v}$
and \eqref{ep1} by $A\overline{u}$ summing the product result and taking real 
part we obtain
\begin{equation} \label{eop1}
\|A^{-\gamma/2}v\|^2 \leq C\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}}.
\end{equation}
Multiplying \eqref{ep2} by $A^{-\gamma}\overline{u}$ and
using \eqref{ep1}, we obtain
$$
\|A^{(1-\gamma)/2}u\|^2= \|A^{-\gamma/2}v\|^2-(Bv,
A^{-\gamma}u)+(A^{-\gamma}v,f)+(g,A^{-\gamma}u).
$$
Since $(1-\gamma)/2 >-\gamma$ and using \eqref{eop1} we obtain that
\begin{eqnarray}\label{eop2}
\|A^{(1-\gamma)/2}u\|^2\leq C \|U\|_{\mathcal{H}}\|F\|_{\mathcal{H}}.
\end{eqnarray}
Applying  $A^{-(1+\gamma)/2}$ on \eqref{ep2}, we obtain
$$
i\beta
A^{-(1+\gamma)/2}v+A^{(1-\gamma)/2}u+A^{-(\gamma+1)/2}Bv=A^{-(\gamma+1)/2}g.
$$
Since $-(3\gamma+1)/2< -\gamma/2$ and using \eqref{eop1}-\eqref{eop2} in  the
above identities, it follows that
\begin{equation} \label{eop4}
|\beta|\| A^{-(1+\gamma)/2}v\|
\leq C\|U\|^{1/2}_{\mathcal{H}}\|F\|^{1/2}_{\mathcal{H}}+c\|F\|_{\mathcal{H}}.
\end{equation}
Multiplying \eqref{ep1} by $A^{s}\overline{u}$  where
$s=(1-2\gamma)/2$ we obtain
$$i\beta
\|A^{s/2}u\|^2=(A^{-\gamma/2}v,A^{\gamma/2+s}u)+(f,A^{s}u).
$$
That is,
\begin{equation} \label{eop5}
|\beta|\|A^{(1-2\gamma)/4}u\|^2
\leq C\|U\|_{\mathcal{H}}\|F\|_{\mathcal{H}}+C\|F\|^2_{\mathcal{H}}.
\end{equation}
Applying $A^{\gamma/2}$ on \eqref{ep1} and \eqref{eop1},
we obtain
\begin{equation} \label{eop6}
|\beta|\, \|A^{-\gamma/2}u\|
\leq C\|U\|^{1/2}_{\mathcal{H}}\|F\|^{1/2}_{\mathcal{H}}+c\|F\|_{\mathcal{H}}.
\end{equation}
Using interpolation,
$$
0=\theta(1-2\gamma)/4 -\gamma/2(1-\theta) \quad\Rightarrow\quad
\theta=2\gamma,
$$
we have
$$
\|u\|\leq
c\|A^{-\gamma/2}u\|^{1-\theta}\|A^{(1-2\gamma)/4}u\|^{\theta}.
$$
Now using \eqref{eop5} and \eqref{eop6}, we obtain
\begin{align*}
\|u\|&\leq  \frac{C}{|\beta|^{1-\theta/2}}(\|U\|^{1/2}_{\mathcal{H}}
\|F\|^{1/2}_{\mathcal{H}}+\|F\|^2_{\mathcal{H}})\\
&= \frac{C}{|\beta|^{1-\gamma}}(\|U\|^{1/2}_{\mathcal{H}}\|F\|^{1/2}_{\mathcal{H}}
+\|F\|^2_{\mathcal{H}}).
\end{align*}
From this,
$$
\|v\|\leq |\beta| \|u\|+\|f\|\leq
\frac{C}{|\beta|^{-\gamma}}(\|U\|^{1/2}_{\mathcal{H}}\|F\|^{1/2}_{\mathcal{H}}
+\|F\|_{\mathcal{H}})+\|F\|_{\mathcal{H}}\,,
$$
and so
\begin{equation} \label{eop7}
|\beta|^{-\gamma}\|v\|\leq C(\|U\|^{1/2}_{\mathcal{H}}\|F\|^{1/2}_{\mathcal{H}}
+\|F\|_{\mathcal{H}}).
\end{equation}
Multiplying \eqref{ep2} by $\overline{u}$ and using \eqref{ep1} we obtain
\[
\|A^{1/2}u\|^2=(B v,u)-(i\beta v,u)+(g,u) =(B v,u)+\|v\|^2-(f,u)+(g,u).
\]
From \eqref{eop1} we conclude that
\begin{equation} \label{eop8}
|\beta|^{-2\gamma}\|A^{1/2}u\|^2\leq
|\beta|^{-2\gamma}\|v\|^2+C|\beta|^{-2\gamma}(\|U\|^{3/2}_{\mathcal{H}}\|F\|^{1/2}
_{\mathcal{H}} +\|F\|\|U\|).
\end{equation}
Adding \eqref{eop7} and \eqref{eop8}, we have
$$
|\beta|^{-2\gamma}\|U\|_{\mathcal{H}}^2\leq \beta^{-2\gamma}\|v\|^2+C\beta^{-2\gamma}\|U\|_{\mathcal{H}}\|F\|_{\mathcal{H}}+C\|F\|_{\mathcal{H}}^2.
$$
%%
Applying Young's inequality  in the last term,
$$
\frac{1}{2}|\beta|^{-2\gamma}\|U\|_{\mathcal{H}}^2\leq
C|\beta|^{-2\gamma}\|F\|^2\leq C\|F\|^2.
$$
Therefore, the semigroup is polinomially stable and decay as $t^{-1/\gamma}$ over $D(A)$.
Finally, to show the optimality.  We suppose that the operators $A$ and $B$ have
an infinite eigenvector in common.
As in section \ref{analytic},
we can assume that $u_\nu =Kw_\nu$, with $K\in\mathbb{C}$.
Substitution of $u_\nu$ yields
$$
(-\beta^2   +\lambda_\nu  + i\beta \lambda_{B\nu})K w_\nu =w_\nu.
$$
Taking $\beta^2  =\lambda_\nu-\lambda_\nu^{(1-\gamma)/2}$ we obtain that
 $\beta\approx\lambda_\nu^{1/2}$ and
$$
(\lambda_\nu^{(1-\gamma)/2}+i\beta \lambda_{B\nu}) K  =1\quad\Rightarrow\quad
u_\nu = K_\nu w_\nu=\frac{1}{1+i\beta \lambda_{B\nu}}w_\nu,
$$
since
$$
u_\nu = K_\nu w_\nu=\frac{1}{\lambda_\nu^{(1-\gamma)/2}+i\beta \lambda_{B\nu}}w_\nu.
$$
Then we have
\begin{align*}
\|U_\nu\|_{\mathcal{H}}&\geq
\|A^{1/2}u_\nu\|=\frac{\lambda_\nu^{1/2}}{\sqrt{\lambda_\nu^{1-\gamma}+\beta^2
\lambda_{B\nu}^2}}\\
&\geq
\frac{\lambda_\nu^{1/2}}{\sqrt{\lambda_\nu^{1-\gamma}+c_0\lambda_{\nu}^{1-2\gamma}}}\\
&\geq
\frac{\lambda_\nu^{\gamma/2}}{\sqrt{1+c_0\lambda_{\nu}^{-\gamma}}}.
\end{align*}
Note that $\beta\approx \lambda_\nu^{1/2}$ as $\nu\to\infty$. From where we obtain
$$
\beta^{-\gamma+\epsilon}\|U_\nu\|_{\mathcal{H}}
\geq \frac{\lambda_\nu^{\epsilon}}{\sqrt{1+c_0\lambda_{\nu}^{-1}}}\to\infty
$$
as $\nu\to\infty$. Therefore is not possible to improve the polynomial rate of decay.
\end{proof}


\section{Applications}

Here we apply our result to several models.

\subsection*{Viscoelastic plates}
Let $\Omega $ be a bounded subset of $\mathbb{R}^n$ with smooth
boundary $\partial \Omega$, and consider the model
\begin{gather*}
\varrho u_{tt}+\kappa\Delta^2 u-\gamma \Delta u_{t}= 0\quad \text{in } \Omega\\
u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad \text{in } \partial\Omega\\
u=\Delta u =0 \quad \text{on } \partial\Omega.
\end{gather*}
Here $A=k/\rho\Delta^2$ and $B=\gamma\rho/k(-\Delta)$.

From Theorem \ref{anali} we conclude that the semigroup that
defines the solution of the above system is analytic. So, in
particular we have that the solution decays exponentially to zero
and there exists smoothing effect on the initial data, that is no
matter where the initial data $u_0$ and $u_1$ is, the solution satisfies
 $u\in C^\infty(]0,T[;C^\infty(\Omega))$.

On the other hand, if we consider the inertial term on the plate we obtain 
the model
\begin{gather*}
\varrho u_{tt}-h\Delta u_{tt}+ \kappa\Delta^2 u-\gamma \Delta u_{t}= 0\quad \text{in } \Omega\\
u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad \text{in } \partial\Omega\\
u=\Delta u =0 \quad \text{on }\partial\Omega.
\end{gather*}
Here $A=k(\rho I-h\Delta)^{-1}\Delta^2$, $B=-\gamma(\rho I-h\Delta)^{-1}\Delta$.
 So there exists positive constants $c_1$ and $c_0$ such that
$$
c_1(A^0w,w)\leq (Bw,w)\leq c_0(A^0w,w).
$$
We conclude that the model is neither analytic nor differentiable.
But is exponentially stable.

\subsection*{Mixtures}

We consider a beam composed by a mixture of two viscoelastic interacting
continually that occupies the interval $(0,L)$. The displacement of
typical particles at time $t$ are $u$ and $w$, where 
$u=u(x,t)$ and $w=w(y,t)$, $x,y \in (0,L) $. 
We assume that the particles
under consideration occupy the same position  at time $t=0$, so
that $x=y$.  We denote by $\rho_i$ the mass
density of each constituent at time $t=0$, $T,S$ the partial
stresses associated with the constituents, $P$  the internal
diffusive force.  In the absent of body forces the system of equations
consists of the equations of motion
\begin{equation} \label{eq1.0}
 \rho_1u_{tt} =T_x-P, \quad  \rho_2w_{tt}=S_x+P,
\end{equation}
and the constitutive equations
\begin{equation} \label{eq1.01}
\begin{gathered}
T=a_{11} u_x+a_{12} w_x+b_{11} u_{xt}+b_{12} w_{xt}\\
 S=a_{12} u_x+a_{22} w_x+b_{21} u_{xt}+b_{22} w_{xt}\\
P=\alpha (u-w).
\end{gathered}
\end{equation}
If we substitute the constitutive equations into the motion
equations and the energy equation, we obtain the system of field
equations
\begin{gather*}
 \rho_1u_{tt}- a_{11}u_{xx} -a_{12} w_{xx}+\alpha(u-w)-b_{11} u_{xxt}-b_{12} w_{xxt}=0
\quad \text{ in } (0,\infty)\times(0,L), \\
\rho_2w_{tt}- a_{12}u_{xx} -a_{22} w_{xx}-\alpha(u-w)-b_{21} u_{xxt}-b_{22} w_{xxt}=0
\quad \text{ in } (0,\infty)\times(0,L),
\end{gather*}
We assume that the constants $\rho_1$, $ \rho_2$ , $c$,  and
$\alpha$ are positive, and that the matrix $A=(a_{ij})$,
$B=(b_{ij})$ are symmetric and  positive definite.
\begin{gather*}
u(0)=u_0,\quad u_t(0)=u_1,\quad w(0)=w_0,\quad w_t(0)=w_1,\\
u(t,0)=u(t,L)=w(t,0)=w(t,L)=0.
\end{gather*}
In vectorial notation, the above system can be written as
$$
U_{tt}+AU_{xx}+BU_{xxt}=0,
$$
where $U=(u,w)^t$. Note that
$$
c_1(Aw,w)\leq (Bw,w)\leq c_0(Aw,w).
$$
Therefore, by Theorem \ref{anali} we
conclude that the solution of the mixture model is defined by an
analytic semigroup.

\subsection*{Elasticity}

Let us denote by $\Omega\subset\mathbb{R}^2$ an open bounded set with smooth boundary. 
Let us consider the plate equation
\begin{gather*}
u_{tt}-\Delta u_{tt}+\Delta^2 u+\gamma u_t=0,\quad\text{in } \Omega\times]0,\infty[  \\
u=\Delta u=0 \quad\text{on }\partial\Omega \\
u(0)=u_0,u_t(0)=u_1\quad\text{in }
\partial\Omega.
\end{gather*}
Letting $A=[I-\Delta]^{-1}\Delta^2$ and $B=[I-\Delta]^{-1}$, with  $H$ and 
$D(\Delta)$ being $L^2(\Omega)$ and $H^1_0(\Omega)\cap H^2(\Omega)$ respectively, 
the above model may be written as \eqref{eq1.1}. Note that
$$
c_1(A^{-1}w,w)\leq (Bw,w)\leq c_0(A^{-1}w,w).
$$
Using Theorem \ref{decpolim} we conclude that the corresponding
semigroup decays polynomial as
$$
\|S_B(t)U_0\|_{\mathcal{H}}\leq \frac{C}{t}\|U_0\|_{D(A)}.
$$ 
Where the rate $1/t$ can not be improved over domain of $D(A)$. This result improves
the rate of decay given in \cite{M3z205}.


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\end{document}