text:
\section{Introduction}\label{s1}

First we consider the evolution equation
\begin{equation}\label{11}
\begin{cases}
U'(t)=AU(t)+k(t)BU(t-\tau)\qtq{in}(0,\infty),\\
U(0)=U_0,\\
BU(t-\tau)=f(t)\qtq{for}t\in(0,\tau),
\end{cases}
\end{equation}
where $A$ generates an exponentially stable semigroup $(S(t))_{t\ge 0}$ in a Hilbert space $H$, $B$ is a  continuous linear operator of $H$ into itself, $k:[0,\infty)\to\mathbb R$ is a function belonging to $L^1_{loc}([0,\infty);\mathbb R)$, and $\tau>0$ is a delay parameter.
The initial data $U_0$ and $f$ are taken in $H$ and $C([0,\tau];H)$, respectively.
By the assumptions on the operator $A$ there exist two numbers $M,\omega>0$ such that
\begin{equation}\label{12}
\norm{S(t)}\le Me^{-\omega t}\qtq{for all}t\ge 0.
\end{equation}
Time delay effects often appear in applications to physical models, and it is well-known (\cite{BP, DLP}) that they can induce instability phenomena.
We are interested in showing that under some mild assumptions on $\tau, k$, the operator $B$ and the constants $M, \omega$ the system \eqref{11} is still exponentially stable.

Stability results for the above abstract  model have been recently obtained in \cite{JEE15, NicaisePignotti18} but only  for a constant delay feedback coefficient $k$.
In these papers  some nonlinear extensions are also  considered.
The arguments of \cite{JEE15, NicaisePignotti18} could be easily extended to $k(\cdot)\in L^\infty$ with  sufficiently small $\norm{k}_\infty$.
Recently, motivated by some applications, $L^1_{loc}$ damping coefficients $k$ have been considered, for instance of intermittent type (see \cite{Haraux, Pignotti, FP}), and stability estimates have been obtained for some particular models.
Here, our aim is to give a well-posedness result  and an exponential decay estimate for the general model \eqref{11} with a damping coefficient $k$ belonging only to $L^1_{loc}$.

As a non-trivial generalization, next we consider the case of multiple time-varying delays.
Namely, let $\tau_i:[0, +\infty)\rightarrow (0, +\infty),$ $i=1, \dots, l,$ be the time delays functions belonging to $W^{1,\infty}(0,+\infty)$.
We assume for each  $i=1, \ldots, l,$ that
\begin{equation}\label{13qqq}
0\le \tau_i(t)\le \overline\tau_i
\end{equation}
and
\begin{equation}\label{14qqq}
\tau_i^\prime (t)\le c_i<1.
\end{equation}
for a.e. $t>0$, with suitable constants $\overline\tau_i$ and $c_i$.
It follows  from \eqref{14qqq} that
\begin{equation*}
(t-\tau_i(t))^\prime = 1-\tau_i^\prime (t) >0, \quad \mbox{a.e.}\ t>0,
\end{equation*}
for a.e. $t>0$, and hence
\begin{equation*}
t-\tau_i(t)\ge -\tau_i(0)\qtq{for all}t\ge 0.
\end{equation*}
Therefore, setting
\begin{equation}\label{15qqq}
\tau^*:=\max_{i=1,..., l} \set{\tau_i(0)}
\end{equation}
we may consider the following abstract model:

\begin{equation}\label{16qqq}
\begin{cases}
U'(t)=AU(t)+\sum_{i=1}^lk_i(t)B_iU(t-\tau_i(t))\qtq{in}(0,\infty),\\
U(0)=U_0,\\
B_iU(t)=f_i(t)\qtq{for}t\in [-\tau^*, 0],
\end{cases}
\end{equation}
where the operator $A$, as before, generates an exponentially stable semigroup $(S(t))_{t\ge 0}$ in a Hilbert space $H$, and for each $i=1,\dots, l$,  $B_i$ is a  continuous linear operator of $H$ into itself, $k_i:[0,\infty)\to\mathbb R$ belongs to ${\mathcal L}^1_{loc}([0,+\infty);\mathbb R )$, and $\tau_i$ is a  variable time delay.
Under some mild assumptions on the involved functions and parameters we will establish the  well-posedness of the problem \eqref{16qqq}, and we will obtain exponential decay estimates for its solutions.

The nonlinear version of previous model

\begin{equation}\label{51qqq}
\begin{cases}
U'(t)=AU(t)+\sum_{i=1}^lk_i(t)B_iU(t-\tau_i(t))+F(U(t))\\
\hspace{7,6 cm}\qtq{in}(0,\infty),\\
U(0)=U_0,\\
B_iU(t)=f_i(t)\qtq{for}t\in [-\tau^*, 0],
\end{cases}
\end{equation}
is also analyzed under some Lipschitz continuity assumption on the nonlinear function $F.$ Exponential  decay of the energy  is obtained
for small initial data under a suitable well-posedness assumption. A quite general class of examples satisfying our abstract setting is exhibited.

The paper is organized as follows.
In Sections \ref{s2qqq} and \ref{s3qqq} we prove the well-posedness and the exponential decay estimate for the model \eqref{11} with a single constant time delay.
Next, in Section \ref{s4qqq} we  analyze the more general system \eqref{16qqq} with multiple time-varying time delays. The nonlinear abstract model \eqref{51qqq} is studied in Section \ref{s5qqq}, where we give an exponential decay estimate for small initial data.  Finally, in Section \ref{s6qqq} we  give some applications of our abstract results to concrete models.

\section{Single constant delay: well-posedness}\label{s2qqq}

The following well-posedness result holds true.

\begin{proposition}\label{p21qqq}
Given $U_0\in H$ and a continuous function $f:[0,\tau]\to H$, the problem \eqref{11} has a unique (weak) solution given by Duhamel's formula
\begin{equation}\label{21qqq}
U(t)=S(t)U_0+\int_0^tk(s)S(t-s)BU(s-\tau)\ ds\qtq{for all}t\ge 0.
\end{equation}
\end{proposition}

\begin{proof}
We  proceed step by step by working on time intervals of length $\tau.$
First we consider $t\in [0,\tau]$.
Setting $G_1(t) =k(t)BU(t-\tau),$ $t\in [0,\tau]$ we observe that $G_1(t)= k(t)f(t),$ $t\in [0,\tau]$.
Then, problem \eqref{11} can be rewritten, in the time interval $[0,\tau]$, as a standard inhomogeneous evolution problem:
\begin{equation}\label{22qqq}
\begin{cases}
U'(t)=AU(t)+G_1(t) \qtq{in}(0,\tau),\\
U(0)=U_0.
\end{cases}
\end{equation}

Since $k\in L^1_{loc}([0,\infty);\mathbb R)$ and $f\in C([0,\tau];H)$, we have that $G_1\in L^1((0,\tau);H)$.
Therefore, applying  \cite[Corollary 2.2]{Pazy} there exists a unique solution $U \in C([0,\tau]; H)$ of \eqref{22qqq} satisfying Duhamel's formula
\begin{equation*}
U(t)= S(t)U_0+\int_0^tS(t-s) G_1 (s) ds,\quad t\in [0,\tau]
\end{equation*}
and hence
\begin{equation*}
U(t)= S(t)U_0+\int_0^tk(s) S(t-s) BU (s-\tau) ds,\quad t\in [0,\tau].
\end{equation*}
Next we consider the time interval $[\tau, 2\tau]$.
Setting $G_2(t)=$$k(t)BU(t-\tau)$ we observe that $U(t-\tau)$ is known for $t\in [\tau, 2\tau]$  from the first step.
Then $G_2$ is a known function and it belongs to $L^1((\tau,2\tau); H)$.
So we can rewrite our model \eqref{11} in the time interval $[\tau,2\tau ]$ as the inhomogeneous evolution problem
\begin{equation}\label{23qqq}
\begin{cases}
U'(t)=AU(t)+G_2(t) \qtq{in}(\tau ,2\tau),\\
U(\tau)=U (\tau^-).
\end{cases}
\end{equation}

By the standard theory of abstract Cauchy problems we have a unique continuous solution $U:[\tau, 2\tau)\rightarrow H$ satisfying
\begin{equation*}
U(t)= S(t-\tau )U(\tau^-) +\int_\tau ^tS(t -s) G_2 (s) ds,\quad t\in [\tau ,2\tau],
\end{equation*}
and hence
\begin{equation*}
U(t)= S(t-\tau )U(\tau^-) +\int_\tau ^tk(s) S(t -s) BU (s-\tau) ds,\quad t\in [\tau ,2\tau].
\end{equation*}
Putting together the partial solutions obtained in the first and second steps we get a unique continuous solution $U:[0,2\tau]\rightarrow \mathbb R$
satisfying  Duhamel's formula
\begin{equation*}
U(t)= S(t)U_0+\int_0^tk(s) S(t-s) BU (s-\tau) ds,\quad t\in [0,2\tau].
\end{equation*}

Iterating this argument we find a unique solution $U\in C([0,\infty);H)$ satisfying the representation formula \eqref{21qqq}.
\end{proof}

\section{Single constant delay: exponential stability}\label{s3qqq}

It follows from Duhamel's formula \eqref{21qqq}
 that
\begin{equation}\label{31qqq}
\begin{array}{l}
\displaystyle{e^{\omega t}\norm{U(t)}
\le M\norm{U_0}+Me^{\omega\tau}\int_0^{\tau}\abs{k(s)}e^{\omega (s-\tau)}\norm{f(s)}\ ds}\\
\qquad\displaystyle{
+M\norm{B}e^{\omega\tau}\int_{\tau}^t\abs{k(s)}e^{\omega (s-\tau)}\norm{U(s-\tau)}\ ds}
\end{array}
\end{equation}
for all $t\ge 0$.
Setting
\begin{align*}
&u(t):=e^{\omega t}\norm{U(t)},\\
&\alpha:=M\norm{U_0}+Me^{\omega\tau}\int_0^{\tau}\abs{k(s)}e^{\omega (s-\tau)}\norm{f(s)}\ ds\intertext{and}
&\beta(t):=M\norm{B}e^{\omega\tau}\abs{k(t+\tau)}
\end{align*}
we may rewrite \eqref{31qqq} in the form
\begin{equation*}
u(t)\le \alpha+\int_0^{t-\tau}\beta(s)u(s)\ ds\qtq{for all}t\ge 0.
\end{equation*}

Since $\beta\ge 0$ and $u\ge 0$, it follows that
\begin{equation*}
u(t)\le \alpha+\int_0^{t}\beta(s)u(s)\ ds\qtq{for all}t\ge 0.
\end{equation*}
Applying Gronwall's inequality we conclude that
\begin{equation*}
u(t)\le \alpha e^{\int_0^t\beta(s)\ ds}\qtq{for all}t\ge 0,
\end{equation*}
i.e.,
\begin{equation}\label{32qqq}
\norm{U(t)}\le \alpha e^{\int_0^t\beta(s)\ ds-\omega t}\qtq{for all}t\ge 0.
\end{equation}

This estimate yields the following result:

\begin{theorem}\label{t31qqq}
Assume that there exist two constants  $\omega'\in(0,\omega)$ and $\gamma\in\mathbb R$ such that
\begin{equation}\label{33qqq}
M\norm{B}e^{\omega\tau}\int_0^t \abs{k(s+\tau)}\ ds\le\gamma+\omega't\qtq{for all}t\ge 0.
\end{equation}
Then there exists a constant $M'>0$ such that the solutions of \eqref{11} satisfy the estimate
\begin{equation}\label{34qqq}
\norm{U(t)}\le M'e^{-(\omega-\omega') t}\qtq{for all}t\ge 0.
\end{equation}
\end{theorem}

\begin{proof}
Using our previous notation $\beta(s)=M\norm{B}e^{\omega\tau}\abs{k(s+\tau)}$ we have
\begin{equation*}
\int_0^t\beta(s)\ ds-\omega t\le\gamma-(\omega-\omega') t\qtq{for all}t\ge 0.
\end{equation*}
Combining this with \eqref{32qqq} the estimate \eqref{34qqq} follows with $M':=\alpha e^{\gamma}$.
\end{proof}

\begin{remark}
The hypothesis \eqref{33qqq} is satisfied, in particular,
if the feedback coefficient $k$ belongs to $L^\infty (0,\infty)$
and
\begin{equation*}
M\norm{B}\cdot\norm{k}_\infty e^{\omega\tau} < \omega.
\end{equation*}
It is also satisfied if $k\in L^1(0,\infty)$ or, more generally, if $k=k_1+k_2$ with $k_1\in L^1(0,\infty)$ and $k_2\in L^\infty (0,\infty)$ with a sufficiently small norm $\norm{k_2}_\infty$.
Thus Proposition \ref{t31qqq} extends the results obtained in
\cite{NicaisePignotti18}, in the linear setting, for constant $k$.
\end{remark}

\section{Time variable delays}\label{s4qqq}

Let us now consider the model \eqref{16qqq} with multiple time varying delays.
First, we study its well-posedness.

\begin{theorem}\label{t41}
Given $U_0\in H$ and  continuous functions $f_i:[-\tau^*, 0]\to H$, $i=1,\dots,l,$  the problem \eqref{16qqq} has a unique (weak) solution given by Duhamel's formula
\begin{equation}\label{41qqq}
U(t)=S(t)U_0+\int_0^tS(t-s)\sum_{i=1}^lk_i(s)B_iU(s-\tau_i(s))\ ds,
\end{equation}
for all $t\ge 0.$
\end{theorem}

\begin{proof}
First we consider the case  with time delays functions $\tau_i(\cdot)$  bounded  from below by some positive constants, i.e., there exists for each $i=1,\dots,l$ a constant $\underline{\tau}_i>0$  such that
\begin{equation*}
\tau_i(t)\ge\underline{\tau}_i\qtq{for all}t\ge 0.
\end{equation*}
Then we may argue step by step, as in the proof of Proposition \ref{p21qqq}, by restricting ourselves each time to a time interval of length
\begin{equation*}
\tau_{min}:=\min \set{\underline {\tau}_i\ :\ i=1, \dots, l}>0.
\end{equation*}
Indeed, we infer from the assumption \eqref{13qqq} that
\begin{equation*}
t-\tau_i(t) \le t-\underline\tau_i\le t-\tau_{min}
\end{equation*}
for all $t\ge 0$ and $i=1,\dots,l$.
Therefore, if $t\in [k \tau_{min}, (k+1)  \tau_{min}],$ then $t-\tau_i(t) \le k\tau_{min}$ for all $i=1,\dots, l$.

Now, we pass to the general case.
For each fixed positive number $\epsilon\le 1$ we set
\begin{equation}\label{42qqq}
\tau_i^\epsilon (t):=\tau_i(t)+\epsilon,\quad t\ge0, \quad i=1,\dots, l.
\end{equation}
Moreover, we extend the initial data $f_i$ to continuous functions $\tilde f_i: [-\tau^* -1, 0]\rightarrow H$ with the constant $\tau^*$ defined in \eqref{15qqq}.

Since $\tau_i^\epsilon(t)\ge \epsilon >0$ for all $t$ and $i$, the corresponding model
\eqref{16qqq} with initial data $U_0,$ $\tilde f_i$ and time delay functions $\tau^{\epsilon}_i(\cdot)$  has a unique solution $U_\epsilon (\cdot)\in C[0, +\infty)\rightarrow H$ satisying the representation formula \eqref{41qqq}.

Now consider two positive parameters $\epsilon_1,\epsilon_2\le 1$.
Since $U_{\epsilon_1}(\cdot)$ and $U_{\epsilon_2}(\cdot)$ satisfy \eqref{41qqq}, we have
\begin{multline*}
U_{\epsilon_1}(t)- U_{\epsilon_2}(t) =\\
\int_0^t S(t-s) \sum_{i=1}^l k_i(s)\left [ B_iU_{\epsilon_1}(s-\tau_i^{\epsilon_1}(s))- B_iU_{\epsilon_2}(s-\tau_i^{\epsilon_2}(s))
 \right ]\, ds
\end{multline*}
and hence
\begin{multline*}
U_{\epsilon_1}(t)- U_{\epsilon_2}(t) =\\
\int_0^t S(t-s) \sum_{i=1}^l k_i(s)\left [ B_iU_{\epsilon_1}(s-\tau_i^{\epsilon_1}(s))- B_iU_{\epsilon_2}(s-\tau_i^{\epsilon_1}(s))
 \right ]\, ds\\
 + \int_0^t S(t-s) \sum_{i=1}^l k_i(s)\left [ B_iU_{\epsilon_2}(s-\tau_i^{\epsilon_1}(s))- B_iU_{\epsilon_2}(s-\tau_i^{\epsilon_2}(s))
 \right ]\, ds.
\end{multline*}
It follows that
\begin{equation}\label{43qqq}
e^{\omega t}\Vert U_{\epsilon_1}(t)- U_{\epsilon_2}(t)\Vert  \le M(I_1+I_2)
\end{equation}
with
\begin{equation*}
I_1:=\sum_{i=1}^l\int_{\tau_i^{\epsilon_1}(0)}^t e^{-\omega s}  \vert k_i\vert \vert B_i\Vert\,\Vert U_{\epsilon_1}(s-\tau_i^{\epsilon_1}(s))- U_{\epsilon_2}(s-\tau_i^{\epsilon_1}(s))
 \Vert\, ds
\end{equation*}
and
\begin{equation*}
I_2:=\sum_{i=1}^l \int_0^t e^{-\omega s}  \vert k_i\vert \Vert B_i\Vert\,\Vert  B_iU_{\epsilon_2}(s-\tau_i^{\epsilon_1}(s))- B_iU_{\epsilon_2}(s-\tau_i^{\epsilon_2}(s))
 \Vert \, ds.
\end{equation*}

Using the changes of variable
\begin{equation*}
s-\tau_i^{\epsilon_1}(s)=\sigma
\end{equation*}
in the integrals in the sum $I_1$ and using the notation
\begin{equation}\label{44qqq}
\varphi_i(s):=s-\tau_i(s)=\epsilon_1+\sigma
\end{equation}
we obtain the estimate
\begin{equation}\label{45qqq}
I_1\le \sum_{i=1}^l \frac {e^{\omega (\overline\tau_i+1)}}{1-c_i}\Vert B_i\Vert \int_0^t e^{\omega s} \vert k_i(\varphi_i^{-1}(s+\epsilon_1)\vert \Vert U_{\epsilon_1}(s)- U_{\epsilon_2}(s)\Vert \,
 ds.
\end{equation}

Furthermore, since $U_{\epsilon_2}(\cdot)\in C([0,+\infty);H)$ is locally uniformly continuous and
\begin{equation*}
\tau_i^{\epsilon_1}(t)- \tau_i^{\epsilon_2}(t)=\epsilon_1-\epsilon_2
\end{equation*}
for all $t$ and $i$, we have for every fixed $T>0$ the estimate
\begin{equation}\label{46qqq}
I_2 \le C(T; \epsilon_1-\epsilon_2),
\end{equation}
with a constant $C(T;\epsilon_1-\epsilon_2)$ tending to zero as $\epsilon_1-\epsilon_2\to 0.$

Using \eqref{45qqq} and \eqref{46qqq} in \eqref{43qqq} and applying Gronwall's Lemma for each fixed $T>0,$ we  conclude that
for $\epsilon\rightarrow 0$ the functions $U_\epsilon(\cdot)$ converge locally uniformly to a function $U\in C([0,+\infty); H)$ which satisfies \eqref{41qqq}. This completes the proof.
\end{proof}

Under an appropriate relation between the problem's parameters the system \eqref{16qqq} is exponentially stable:

\begin{theorem}\label{t42qqq}
Assume that there exist two constants $\omega'\in(0,\omega)$ and $\gamma\in\mathbb R$ such that
\begin{equation}\label{47qqq}
M \sum_{i=1}^l\frac {e^{\omega\overline \tau_i}}{1-c_i}\Vert B_i\Vert \int_0^t\abs{k_i(\varphi_i^{-1}(s))} \ ds\le \gamma+\omega^\prime t \quad\mbox{for all}\quad t\ge 0.
\end{equation}
Then there exists a constant $M'>0$ such that the solutions of \eqref{16qqq} satisfy the estimate
\begin{equation}\label{48qqq}
\norm{U(t)}\le M'e^{-(\omega-\omega') t}\qtq{for all}t\ge 0.
\end{equation}
\end{theorem}

\begin{proof} It follows from  Duhamel's formula \eqref{41qqq} that
\begin{align*}
e^{\omega t}\norm{U(t)}
&\le M\norm{U_0}+M\sum_{i=1}^l \int_0^{t}e^{\omega s}\vert k_i(s)\vert  \,\Vert B_i U(s-\tau_i)\Vert\ ds\\
&\le M\norm{U_0}
+M\sum_{i=1}^l e^{\omega\overline\tau_i}\int_{0}^t e^{\omega (s-\tau_i(s))}\vert k_i(s)\vert \,\Vert B_i U(s-\tau_i)\Vert\ ds
\end{align*}
for all $t\ge 0$.

Now we make the change of variable  $\varphi_i(s):= s-\tau_i(s)=\sigma$ for every $i=1,\dots, l$.
Note that the functions $\varphi_i(\cdot) $ are invertible by \eqref{14qqq}.
We have the estimates
\begin{multline*}
\int_0^t e^{\omega (s-\tau_i(s))}\vert k_i(s)\vert  \,\Vert B_i U(s-\tau_i)\Vert\ ds\\
\le \frac 1 {1-c_i}\int_{-\tau_i(0)}^{t-\tau_i(t)} e^{\omega\sigma}\vert k_i(\varphi_i^{-1}(\sigma)\vert\,\Vert B_i U(\sigma )\Vert\ d\sigma
\end{multline*}
and hence
\begin{multline*}
e^{\omega t}\norm{U(t)}
\le M\norm{U_0}+M\sum_{i=1}^l \frac {e^{\omega\overline\tau_i}} {1-c_i} \int_{-\tau_i(0)}^{0}e^{\omega s}\vert k_i(\varphi_i^{-1}(s))\vert  \,\Vert f_i(s)\Vert\ ds\\
+M\sum_{i=1}^l \frac {e^{\omega\overline\tau_i}}{1-c_i}\int_{0}^t e^{\omega s}\vert k_i(\varphi_i^{-1}(s))\vert \,\Vert B_i \Vert \norm{U(s)}\ ds
\end{multline*}
for all $t\ge 0$.
Setting
$\tilde u(t):=e^{\omega t}\norm{U(t)}$ and
\begin{equation}\label{49qqq}
\tilde \alpha:=M\Big  (\norm{U_0}+
\sum_{i=1}^l \frac {e^{\omega\overline\tau_i}} {1-c_i} \int_{-\tau_i(0)}^{0}e^{\omega s}\vert k_i(\varphi_i^{-1}(s))\vert  \,\Vert f_i(s)\Vert\ ds
\Big ),
\end{equation}
\begin{equation}\label{410qqq}
\tilde \beta (t)=M \sum_{i=1}^l \frac {e^{\omega\overline\tau_i}}{1-c_i}\vert k_i(\varphi_i^{-1}(s))\vert \Vert B_i\Vert,
\end{equation}
we may rewrite the above estimate in the form
\begin{equation*}
\tilde u(t)\le \tilde \alpha+\int_0^{t}\tilde \beta(s)\tilde u(s)\ ds\qtq{for all}t\ge 0.
\end{equation*}
Applying Gronwall's inequality we conclude that
\begin{equation*}
\tilde u(t)\le \tilde \alpha e^{\int_0^t\tilde \beta(s)\ ds}\qtq{for all}t\ge 0,
\end{equation*}
i.e.,
\begin{equation}\label{411qqq}
\norm{U(t)}\le \alpha e^{\int_0^t\tilde \beta(s)\ ds-\omega t}\qtq{for all}t\ge 0.
\end{equation}
Now we can conclude as in the proof of Proposition \ref{t31qqq} provided that \eqref{47qqq} is satisfied.
\end{proof}

\begin{remark}
The hypothesis \eqref{47qqq} is satisfied in particular if the feedback coefficients $k_i, i=1,\dots, l,$ belong to $L^\infty (0,+\infty )$ and
\begin{equation*}
M \sum_{i=1}^l\frac {e^{\omega\overline \tau_i}}{1-c_i}\Vert B_i\Vert\,\Vert k_i\Vert_\infty <\omega.
\end{equation*}
It is also satisfied if $k_i(\varphi^{-1}_i(\cdot ))\in L^1(0,+\infty),$ $i=1,\ldots, l,$ or, more generally if $k_i\circ\varphi^{-1}_i =k_i^1+k_i^2$ with $k_i^1\in L^1(0,+\infty)$ and $ k_i^2\in L^\infty (0,+\infty )$ with sufficiently small norms $\Vert k_i^2\Vert_\infty$.
\end{remark}

\section{A nonlinear model}\label{s5qqq}

We now consider the nonlinear model
\eqref{51qqq},
where the operator $A$, as before, generates an exponentially stable semigroup $(S(t))_{t\ge 0}$ in a Hilbert space $H$, and for each  $i=1,\dots, l,$
\begin{itemize}
\item $B_i$ is a  continuous linear operator of $H$ into itself,
\item $k_i\in {\mathcal L}^1_{loc}([0,+\infty);\mathbb R )$,
\item $\tau_i$ is a  variable time delay satisfying \eqref{13qqq} and \eqref{14qqq},
\item $f_i\in C([-\tau^*,0]; H)$ with  $\tau^*$ defined in  \eqref{15qqq}.
\end{itemize}
Furthermore, the nonlinear function $F$ satisfies some local Lipschitz assumption as precised below.

We will prove an exponential stability result for {\em small} initial data under a suitable well-posedness assumption. Then, we will give a class of examples for which this assumption is satisfied.

Concerning $F$ we assume that $F(0)=0$, and that the following local Lipschitz condition is satisfied: for each constant $r>0$ there exists a constant $L(c)>0$  such that
\begin{equation}\label{52qqq}
\norm{F(U)-F(V)}_H\le L(r)\norm{U-V}_H
\end{equation}
whenever $\norm{U}_H\le r$ and  $\norm{V}_H\le r$.

Furthermore, we assume tha the system is well posed in the following sense:

\begin{wpa}
There exist two constants $\omega'\in(0,\omega)$ and $\gamma\in\mathbb R$ such that
\eqref{47qqq} is satisfied.
Furthermore, there exist two positive constants $\rho$ and $C_{\rho}$ with $L(C_\rho ) <\frac {\omega -\omega'}M$ such that if $U_0\in H$ and $f_i\in C([0,\tau^*];H)$ for $i=1, \ldots,l$ satisfy the inequality
\begin{equation}\label{53qqq}
\norm{U_0}^2_H+ \sum_{i=1}^l \int_0^{\tau^*}\abs{k_i(s)} \cdot\norm{f_i(s)}_H^2\ ds < \rho^2,
\end{equation}
then \eqref{51qqq} has a unique solution $U\in C([0,+\infty );H)$ satisfying $\norm{U(t)}\le C_\rho$ for all $t>0$.
\end{wpa}

\begin{theorem}\label{t51qqq}
Under the above well-posedness assumption,
there exists a constant $\tilde M>0$ such that all these solutions satisfy the estimate
\begin{equation*}
\norm{U(t)}\le \tilde M e^{-(\omega-\omega'-ML(C_\rho)) t}\qtq{for all}t\ge 0.
\end{equation*}
\end{theorem}

\begin{proof} By  Duhamel's formula \eqref{41qqq} we have
\begin{multline*}
e^{\omega t}\norm{U(t)}
\le M\norm{U_0}
+M\sum_{i=1}^l e^{\omega\overline\tau_i}\int_{0}^t e^{\omega (s-\tau_i(s))}\abs{k_i(s)} \cdot\norm{B_i U(s-\tau_i)}\ ds\\
+M L(C_\rho) \int_0^t e^{\omega s} \norm{U(s)} \, ds
\end{multline*}
for all $t\ge 0$.

Now, as before, we make the change of variable  $\varphi_i(s):= s-\tau_i(s)=\sigma$ for every $i=1,\dots, l$ to get the estimate
\begin{multline*}
e^{\omega t}\norm{U(t)}
\le M\norm{U_0}+M\sum_{i=1}^l \frac {e^{\omega\overline\tau_i}} {1-c_i} \int_{-\tau_i(0)}^{0}e^{\omega s}\vert k_i(\varphi_i^{-1}(s))\vert\cdot\Vert f_i(s)\Vert\ ds\\
+M\sum_{i=1}^l \frac {e^{\omega\overline\tau_i}}{1-c_i}\int_{0}^t e^{\omega s}\vert k_i(\varphi_i^{-1}(s))\vert\cdot\Vert B_i \Vert\cdot\norm{U(s)}\ ds\\
 +M L(C_\rho) \int_0^t e^{\omega s} \norm{U(s)} \, ds\hspace{3 cm}
\end{multline*}
for all $t\ge 0$.
Setting $\tilde u(t)=e^{\omega t}\Vert U(t)\Vert$
we may rewrite it in the form
\begin{equation*}
\tilde u(t)\le \tilde \alpha+\int_0^{t}\tilde \beta(s)\tilde u(s)\ ds+M L(C_\rho) \,\int_0^t\tilde u(s)\, ds \qtq{for all}t\ge 0,
\end{equation*}
where $\tilde \alpha, \tilde \beta$ are defined as in \eqref{49qqq} and \eqref{410qqq}.
Applying Gronwall's inequality we conclude that
\begin{equation*}
\tilde u(t)\le \tilde \alpha e^{\int_0^t \left [\tilde \beta(s)+ ML(C_\rho)\right ]\ ds}\qtq{for all}t\ge 0,
\end{equation*}
i.e.,
\begin{equation}\label{54}
\norm{U(t)}\le \tilde\alpha e^{\int_0^t\tilde \beta(s)\ ds +M L(C_\rho)t -\omega t}\qtq{for all}t\ge 0.
\end{equation}
Now we can conclude as in the proof of Theorem \ref{t31qqq} provided that \eqref{47qqq} is satisfied.
\end{proof}

Now we consider a class of examples for which the above well-posedness assumption is satisfied.

Let $W$ be a real Hilbert space and  $A_0: {\mathcal D}(A_0)\rightarrow W$  a positive self-adjoint operator with a compact inverse in $W$.
We denote by $\tilde W:=  {\mathcal D}(A_0^{1/2})$ the domain of the operator $A_0^{1/2}.$
Furthermore, let $W_0, \ldots, W_l$  be  real Hilbert spaces
and  $C:W_0\rightarrow W,$ $D_i:W\rightarrow W_i,$ bounded linear operators such that
\begin{equation}\label{55}
\Vert D_i^*u\Vert_{W_i}^2\le d_i\Vert u\Vert_W^2
\end{equation}
and
\begin{equation}\label{56}
a\Vert u\Vert^2_{W_i}\le \Vert C^*u\Vert^2_{W_0}
\end{equation}
for all $u\in W$ and $i=1,\dots, l$, with
suitable positive constants $d_i$ and $a$.

Furthermore, let $ \Psi : \tilde W\rightarrow \mathbb R$ be a functional having a G\^{a}teaux derivative $D \Psi (u)$ in every $u\in \hat W.$
In the same spirit as in \cite{ACS}, we assume the following:

\begin{enumerate}[\upshape (i)]
\item For every $u\in\tilde W$ there is a constant $c(u)$ such that
\begin{equation*}
\vert D \Psi (u) (v)\vert\le c(u)\Vert v\Vert_W,\quad \forall\ v\in\tilde W,
\end{equation*}
where $D\Psi (u)$ is the G\^ateaux derivative of the functional $\Psi$ at $u.$ Thus, $\Psi$ can be extended to the whole $W$ and we denote by $\nabla\Psi (u)$ the unique vector representing $D\Psi (u)$ in the Riesz isomorphism, namely
\begin{equation*}
\langle \nabla\Psi (u), v\rangle_W=D\Psi (u) (v),\quad \forall\ v\in W.
\end{equation*}
\item For all $r>0$ there exists a constant $L(r)>0$ such that
\begin{equation*}
\Vert \nabla\Psi (u)-\nabla\Psi (v)\Vert_W\le L(r)\Vert A_0^{1/2}(u-v)\Vert_W,
\end{equation*}
for all $u,v\in \hat W$ satisfying $\Vert  A_0^{1/2}u\Vert_W\le r$ and $\Vert  A_0^{1/2}v\Vert_W\le r.$
\item $\Psi (0)=0, \nabla\Psi (0)=0,$ and there exists an increasing continuous function $ h$ such that, $\forall \ u \in \tilde W,$
\begin{equation*}
\Vert \nabla \Psi (u)\Vert_W \le h(\Vert A_0^{1/2}u\Vert_W)\Vert A_0^{1/2}u\Vert_W
\end{equation*}
for all  $\ u \in \tilde W$.
\end{enumerate}

In this framework, let us consider the following second-order model:

\begin{equation}\label{57}
\begin{cases}
u_{tt}+ A_0 u + C C^* u_t=\nabla \Psi (u) +\sum_{i=1}^l k_iD_iD_i^*u_t (t-\tau_i(t)), \\
\hspace{7,6 cm}\qtq{in}(0,\infty),\\
u(0)=u_0,\ u_t(0)=u_1, \\
D_i^*u_t(t)=g_i(t)\qtq{for}t\in [-\tau^*, 0],\ i=1,\dots, l,
\end{cases}
\end{equation}
with $(u_0,u_1)\in \tilde W\times W,$ $g_i \in C([-\tau^*,0]; W).$ Here,  the functions $k_i(\cdot),$ $\tau_i(\cdot),$  and the constant $\tau^*$ as defined as before.
In order to
 deal with the nonlinear model we assume
\begin{equation*}
\tau_i(t)\ge\underline{\tau}_i,\quad i=1,\dots,l,
\end{equation*}
and let us set
\begin{equation}\label{58}
\tau_{min}=\min\  \{\, \underline{\tau}_i\,:\, i=1,\dots,l\,\}.
\end{equation}

If we denote $v:=u_t$ and $U:= (u,v)^T,$ \eqref{57} can be recast in the more abstract form \eqref{51qqq} where operator $A$ is defined in $H:= \tilde W\times W$ by
\begin{equation*}
A:=
\begin{pmatrix}
0&1\\
-A_0&-CC^*
\end{pmatrix}
,
\end{equation*}
while $F$ and $B_i$ for $i=1,\dots, l$ are defined by
\begin{equation*}
F(U):=(0,\nabla \Psi (u))^T
\qtq{and}
B_i U:=(0, D_iD_i^*v)^T.
\end{equation*}
Under some conditions on the damping operator $CC^*$ (see, e.g., \cite{BLR} or \cite[Chapter 5]{Komornikbook}) we know that $A$ generates an exponentially stable  semigroup.
Moreover, the above assumptions on $\Psi$  imply that $F(0)=0$ and $F$ satisfies \eqref{52qqq}.

We define the energy functional for the model \eqref{57} as
\begin{multline*}
E(t):= E(t, u(\cdot ))= \frac 12 \Vert u_t\Vert_W^2 +\frac 12 \Vert A_0^{1/2}u\Vert_W^2-\Psi (u)\\
+\frac 12\sum_{i=1}^l \frac 1 {1-c_i}\int_{t-\tau_i(t)}^t \vert k_i(\varphi_i^{-1}(s))\vert\cdot \Vert D_i^*u_t(s)\Vert^2_{W_i} ds.
\end{multline*}
We are going to show that the problem \eqref{57} satisfies the well-posedness assumption and the exponential decay estimate of Theorem \ref{t51qqq} for {\em small} initial data under a suitable compatibility condition between the functions $k_i$ and the constant $a$ in \eqref{56}.
We need a preliminary lemma.

\begin{lemma}\label{l52qqq}
Assume that  $k_i(t)=k^1_i(t) + k^2_i(t)$ with
$k^1_i\in L^1([0, +\infty))$ and $k^2_i\in L^\infty (0,+\infty)$ for $i=1,\dots ,l$. Furthermore, assume that
\begin{equation}\label{59}
\Vert k^2_i\Vert_\infty  \le \frac {2a} l\cdot\frac {1-c_i}{2-c_i},\quad i=1,\dots,l.
\end{equation}
Then, for any solution $u$ of problem \eqref{57}, defined on $[0, T)$ for some $T>0$, and satisfying $E(t)\ge \frac 1 4 \Vert u_t(t)\Vert^2$ for all $t\in [0, T),$  we have
\begin{equation}\label{510}
E(t)\le\bar{C} E(0)\qtq{for all}t\in [0,T)
\end{equation}
with
\begin{equation}\label{511}
\bar{C}= e^{2\sum_{i=1}^l d_i\int_0^{+\infty}
( \frac 1 {1-c_i} \vert k_i^1(\varphi_i^{-1}(s))\vert + \vert k_i^1(s)\vert )\ ds
}.
\end{equation}
\end{lemma}

\begin{proof}
Differentiating, we have
\begin{align*}
E^\prime(t)
&= -\Vert C^*u_t(t)\Vert_{W_0}^2 +\sum_{i=1}^lk_i(t)\langle D_i^*u_t(t), D_i^* u_t(t-\tau_i(t))\rangle \\
&\qquad +\sum_{i=1}^l \frac {\vert k_i(\varphi^{-1}(t)\vert }{2 (1-c_i)}\Vert D_i^*u_t(t)\Vert^2_{W_i}\\
&\qquad -\sum_{i=1}^l \frac {\vert k_i(t)\vert }{2 (1-c_i)}(1-\tau_i^\prime(t))\Vert D_i^*u_t(t-\tau_i(t))\Vert^2_{W_i}.
\end{align*}
Recalling \eqref{14qqq} and using the
 Cauchy--Schwarz inequality hence we infer that
\begin{align*}
E^\prime(t)
&\le  -\Vert C^*u_t(t)\Vert_{W_0}^2 +\frac 1 2 \sum_{i=1}^l \Big ( \frac 1 {1-c_i} \vert k_i(\varphi^{-1}(t))\vert +\vert k_i(t)\vert    \Big )\Vert D_i^*u_t(t)\Vert_{W_i}^2\\
&\le -\Vert C^*u_t(t)\Vert_{W_0}^2 + \frac 1 2 \sum_{i=1}^l \Big ( \frac 1 {1-c_i} \vert k_i^2(\varphi^{-1}(t))\vert +\vert k_i^2(t)\vert    \Big )\Vert D_i^*u_t(t)\Vert_{W_i}^2\\
&\qquad +\frac 1 2 \sum_{i=1}^l \Big ( \frac 1 {1-c_i} \vert k_i^1(\varphi^{-1}(t))\vert +\vert k_i^1(t)\vert    \Big )\Vert D_i^*u_t(t)\Vert_{W_i}^2,
\end{align*}
and then, using \eqref{56} and \eqref{59}, that
\begin{equation}\label{512}
E^\prime (t) \le \frac 1 2 \sum_{i=1}^l  \Big ( \frac 1 {1-c_i}\vert k_i^1(\varphi_i^{-1}(t)\vert +\vert k_i^1(t)\vert \Big   )\, \Vert D_i^* u_t(t)\Vert_{W_i}^2\,.
\end{equation}
From \eqref{512}, recalling \eqref{55}, we obtain
\begin{equation*}
E(t)\le E(0)+\frac 14 \int_0^t K(s) \Vert u_t(s)\Vert^2\, ds
\end{equation*}
with the notation
\begin{equation*}
K(t):=2\sum_{i=1}^l d_i \Big (\frac 1 {1-c_i} \vert k_i^1(\varphi_i^{-1}(t))\vert +\vert k_i^1(t)\vert \Big )\,.
\end{equation*}
Now Gronwall's inequality yields
\begin{equation*}
E(t)\le E(0) e^{\int_0^t K(s)\ ds },
\end{equation*}
proving \eqref{510} with $\bar{C}$ defined by \eqref{511}.
\end{proof}
\begin{proposition}\label{p53qqq}
Assume that  $k_i(t)=k^1_i(t) + k^2_i(t), i=1,\dots ,l,$ with
$k^1_i\in L^1([0, +\infty))$ and $k^2_i\in L^\infty (0,+\infty).$
Moreover, assume \eqref{59}.
Then, the model \eqref{57} satisfies the well posedness assumption of Theorem \ref{t51qqq}.
\end{proposition}

\begin{proof}
First we restrict ourselves to the time interval $[0, \tau_{min}]$  where $\tau_{min}$ is the constant defined in \eqref{58}.
In such an interval the model can be rewritten in the abstract form

\begin{equation}\label{513}
\begin{cases}
U'(t)=AU(t)+\sum_{i=1}^lk_i(t) G_i(t)+F(U(t))\qtq{in}(0,\infty),\\
U(0)=U_0,
\end{cases}
\end{equation}
where $G_i(t)=(0, g_i(t-\tau_i(t))), $ $i=1,\dots,l.$
Then one can apply the classical theory of nonlinear semigroup to deduce the existence of a unique mild solution defined on a maximal interval $[0, \delta )$ with $\delta\le\tau_{min}.$
We will show that for suitably small initial data the solution is global and it satisfies a certain bound.
Our argument is inspired by \cite{ACS} but here additional difficulties appear due to the fact that, being $k_i(\cdot)$ variable in time, we do not have a decreasing energy.  First we observe that if $h\left ( \Vert A_0^{1/2}u_0\Vert_W \right ) <\frac 12,$ then $E(0)>0.$
Indeed,  we deduce from the assumption (iii) on $\Psi$ that
\begin{multline}\label{514}
\vert \Psi(u) \vert\le \int_0^1\vert \langle \nabla \Psi (su), u\rangle\vert\, ds\\
\le
\Vert A_0^{1/2} u\Vert_W^2\, \int_0^1 h (s\Vert A_0^{1/2} u\Vert_W ) s\, ds
\le \frac 12 h (\Vert A_0^{1/2} u\Vert_W  ) \Vert A_0^{1/2} u\Vert^2_W
\end{multline}
Then we have the estimate
\begin{align*}
E(0)&= \frac 12 \Vert u_1\Vert_W^2 +\frac 12 \Vert A_0^{1/2} u_0\Vert_W^2 -\Psi (u_0)\\
&\qquad +\frac 1 2\sum_{i=1}^l \frac 1 {1-c_i} \int_{-\tau_i(0)}^0 \vert k_i(\varphi_i^{-1}(s))\vert\, \Vert D_i^*u_t(s)\Vert^2_{W_i}\ ds\\
&\ge\frac 12 \Vert u_1\Vert_W^2 +\frac 14 \Vert A_0^{1/2} u_0\Vert_W^2\\
&\qquad \frac 1 2\sum_{i=1}^l \frac 1 {1-c_i} \int_{-\tau_i(0)}^0 \vert k_i(\varphi_i^{-1}(s))\vert\, \Vert D_i^*u_t(s)\Vert^2_{W_i}\ ds>0.
\end{align*}

Now  we prove that if
\begin{equation}\label{515}
h\left ( \Vert A_0^{1/2}u_0\Vert_W \right ) <\frac 12 \quad \mbox{and}\quad h\left ( 2 \bar{C}^{1/2}E^{1/2}(0) \right ) <\frac 12,
\end{equation}
where
$\bar{C}$ is the constant defined in \eqref{511},
then
\begin{multline}\label{516}
E(t)> \frac 14 \Vert u_t(t)\Vert_W^2 +\frac 14 \Vert A_0^{1/2} u(t)\Vert^2_W\\
+\frac 1 4\sum_{i=1}^l \frac 1 {1-c_i} \int_{t-\tau_i(t)}^t \vert k_i(\varphi_i^{-1}(s))\vert\, \Vert D_i^*u_t(s)\Vert^2_{W_i}\ ds
\end{multline}
for all $t\in [0,\delta )$.
Indeed, let $r$ be the supremum of all $s\in [0,\delta )$ such that \eqref{516} holds true for every $t\in [0,s].$
Arguing by contradiction, suppose that$r<\delta.$
Then by continuity we have
\begin{multline}\label{517}
E(r)= \frac 14 \Vert u_t(r)\Vert_W^2 +\frac 14 \Vert A_0^{1/2} u(r)\Vert^2_W\\
+\frac 1 4\sum_{i=1}^l \frac 1 {1-c_i} \int_{r-\tau_i(r)}^r \vert k_i(\varphi_i^{-1}(s))\vert\, \Vert D_i^*u_t(s)\Vert^2_{W_i}\ ds.
\end{multline}
Therefore we infer from \eqref{517} and Lemma \ref{l52qqq} that
\begin{equation*}
h( \Vert A_0^{1/2}u(r)\Vert_W)\le h(2 E^{1/2}(r))\le h( 2 \bar{C}^{1/2} E^{1/2}(0))<\frac 12\,.
\end{equation*}
Using \eqref{514} in the definition of $E(r),$ this gives
\begin{align*}
E(r)&= \frac 12 \Vert u_t(r)\Vert_W^2 +\frac 12 \Vert A_0^{1/2} u(r)\Vert^2_W-\Psi (u(r))\\
&\qquad +\frac 1 2\sum_{i=1}^l \frac 1 {1-c_i} \int_{r-\tau_i(r)}^r \vert k_i(\varphi_i^{-1}(s))\vert\cdot\Vert D_i^*u_t(s)\Vert^2_{W_i}ds
\\
&>\frac 14 \Vert u_t(r)\Vert_W^2 +\frac 14 \Vert A_0^{1/2} u(r)\Vert^2_W\\
&\qquad +
\frac 1 4\sum_{i=1}^l \frac 1 {1-c_i} \int_{r-\tau_i(r)}^r \vert k_i(\varphi_i^{-1}(s))\vert\cdot\Vert D_i^*u_t(s)\Vert^2_{W_i}ds,
\end{align*}
contradicting the maximality of $r.$
This implies $r=\delta.$

Now, let us define
\begin{equation}\label{518}
\rho:=
\frac 1 {2\bar{C}^{1/2}} h^{-1}(\frac 12).
\end{equation}
We show that \eqref{515} is satisfied for all $u_0\in \tilde W,$ $u_1\in W,$ $g_i\in C([-\tau^*,0], W_i),$ $i=1,\dots,l,$ satisfying
\begin{equation*}
\Vert A_0^{1/2}u_0\Vert_W^2+\Vert u_1\Vert_W^2+ \frac 12 \sum_{i=1}^l\frac 1 {1-c_i}\int_{-\tau_i(0)}^0 \vert k_i(\varphi_i^{-1} (s))\vert\, \Vert g_i(s)\Vert^2_{W_i}\, ds
<\rho^2.
\end{equation*}
Indeed, this assumption implies $\Vert A_0^{1/2}u_0\Vert_W <\rho$ and then, observing that $\bar{C}>1,$ we have
$$ h \Big ( \Vert A_0^{1/2}u_0\Vert_W\Big ) < h(\rho )=h \Big (  \frac 1 {2\bar{C}^{1/2}} h^{-1}(\frac 12  )\Big )< \frac 12\,.$$
Moreover, from \eqref{514} we get the estimate
\begin{multline*}
E(0)\le \frac 34 \Vert A_0^{1/2} u_0\Vert_W^2+\frac 12\Vert u_1\Vert_W^2\\
+ \frac 12 \sum_{i=1}^l\frac 1 {1-c_i}\int_{-\tau_i(0)}^0 \vert k_i(\varphi_i^{-1} (s))\vert\, \Vert g_i(s)\Vert^2_{W_i}\, ds<\rho^2,
\end{multline*}
and thus, from \eqref{518} we infer that
\begin{equation*}
h\left ( 2 \bar{C}^{1/2}E^{1/2}(0) \right ) < h (2\bar{C}^{1/2}\rho ) <h(h^{-1}(1/2))= \frac 1 2.
\end{equation*}
We  conclude that \eqref{515} holds, and that
\begin{multline*}
0< \frac 14 \Vert u_t(t)\Vert^2_W +\frac 14 \Vert A_0^{1/2} u(t)\Vert^2_W\\
+\frac 1 4\sum_{i=1}^l \frac 1 {1-c_i} \int_{t-\tau_i(t)}^t \vert k_i(\varphi_i^{-1}(s))\vert\, \Vert D_i^*u_t(s)\Vert^2_{W_i}ds\\
\le E(t)\le \bar{C} E(0)\le \bar{C}\rho^2, \ \forall \ t\in [0,\delta].
\end{multline*}
One can extend the solution of problem (\ref{513}) by considering as initial datum the solution at time $t=\delta\,.$
Arguing as above, we can extend the solution to the whole $[0,\tau_{min}]$ and the solution satisfies
$$h\left (\Vert A_0^{1/2}u(\tau_{min})\Vert_W \right )\le h\left (2 E^{1/2}(\tau_{min})\right )\le h\left (2 \bar{C}^{1/2}E^{1/2}(0) \right )<\frac 12,$$
where we have applied estimate \eqref{510}
on the whole interval $[0,\tau_{min}].$ Once we have the solution $U(\cdot)$ on the interval $[0, \tau_{min}],$ then on the second interval $[\tau_{min}, 2\tau_{min}]$ one  can rewrite  again
our problem in the abstract form  (\ref{513}) with $G_i(t)=(0, D_iD_i^*u_t(t-\tau_i(t)))$ (note that, for $t\in     [\tau_{min}, 2\tau_{min}]$ it results $t-\tau_i(t)\le\tau_{min}$). One can repeat the same argument on every interval of length $\tau_{min}$ to get a global solution satisfying \eqref{510}.
\end{proof}

\section{Examples}\label{s6qqq}

In the following examples we consider a non-empty bounded domain $\Omega$ in $\mathbb R^n$ with a boundary $\Gamma $ of class $C^2$.

\subsection{The wave equation with localized frictional damping}

Let $O\subset \Omega$ be a nonempty open subset satisfying the \emph{geometrical control  property} in \cite{BLR}.
For instance, denoting by $m$ the standard multiplier $m(x)=x-x_0,$ $x_0\in\mathbb R^n,$ as in \cite{Lions-1988}, $O$ can be the intersection of $\Omega$ with an open neighborhood of the set
\begin{equation*}
\Gamma_0=\set{x\in\Gamma \:\ m(x)\cdot\nu (x)>0}.
\end{equation*}
Denoting by $\chi_D$ the characteristic function of a set $D,$
let us consider the following system:

\begin{equation}\label{61qqq}
\begin{cases}
u_{tt}(x,t)-\Delta u(x,t)+a\chi_O(x) u_t(x,t) \\
\hspace{2cm} {-k(t)\chi_{\tilde O}(x) u_t(x,t-\tau)=0}\qtq{in}\Omega\times (0, \infty),\\
{u(x,t)=0\qtq{on}\Gamma\times (0,\infty )},\\
{u(x,t-\tau)=u_0(x, t) \qtq{in}\Omega\times (0,\tau]},\\
{u_t(x,t-\tau)=u_1(x,t)}
{\qtq{in}\Omega\times (0,\tau]},
\end{cases}
\end{equation}
where $a$ is a positive constant,
$\tau >0$ is the time delay, and the damping coefficient $k$ belongs to $L^1_{loc}(0,\infty)$.
The set $\tilde O\subset\Omega$ where the delay feedback is localized can be any measurable subset of $\Omega.$

Setting $U= (u, u_t)^T,$ this problem can be rewritten in the form \eqref{11}
with $H=  H^1_0(\Omega)\times L^2(\Omega),$
\begin{equation*}
A=
\begin{pmatrix}
0&1\\\Delta&\chi_O
\end{pmatrix}
\end{equation*}
and
$B: H\to H$ defined by
\begin{equation*}
B (u,v)^T= (0, \chi_{\tilde O}v)^T.
\end{equation*}
It is well-known that ${A}$ generates a strongly continuous semigroup which is exponentially stable (see e.g. \cite{zuazua, Komornikbook}).
Since $\norm{B}\le 1$, we have exponential stability result under the assumption
\begin{equation}\label{62qqq}
Me^{\omega\tau}\int_0^t\abs{k(t+\tau )}\ ds \le\gamma+\omega't\qtq{for all}t\ge 0
\end{equation}
for some $\omega'<\omega$ and $\gamma\in\mathbb R$, where $M$ and $\omega$ denote the positive constants in the exponential estimate \eqref{12} for the semigroup generated by $A.$
This extends to more general delay feedbacks a previous result of the second author \cite{SCL12}.

\subsection{The wave equation with memory }\label{ss62qqq}
Given an arbitrary open subset $O$ of $\Omega$, we consider the system

\begin{equation}\label{63qqq}
\begin{cases}
u_{tt}(x,t) -\Delta u (x,t)+
\int_0^\infty \mu (s)\Delta u(x,t-s) ds\\
\hspace{2,5 cm}
 =k(t)\chi_O (x) u_t(x,t-\tau) )\qtq{in} \Omega\times
(0,\infty),\\
u (x,t) =0\qtq{on}\Gamma \times
(0,\infty),\\
u(x,t)=u_0(x, t)\qtq{in}\Omega\times (-\infty, 0]
\end{cases}
\end{equation}
with a constant time delay $\tau >0$, and a locally absolutely continuous memory kernel $\mu :[0,\infty)\rightarrow [0,\infty)$,
satisfying the following three conditions:
\begin{enumerate}[\upshape (i)]
\item $\mu (0)=\mu_0>0;$
\item $\int_0^{\infty} \mu (t) dt=\tilde \mu <1;$
\item $\mu^{\prime} (t)\le -\alpha \mu (t) \qtq{for some}\alpha >0.$
\end{enumerate}

As in \cite{Dafermos}, we introduce the notation
\begin{equation*}
\eta^t(x,s):=u(x,t)-u(x,t-s).
\end{equation*}
Then we can restate  \eqref{63qqq}
in the following form:
\begin{equation}\label{64qqq}
\begin{cases}
u_{tt}(x,t)= (1-\tilde \mu)\Delta u (x,t)+
\int_0^\infty \mu (s)\Delta \eta^t(x,s) ds\\
\hspace{2.5 cm}
=k(t)\chi_O (x) u_t(x,t-\tau))\qtq{in} \Omega\times
(0,\infty)\\
\eta_t^t(x,s)=-\eta^t_s(x,s)+u_t(x,t)\qtq{in}\Omega\times
(0,\infty)\times (0,\infty ),\\
u (x,t) =0\qtq{on}\Gamma \times
(0,\infty)\\
\eta^t (x,s) =0\qtq{in}\Gamma \times
(0,\infty) \qtq{for} t\ge 0,\\
u(x,0)=u_0(x)\qtq{and}\quad u_t(x,0)=u_1(x)\qtq{in}\Omega,\\
\eta^0(x,s)=\eta_0(x,s) \qtq{in}\ \Omega\times
(0,\infty),
\end{cases}
\end{equation}
where
\begin{equation}\label{65qqq}
\begin{cases}
u_0(x)=u_0(x,0), \quad x\in\Omega,\\
u_1(x)=\frac {\partial u_0}{\partial t}(x,t)\vert_{t=0},\quad x\in\Omega,\\
\eta_0(x,s)=u_0(x,0)-u_0(x,-s),\quad x\in\Omega,\  s\in (0,\infty).
\end{cases}
\end{equation}

Let us introduce the Hilbert space
$L^2_{\mu}((0, \infty);H^1_0(\Omega ))$
of $H^1_0$-valued functions on $(0,\infty),$
endowed with the inner product
\begin{equation*}
\langle \varphi, \psi\rangle_{L^2_{\mu}((0, \infty);H^1_0(\Omega ))}=
\int_{\Omega}\left (\int_0^\infty \mu (s)\nabla \varphi (x,s)\nabla \psi (x,s) ds\right )dx,
\end{equation*}
and then the the Hilbert space
\begin{equation*}
H:=
H^1_0(\Omega)\times L^2(\Omega)\times L^2_{\mu}((0, \infty);H^1_0(\Omega ))
\end{equation*}
equipped  with the inner product
\begin{multline*}\label{innerd}
\left\langle
\begin{pmatrix}
u\\ v\\ w
\end{pmatrix}
,
\begin{pmatrix}
\tilde u\\ \tilde v\\ \tilde w
\end{pmatrix}
\right\rangle_H
:=  (1-\tilde\mu )\int_\Omega \nabla u\nabla\tilde u dx + \int_\Omega v\tilde v dx\\
+\int_{\Omega} \int_0^\infty \mu (s)\nabla w\nabla\tilde w ds dx.
\end{multline*}

Setting
$U:= (u,u_t,\eta^t)^T$
we may rewrite   the problem \eqref{64qqq}--\eqref{65qqq}
in the abstract form
\begin{equation*}\label{abstractd}
\begin{cases}
U_t(t)={ A} {U}(t)+k B U(t-\tau),\\
U(0)=(u_0,u_1, \eta_0)^T,
\end{cases}
\end{equation*}
where the operator $A$ is defined by
\begin{equation*}\label{Operator}
A
\begin{pmatrix}
u\\ v\\ w
\end{pmatrix}
:=
\begin{pmatrix}
v\\
(1-\tilde\mu)\Delta u+\int_0^{\infty}\mu (s)\Delta w(s)ds\\
-w_s+v
\end{pmatrix}
\end{equation*}
with domain
\begin{align*}\label{dominioOpd}
D(A):=
&\left\{(u,v,\eta )\in   H^1_0(\Omega)\times H^1_0(\Omega)
\times L^2_{\mu}((0,\infty);H^1_0(\Omega))\, :\right.\\
&(1-\tilde\mu)u+\int_0^\infty \mu (s)\eta (s) ds \in H^2(\Omega)\cap H^1_0(\Omega),\\
&\left.\eta_s\in  L^2_{\mu}((0\infty);H^1_0(\Omega))\right\}
\end{align*}
in the Hilbert space $H$,
and the bounded operator $B:H\to H$ is defined by the formula
\begin{equation*}
B(u,v, \eta^t)^T:= (0, \chi_O v, 0)^T.
\end{equation*}

It is well-known (see e.g. \cite{GRP}) that the operator $A$ generates an exponentially stable semigroup.
Since $\norm{B}\le 1$, Theorems \ref{p21qqq} and  \ref{t31qqq} guarantee the well-posedness and exponential stability of \eqref{64qqq}--\eqref{65qqq} if the condition \eqref{62qqq} is satisfied
for some $\omega'<\omega$ and $\gamma\in\mathbb R$, where $M$ and $\omega$ denote the positive constants in the exponential estimate \eqref{12} for the semigroup generated by $A.$

\subsection{The wave equation with frictional damping and source}

Let $O\subset \Omega$ be a nonempty open subset satisfying the \emph{geometrical control  property} in \cite{BLR} and let  $\tilde O\subset O.$
As an explicit example of (\ref{513}), let us consider the following system:

\begin{equation}\label{66qqq}
\begin{cases}
u_{tt}(x,t)-\Delta u(x,t)+a\chi_O(x) u_t(x,t)\\
\hspace{6cm} -k(t)\chi_{\tilde O}(x) u_t(x,t-\tau) \\
\qquad =\vert u(x,t)\vert^\mu u(x,t)\qtq{in}\Omega\times (0, \infty),\\
u(x,t)=0\qtq{on}\Gamma\times (0,\infty ),\\
{u(x,t-\tau)=u_0(x, t) \qtq{in}\Omega\times (0,\tau]},\\
{u_t(x,t-\tau)=u_1(x,t)}
{\qtq{in}\Omega\times (0,\tau]},
\end{cases}
\end{equation}
where $a$ is a positive constant,
$\tau >0$ is the time delay, and the damping coefficient $k$ belongs to $L^1_{loc}(0,\infty)$.
Moreover, we assume $k= k^1+k^2$ with $k_1\in L^1([0,+\infty))$ and $k_2\in L^\infty (0,+\infty)$ satisfying $\Vert k^2\Vert_\infty < a.$
Setting $U= (u, u_t)^T,$ this problem can be rewritten in the form \eqref{51qqq}
with $H=  H^1_0(\Omega)\times L^2(\Omega),$
\begin{equation*}
A=
\begin{pmatrix}
0&1\\\Delta&\chi_O
\end{pmatrix}
\end{equation*}
and
$B: H\to H$ defined by
\begin{equation*}
B (u,v)^T= (0, \chi_{\tilde O}v)^T.
\end{equation*}
It is well-known that ${A}$ generates a strongly continuous semigroup which is exponentially stable (see e.g. \cite{zuazua, Komornikbook}).
Next, consider the functional
$$\Psi (u):= \frac 1 {\mu+2} \int_{\Omega} \vert u(x)\vert^{\mu+2} dx,\quad u\in H_0^1(\Omega),$$
which is well-defined, for $0<\mu\le \frac 4 {n-2},$ by Sobolev's embedding theorem. Note that $\Psi$ is G\^{a}teaux differentiable at every $u\in
H_0^1(\Omega)$ and its G\^{a}teaux  derivative is given by
$$D\Psi (u)(v)= \int_{\Omega} \vert u(x)\vert^\mu u(x) v(x)\, dx,\quad v \in H_0^1(\Omega).$$
As showed in \cite{ACS}, assuming
$0<\mu<\frac 4 {n-2},$ then $\Psi$ satisfies previous assumptions (i), (ii), (iii), and then problem \eqref{66qqq} is included in the abstract form (\ref{513}). Since the assumptions of Lemma \ref{l52qqq} are satisfied then Theorem \ref{t51qqq} holds
for small initial data if
\eqref{62qqq}
is satisfied
for some $\omega'<\omega$ and $\gamma\in\mathbb R$, where $M$ and $\omega$ denote the positive constants in the exponential estimate \eqref{12} for the semigroup generated by $A.$

\begin{remark}
A  large variety of other examples could be considered, e.g., the wave equation with standard dissipative boundary conditions and internal delays, plate equations with internal/boundary/viscoelastic dissipative feedbacks and internal delays,  elasticity systems with different kinds of feedbacks.
\end{remark}

redpajama_set_name: RedPajamaArXiv


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One Week of Adventure, A Lifetime of Memories!
When: March 25 – 29, 2019 from 8AM – 5PM. Before and After Care from 7:30AM to 6:00PM is an additional $25 per service for the week.
Where: 975 85th Avenue, Oakland, CA, 94621. Click here for directions.
Acta Non Verba and Oakland Parks and Recreation are teaming up again to bring Action, Education, Excitement and FUN to your children this Spring! Join us this Spring Break for gardening, swimming, cooking, art projects, storytelling, reading and games!
**Before and After Care starting at 7:30AM and ending at 6:00PM is just an additional $25 for the week! Limited Spaces Available.
Learn how to grow your own food. Your child will be planning, planting, harvesting and selling tomatoes, carrots, greens, strawberries and more. 100% of the money your child earns will be placed into savings accounts for the children that attend at least 100 hours of camp this summer!
Your child will come home with fresh produce AND know how to cook with it. This spring, your child will be the master of her own dining destiny!! Your kids will have a ball smashing tomatoes and rolling dough as our amazing guess chef helps them plan and execute the lunch and snack menu for camp.


Wasn't everyone raised to be humble?! That you quietly (regardless of your inner pride) celebrate your accomplishments and silently wait for the accolades and compliments. I certainly was not raised to shout my achievements from a mountaintop. My mother would have been sorely disappointed in my "bragging".
Sometimes we need to SHOUT IT OUT! Celebrate. Scream. Screw how we were raised. We need to honor our major feats and triumphs. It is necessary for our self-empowerment. We can't always wait for others to recognize our significant milestones. We need to be our own cheerleaders! Only we are aware of how hard we worked and how much effort was put forth to get to where we are. Celebrate yourself! Shout it from rooftops or the proverbial mountaintop aka social media. Rock on people!
So to completely and proudly toot my own horn, Maikana Foods is celebrating six months of being open. Six months of being a part of your lives. AND I AM SO GRATEFUL! Thank you for allowing Maikana to be a part of your healthy regimen. My only hope at this six-month mark is that I have grown and slowly begun to roll out the bumps and lumps of my business. Maikana is by no means perfect (yet-hehe) but I am constantly working on it and evolving. And I am so happy that you are along for the ride.
Thank you so so so much to all our clients who have stuck by Maikana or helped us along the way. You are the reason Maikana continues every day.
Thank you all for being a part of this amazing journey. Have a beautiful weekend!


text:
package com.example.shridharmali.alertdialog;

import android.app.Dialog;
import android.content.DialogInterface;
import android.os.Bundle;
import android.support.annotation.NonNull;
import android.support.v4.app.DialogFragment;
import android.support.v7.app.AlertDialog;
import android.widget.Toast;

/**
 * Created by shridharmali on 1/30/17.
 */

public class AlertFragment extends DialogFragment {
    @NonNull
    @Override
    public Dialog onCreateDialog(Bundle savedInstanceState) {
        AlertDialog.Builder builder =  new AlertDialog.Builder(getActivity());
        builder.setMessage("All fields are mandatory");
        builder.setPositiveButton("Yes", new DialogInterface.OnClickListener() {
            @Override
            public void onClick(DialogInterface dialog, int which) {
                 Toast.makeText(getActivity(),"Yes button clicked", Toast.LENGTH_LONG).show();
            }
        });

        builder.setNegativeButton("No", new DialogInterface.OnClickListener() {
            @Override
            public void onClick(DialogInterface dialog, int which) {
                 Toast.makeText(getActivity(),"No button clicked", Toast.LENGTH_SHORT).show();
            }
        });

        AlertDialog alert = builder.create();
        return (alert);
    }
}

redpajama_set_name: RedPajamaGithub


At Hedges Estate, we take your dreams, aspirations and wildest ideas for your Wedding day and make them a reality. We pride ourselves on being a venue that works with our couples to ensure we are reflecting their relationship in their wedding in the best way possible. If you're after a beautiful venue set in rural kiwi countryside then look no further and come check us out today.


text:
abiasedperspective Luke Phillips's Blog
"Luke, Do You Want to Be President?"
In which I ignore what little prudence I have, and set out to piss people off
To a Mentor-
I wrote up this little reflection on my ambitions, initially for my blog and Facebook, but am now unsure whether or not I'll share it publicly. I think I have important things for people in here to hear, but it would be uppity and arrogant rather than sage and wise for 22-year old me to counsel my friends to be Noble Romans of Old.
Who knows, by the time you read this I might have jumped the shark and shared it and started taking flak already. But I just figured, since you're my primary political mentor, that I should send you my thoughts on the subject of what I want to do with my life and how that relates to my personal development.
So I keep getting the "Luke do you want to be President" kind of questions over and over again and I keep giving my same standard self-deprecating canned response: "God save the country if that ever happens." Sometimes I tell people my true ambitions and say "no but I want to be the Alexander Hamilton to someone's George Washington." Sometimes I go a bit deeper and say "I want to be the kind of person about whom people ask 'is he going to run for President?'"
But I've been reflecting on it deeper and deeper, and realized something that's been driving me for a while. And I would not trust people who were not driven by the same force, with the reins of public life (hence my antipathy to Trump and Clinton, as well as Gary Johnson.)
Since FDR, arguably since Lincoln, and probably since the days of George Washington himself, we've inhabited a Presidency-heavy constitutional system- in the White House has been vested the symbolic sanctity of the Union and Republic, of our Liberty itself. It's always been a lot more than a mere administrative or policy-advocacy office. It's been a Cincinnatian, perhaps at times a Catonian or Ciceronian office of republican splendor. The President, for better or for worse, has been the face of the nation.
In the last century it has increasingly become an Imperial Presidency, overstepping the old constitutional boundaries that once precluded aspiring Lincolns and TRs from doing great things. That's probably been for the best- with the sheer complexity of modern industrial society, it's unclear how else we could have organized things. But that power must be exercised responsibly, with both history and posterity in mind.
And as the power and glory of the Presidency has increased, both in political fact and in the public imagination, so has increased the necessity of true greatness in the souls of those inhabiting the Oval Office. Greatness of the sort with which Cincinnatus, Cato, Cicero, and all those other dead white males would have been intimately familiar.
But partly because of the deconstructionist monstrosity of the fruits of the raging 60s, partly because of the emphasis of ideological principle over networked character that has characterized the parties since Goldwater and McGovern, and partly because of the increasing democratization and reformation of politics since Watergate, it's been harder for men and women of the character of George Washington or Abraham Lincoln to ascend or even aspire to high office in our Republic. They're there, but they tend to stick to unelected positions in the Military-Industrial Complex and the Commanding Heights of the economy. When they DO seek public office- and I would say the last two great examples were Daniel Patrick Moynihan and the John McCain of 2000- they're either largely impotent (Moynihan) or they resort to selling their souls, principles, and character in fraught quests for power (hence McCain's Palin desperation.)
America since midcentury has been fraught by various social conditions and political reforms that make the crucial mix of Thumos, Arete, and Pietas absolutely impossible in modern statesmen and stateswomen in power. Thus the decline of our politicians from Truman, Johnson, and Nixon (all jerks, but all exceptionally skilled and nonideological politicos) to Clinton, Bush, Obama, and Clinton again (all nice guys/gals, but not particularly competent politicos and certainly not great leaders.) Straight Talk Express McCain 2000 was the closest we got to greatness. It's a tragedy he chose to change.
I'm not bemoaning the fall of the Republic- it isn't forthcoming, even in the terror of the present crisis and the coming ones- but its health and glory are not served by the caliber of Presidents we have as options nowadays. And I think there's a direct correlation between the increasingly ideologized, increasingly democratized politics of the early 21st Century, and the declining character and talent of our leaders.
So what's the answer?
Quite simple really. We need old-fashioned republican statesmen and stateswomen updated for the 21st Century, "men/women with empires in their purpose and new eras in their brains/branching toward the skyey future, rooted in the fertile past," who pay more than lip service to the great heroes of Americana- who pay tribute to them in their lives, choices, actions, and characters. Perhaps we need to make systemic reforms that encourage and incentivize such Roman Souls to enter the hubbub of democratic politics in an institutional Republic with the spirit of a Democracy; but ultimately their entry into the political process is their choice, not the result of machinations by people who want them in.
We need less people who want to be President of the United States, and more people who are worthy of being President of the United States.
So what's my real ambition? What's my real goal in public life? (Aside from "turn America into a great space-faring Republic, an Empire of the Stars?")
To be worthy in my soul of assuming the office of President of the United States. Not to actually assume it- to desire assuming that office, particularly nowadays and at my age, is far more vainglorious than noble.
But to so cultivate my character, tend to my intellect, and master the craftsmanship of politics that, should my country ever call, I would be standing by, ready to act, a soul worthy of the stewardship of the Republic. That by no means means I want to be President- and I think our best public servants don't- but it does mean that in whatever capacity I were serving the public, be it in our national economic apparatus, amid the defense and intelligence community, in the Cabinet or Congress or some statehouse- I would be a leader worthy of heeding my country's call should it ever come.
In my view there are a few character traits and habits of mind that disqualify anyone from being such a worthy statesman or stateswoman, and I'm not sure that I don't show them right now-
1-Basing your love of country off of a vision of a golden future or a return to a golden past, and lacking the moral nuance to accept the evil in your country's heritage- and future- alongside the good. That is to say, being a conservative first and an American second, or a global citizen first and an American second. Conservatives, you need to accept Franklin Roosevelt as an American; Global Citizens, you need to accept Andrew Jackson.
2- Having "becoming President" or something as your primary goal, consciously or subconsciously, and "serving your country" as a second and related goal. No Man or Woman is born to rule. Trump, Hillary.
3- Not easily answering "yes" to "would you die for your country, your countrymen, and your flag?" That, and you need to mean it, and there are few things that suggest you mean it more than actually putting yourself on the line or preparing to put yourself on the line in some form of military service. Not to romanticize military life of course (DuffelBlog exists for a reason) but I continue to hold that it means a lot.
But let's be real- "be worthy" (and I mean a very different word from "qualified") is just as great a mountain to climb as "become President of the United States." The qualities of character and soul requisite of worthiness are a lifetime's work of action and a lifetime's thought and reflection, not habits practicable after a week's cultivation.
Perhaps I'm embarking on a quixotic quest seeking to be worthy. But I'd rather cultivate my soul and serve my country, and fail, than drain my soul and serve myself, and succeed.
A Farewell to ABiasedPerspective: Goodbye, Old Friend | abiasedperspective - November 30, 2017
A Farewell to ABiasedPerspective: Goodbye, Old Friend
America: The Western Melting Pot of the 21st Century
Rambling Notes on Bush, Obama, Trump, and Civic Culture, and Passages from Kelly and McCain
The Case of the Incredible Shrinking Circular Firing Squad- the California GOP
Notes on the Current Wave of Violence in America
Welcome to My New Bl… on A Farewell to ABiasedPerspecti…
A Farewell to ABiase… on Message to Friends and Coworke…
A Farewell to ABiase… on The Case for Universal Childho…
A Farewell to ABiase… on Alexander Hamilton and Me
A Farewell to ABiase… on Parable of the Statue of …
abiasedperspective

redpajama_set_name: RedPajamaCommonCrawl


PUTRAJAYA: Malaysia has issued a travel advisory for 26 countries and territories where cases of Zika virus infections have been reported.
They are Barbados, Bolivia, Brazil, Colombia, Costa Rica, Curacao, Dominican Republic, Ecuador, El Salvador, French Guiana, Guadeloupe, Guatemala, Guyana and Haiti.
Malaysians are also advised to avoid travelling to Honduras, Jamaica, Martinique, Mexico, Nicaragua, Panama, Paraguay, Puerto Rico, Saint Marti, Suriname, US Virgin Islands and Venezuela.
Health Minister Datuk Seri Dr S. Subramaniam said that although the World Health Organisation had not issued any travel restrictions, health authorities felt that it was necessary to advice the public to postpone travel to the countries on the list, if it could be avoided.
"If travelling cannot be avoided, the public should ensure they bring mosquito repellents and wear clothes that cover their body, including the arms and legs, to prevent getting bitten by mosquito," he said on Wednesday.
Dr Subramaniam stressed that the travel advisory was only meant for the 26 countries and territories with reported cases and Malaysians were free to travel to countries that were not on the list.
The minister added that to date, there were no reported cases of Zika virus in the country and that his ministry had taken necessary precautionary measures.


text:
Faculty of Health Sciences and Sport
Faculty of Health Sciences and Sport Journal Articles
Appears in Collections: Faculty of Health Sciences and Sport Journal Articles
Title: A realist evaluation of an antenatal programme to change drinking behaviour of pregnant women
Author(s): Doi, Lawrence
Jepson, Ruth
Cheyne, Helen
Contact Email: h.l.cheyne@stir.ac.uk
Keywords: Realist evaluation
Alcohol screening
Brief interventions
Issue Date: Oct-2015
Citation: Doi L, Jepson R & Cheyne H (2015) A realist evaluation of an antenatal programme to change drinking behaviour of pregnant women, Midwifery, 31 (10), pp. 965-972.
Abstract: Objective: to use realist evaluation to describe and explain how and in what circumstances screening and alcohol brief interventions work in routine antenatal care. Design: a realist evaluation incorporating systematic reviews and qualitative data. Setting: NHS Lothian, which is one of the 14 Scottish health boards. Participants: participants were recruited from two maternity units. In phase one, interviews were conducted with four participants responsible for policy implementation. These data were supported by two systematic reviews. In phase two, 17 pregnant women and 15 midwives participated in interviews, with a further six midwifery team leaders involved in a focus group. Findings: training and resources provided to midwives as part of the programme acted as facilitating mechanisms that improved their skills and confidence to screen and deliver alcohol brief interventions. The programme elicited positive change in attitudes to drinking in pregnancy and possibly stimulated drinking behaviour change amongst pregnant women. However, the small numbers of pregnant women being identified for alcohol brief interventions meant delivery was infrequent and resulted in the programme not working as anticipated. The findings also revealed contextual issues around midwife-pregnant woman relationship and the challenges of negotiating the timing of screening and alcohol brief interventions delivery. Conclusions: Drinking in pregnancy is an emotive issue, therefore delivering alcohol brief interventions at the first antenatal appointment when they are more likely to achieve the most benefits poses challenges. When training midwives to screen and deliver alcohol brief interventions, special attention is needed to improve person-centred communication skills to overcome barriers associated with discussing sensitive prenatal alcohol use and enhance early identification and delivery of alcohol brief interventions at the first antenatal appointment.
DOI Link: http://dx.doi.org/10.1016/j.midw.2015.06.007
Rights: This article is open-access. Open access publishing allows free access to and distribution of published articles where the author retains copyright of their work by employing a Creative Commons attribution licence (CC BY-NC-ND). Proper attribution of authorship and correct citation details should be given. Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. NonCommercial — You may not use the material for commercial purposes. NoDerivatives — If you remix, transform, or build upon the material, you may not distribute the modified material.
Doi et al_Midwifery_2015.pdf 275.76 kB Adobe PDF View/Open


text:
Dipl.-Ing. Environmental Technology
Mikael P. Henzler is a co-founder and Managing Director of adelphi consult and a managing director of adelphi research. The core topics of his research and consultancy work are: Climate and Energy, Technology and Innovation, and Companies and Finances.
He is also keenly involved in issues concerning environment and energy management by small and mid-sized companies, corporate responsibility, energy efficiency and environmental technology & technology cooperation. He principally advises aid agencies within the development and cooperation sector, international organisations and the European Commission. Since 1995, he has managed more than 100 national and international advisory projects with inter-disciplinary teams. His regional focuses are: South and South East Asia, the Mediterranean region, the Maghreb countries, Southern Africa and Turkey.
Mikael P. Henzler moderates international dialogue processes and plans and oversees a large number of climate, environment and energy-related qualification measures around the world for public-sector clients and banks. He is fully trained in Capacity WORKS and obtained a lead auditor certification for environment management systems in accordance with ISO 14001. He is also a certified sustainability assurance practitioner (CSAP) authorised under AA 1000 to audit the sustainability strategies and reports of companies.
He has developed comprehensive information platforms for the European Commission and the German Federal Environment Agency on green technologies, energy efficiency, corporate social responsibility and the promotion of exports. In addition to formulating regional and sectoral programmes for national and international donor agencies, he regularly evaluates development policy projects. Mikael P. Henzler is the author of a great number of publications on environment technology and energy, sustainability management, and technology cooperation – particularly in emerging economies and developing countries.
Before joining adelphi, Mikael P. Henzler worked in development cooperation. From 1997 to 2000, he managed the environmental division of the Asian and Pacific Center for Transfer of Technology of the United Nations (UN ESCAP) in New Delhi. Whilst in this post, he helped implement various multilateral and bilateral environmental and energy projects and coordinated the cooperation between various departments within the United Nations. He was the Head of Division for Asia, the Middle East and South Africa at the International Environmental Transfer Centre (ITUT) in Leipzig from 2000 to 2001 and has advised the United Nations, the Deutsche Gesellschaft fuer Internationale Zusammenarbeit (GIZ) and the World Bank on globalisation and sustainability.
Mikael P. Henzler studied environmental engineering and management at Technical University Berlin.
Approaches to tackle marine litter in India
Client: Deutsche Gesellschaft für Internationale Zusammenarbeit (GIZ) GmbH
Term: 08/2019 - 10/2019
Advisory services for sustainable urban development in selected smart cities in India
Promoting financing of sustainable production in the textile sector
Innovative and sustainable: Promoting international partnerships with CLIENT II
Client: Federal Ministry of Education and Research (BMBF)
Waste solutions for a circular economy in India
Development of national energy efficiency standards for residential buildings in India
Closing Material Loops through Extended Producer Responsibility: Implementing the Waste Management Code in Georgia
Client: Georgia's Environmental Outlook (GEO)
Technical Assistance to the Indian Resource Efficiency Initiative (EU-REI)
Promoting sustainable production across Indian industries
Conceptualising a project to support the G7 states' InsuResilience Initiative
CTI Workshop 2016 on Carbon Market Mechanisms
Client: Federal Ministry for the Environment, Nature Conservation and Nuclear Safety (BMU)
Development of a Nationally Appropriate Mitigation Action in the Indian waste sector
Brief studies on access to Green Finance for small and medium-sized enterprises in Asia
Client: European Commission, EuropeAid - Co-operation Office, SWITCH-Asia Programme, GFA Consulting Group GmbH
Technical advisory for the implementation of a credit line for renewable energies in India
Client: Indian Renewable Energy Development Agency Ltd (IREDA)
CTI Workshop 2015 on Climate Change Mitigation in Cities
New financing opportunities for innovative enterprises in India
Client: Small Industries Development Bank of India (SIDBI)
Supporting an energy efficiency credit line for residential housing in India
Client: KfW-Bankengruppe
Transnational Dialogue-Forum Climate Investment
Supporting a credit line for energy efficiency measures in Indian MSMEs
Short–term Expert Pools for TRANSfer Project
Capacity building and awareness raising on climate change adaptation in the private sector
Supporting an environmental credit line for Indian MSMEs
International Approaches to Sustainability Reporting – Sustainability Reporting Landscape Profiles of Selected Countries Around the World: An Overview for Policy Makers
Henzler, Mikael und Christian Metzler 2021: International Approaches to Sustainability Reporting. Sustainability Reporting Landscape Profiles of Selected Countries Around the World: An Overview for Policy Makers. Berlin: adelphi.
Laying the groundwork for the development of Long-term Strategies in ASEAN – Policy Brief: Strengthening Science/ Policy Interface in Climate Change related Decision-Making Process
Groß, Julia; Dennis Taenzler and Mikael Henzler 2021: Laying the groundwork for the development of Long-term Strategies in ASEAN. Policy Brief: Strengthening Science/ Policy Interface in Climate Change related Decision-Making Process. Jakarta: ASEAN Secretariat.
Strengthening Science and Policy Interface in Climate Change related Decision-Making Process – Laying the groundwork for the development of long-term strategies (LTS) in ASEAN
Groß, Julia; Dennis Taenzler and Mikael Henzler 2021: Strengthening Science and Policy Interface in Climate Change related Decision-Making Process. Laying the groundwork for the development of long-term strategies (LTS) in ASEAN. Jakarta: ASEAN Secretariat.
A Snapshot of the Chinese Cleantech Financing Ecosystem – ACMFN Flagship Report Series
Kirsch, Kathrin; Mikael Henzler, Mirko Zuerker, Rainer Agster, Ruijnun Palahan and Walter Ge 2019: A Snapshot of the Chinese Cleantech Financing Ecosystem . ACMFN Flagship Report Series.
A Snapshot of the Indian Cleantech Financing Ecosystem – ACMFN Flagship Report Series
Rose, Benita; Kathrin Kirsch, Mirko Zuerker, Akshay Kaul, Mikael Henzler and Rainer Agster 2019: A Snapshot of the Indian Cleantech Financing Ecosystem. ACMFN Flagship Report Series. Berlin: adelphi.
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Kirsch, Kathrin; Mikael Henzler, Mirko Zuerker, Kristanto Budi and Rainer Agster 2019: A Snapshot of the Indonesian Cleantech Financing Ecosystem. ACMFN Flagship Report Series. Berlin: adelphi.
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Henzler, Mikael P., Morton Hemkhaus, Frederik Eisinger, Sinha Satish, Priti Mahesh and Gautam Mehra 2018: Creating Successful Formal-informal Partnerships in the Indian E-waste Sector. Practical Guidance for Implementation under the Indian E-Waste Rules. Bonn/Eschborn: GIZ.
Building the Link: Leveraging Formal-Informal Partnerships in the Indian E-Waste Sector
Henzler, Mikael; Fredrik Eisinger, Jai Gaurav, Morton Hemkhaus, Satish Sinha, Priti Mahesh and Gautam Mehra 2017: Building the Link: Leveraging Formal-Informal Partnerships in the Indian E-Waste Sector. Eschborn: GIZ.
Corporate Green Strategies: Green EEE Products and the Relevance of Recycling – Re-furbishment and Reuse, Report (1/2)
Henzler, Mikael; Frederik Eisinger, Ronjon Chakrabarti und Christine Krüger 2017: Corporate Green Strategies: Green EEE Products and the Relevance of Recycling. Re-furbishment and Reuse, Report (1/2). Berlin:
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text:
Mexican-American, b. 1965
Acaba, 2008-09, Oil and beeswax on birch panel, 18 x 20 inches
Alexis Portilla's layered compositions create an expressive sense of depth that is both wholly contemporary and suggestive of the New York School.
Alexis Portilla's layered compositions create an expressive sense of depth that is both wholly contemporary and suggestive of the New York School. His meticulous consideration of surface results in vibrating, active planes of color that interact in interesting and unexpected ways. In a process that is equal parts planning and experimentation, he creates colors that are not mixed, but layered, often many times over. Visible passages of these multiple under layers give his work an atmospheric feeling of space. Portilla's predominately grey palette, with undertones of blue and slate juxtaposed against bright passages of red and yellow, reflects the urban landscape in which he lives and works: "I've been living in New York for a long time and it's natural for an artist to absorb images from the place they see daily." (1)
Alexis Portilla was born in Mexico City in 1965 and currently lives and works in New York. He holds an MFA from Columbia University and a BFA from Pratt Institute. His work is in the private and corporate art collections of over 250 individuals and institutions including The New York Public Library, Penn State University, Warner Brothers Records, and Pepsi Co.
Portilla's work has been the subject of several solo exhibitions in New York and New Jersey, as well as a two-person exhibition in Hong Kong. His work has been exhibited in group exhibitions throughout the United States including the Grand Rapids Art Museum in Michigan and at prominent venues in Miami, Chicago, Atlanta, Philadelphia and New York. Portilla's work has been reviewed in several publications, including ARTNews.
1. Alexis Portilla, quoted in Pat Rogers, "Alexis Portilla's Layered Color and Circular Forms Create 'Surface Tension,'" Hamptons Art Hub. 20 December 2014.
Summer Paintings #2, 2016
Oil on canvas, 14 x 12 inches
Pale Fire, 2016
Oil on linen, 68 x 72 inches
Red Thrush, 2015
Sedona, 2015
Cat and Canary, 2015
Acaba, 2008-09
Oil and beeswax on birch panel, 18 x 20 inches
Selections from our Contemporary Collection
16 January - 1 March 2019
Alexis Portilla: Poet in Space
1 June - 7 July 2017
In the Absence of Color: Artists Working in Black and White
Poet in Space
Published on the occasion of the exhibition Alexis Portilla: Poet in Space held at Hollis Taggart Galleries from June 1–30, 2017.
British, b. 1960
Stay up to date with Alexis Portilla


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Q: NetSuite System Notes CASE formula returning all notes I've added a formula(date/time) column to a saved search in NetSuite, to return a system notes' date.
My CASE formula is returning all the system notes row's, and I would like a specific row's date i.e. 'POP Host Int ID' date.
How can I specify the row to return the date from, or remove the rows with no date that are not relevant?
CASE
WHEN {systemnotes.field} = 'POP Host Int ID' AND {systemnotes.type} = 'Set'
THEN {systemnotes.date}
ELSE NULL
END

It appears that my WHEN logic works to identify the record's system notes do contain an entry for 'POP Host Int ID' but in THEN I'm not specifying which row to get the date from so it returns all rows. And I could be wrong on this part.
Example results

Example System Notes for 1 record

Thank you for your assistance.

A: The CASE statement doesn't determine which rows are returned, only what data is returned for that field.  On the other hand, the reference to the systemnotes table creates a join that causes each record result to be repeated for every system note entry.
To avoid this, add {systemnotes.field} = 'POP Host Int ID' and {systemnotes.type} = 'Set' as Filters in the Criteria tab instead of in the WHEN conditions. You can then just add the field under results instead of needing a formula.
Edit in response to comment below:
In cases where you need one result per base record (user), but they don't all have valid values from the joined table (system notes), I'd suggest grouping the results by user, and using aggregation functions for all the columns.  EG: For the column in question I'm assuming you are getting one valid result and a lot of blanks per user.  If you group by user and set the Summarize function to MAX, you should just get one result where the valid value is returned.  If no valid value exists from the system notes, you would still get a result from the user and that field will be blank.

A: If you are creating a saved search the place to do this is in the criteria section.
The views you've shared are for the System Notes pertaining to a single record.
For those views you could just use the Field selector in the Filters section to select your POP Host Ing ID field.
For a saved search you would use the Advanced view and scroll down the criteria field list. Near the bottom are the System Notes. You can filter on Field, Date etc

redpajama_set_name: RedPajamaStackExchange


text:
package net.digitalid.utility.logging.filter;

import java.util.Objects;

import javax.annotation.Nonnull;
import javax.annotation.Nullable;

import net.digitalid.utility.annotations.method.Pure;
import net.digitalid.utility.annotations.ownership.NonCaptured;
import net.digitalid.utility.annotations.parameter.Unmodified;
import net.digitalid.utility.circumfixes.Quotes;
import net.digitalid.utility.logging.Level;
import net.digitalid.utility.validation.annotations.elements.NonNullableElements;
import net.digitalid.utility.validation.annotations.type.Immutable;

/**
 * A logging rule defines which messages are accepted and thus logged.
 */
@Immutable
public class LoggingRule {

    /* -------------------------------------------------- Threshold -------------------------------------------------- */

    private final @Nonnull Level threshold;

    /**
     * Returns the threshold at and above which messages are accepted.
     */
    @Pure
    public @Nonnull Level getThreshold() {
        return threshold;
    }

    /* -------------------------------------------------- Caller Prefix -------------------------------------------------- */

    private final @Nullable String callerPrefix;

    /**
     * Returns the prefix that the caller has to have in order to be accepted.
     */
    @Pure
    public @Nullable String getCallerPrefix() {
        return callerPrefix;
    }

    /* -------------------------------------------------- Thread Prefix -------------------------------------------------- */

    private final @Nullable String threadPrefix;

    /**
     * Returns the prefix that the thread has to have in order to be accepted.
     */
    @Pure
    public @Nullable String getThreadPrefix() {
        return threadPrefix;
    }

    /* -------------------------------------------------- Message Regex -------------------------------------------------- */

    private final @Nullable String messageRegex;

    /**
     * Returns the regex that the message has to match in order to be accepted.
     */
    @Pure
    public @Nullable String getMessageRegex() {
        return messageRegex;
    }

    /* -------------------------------------------------- Acceptance -------------------------------------------------- */

    /**
     * Returns whether this rule accepts the given message with the given arguments.
     */
    @Pure
    public boolean accepts(@Nonnull Level level, @Nonnull String caller, @Nonnull String thread, @Nonnull String message) {
        return level.getValue() >= threshold.getValue() &&
                (callerPrefix == null || caller.startsWith(callerPrefix)) &&
                (threadPrefix == null || thread.startsWith(threadPrefix)) &&
                (messageRegex == null || message.matches(messageRegex));
    }

    /* -------------------------------------------------- Constructors -------------------------------------------------- */

    protected LoggingRule(@Nonnull Level threshold, @Nullable String callerPrefix, @Nullable String threadPrefix, @Nullable String messageRegex) {
        this.threshold = threshold;
        this.callerPrefix = callerPrefix;
        this.threadPrefix = threadPrefix;
        this.messageRegex = messageRegex;
    }

    /**
     * Returns a logging rule with the given threshold, caller prefix, thread prefix and message regex.
     */
    @Pure
    public static @Nonnull LoggingRule with(@Nonnull Level threshold, @Nullable String callerPrefix, @Nullable String threadPrefix, @Nullable String messageRegex) {
        return new LoggingRule(threshold, callerPrefix, threadPrefix, messageRegex);
    }

    /**
     * Returns a logging rule with the given threshold, caller prefix and thread prefix.
     */
    @Pure
    public static @Nonnull LoggingRule with(@Nonnull Level threshold, @Nullable String callerPrefix, @Nullable String threadPrefix) {
        return new LoggingRule(threshold, callerPrefix, threadPrefix, null);
    }

    /**
     * Returns a logging rule with the given threshold and caller prefix.
     */
    @Pure
    public static @Nonnull LoggingRule with(@Nonnull Level threshold, @Nullable String callerPrefix) {
        return new LoggingRule(threshold, callerPrefix, null, null);
    }

    /**
     * Returns a logging rule with the given threshold.
     */
    @Pure
    public static @Nonnull LoggingRule with(@Nonnull Level threshold) {
        return new LoggingRule(threshold, null, null, null);
    }

    /* -------------------------------------------------- Encoding and Decoding -------------------------------------------------- */

    /**
     * Encodes this logging rule.
     */
    @Pure
    public @Nonnull String encode() {
        final @Nonnull StringBuilder result = new StringBuilder(threshold.toString());
        if (callerPrefix != null || threadPrefix != null || messageRegex != null) {
            result.append(";");
            if (callerPrefix != null) { result.append(" ").append(callerPrefix); }
            if (threadPrefix != null || messageRegex != null) {
                result.append(";");
                if (threadPrefix != null) { result.append(" ").append(threadPrefix); }
                if (messageRegex != null) { result.append("; ").append(messageRegex); }
            }
        }
        return result.toString();
    }

    /**
     * Returns the token with the given index or null if the array is not long enough or the token is empty.
     */
    @Pure
    private static @Nullable String getNonEmpty(@NonCaptured @Unmodified @Nonnull @NonNullableElements String[] tokens, int index) {
        if (tokens.length > index) {
            final @Nonnull String token = tokens[index].trim();
            if (!token.isEmpty()) { return token; }
        }
        return null;
    }

    /**
     * Decodes the given line and returns the corresponding rule.
     *
     * @throws IllegalArgumentException if a rule has an invalid level.
     */
    @Pure
    public static @Nonnull LoggingRule decode(@Nonnull String line) throws IllegalArgumentException {
        final @Nonnull @NonNullableElements String[] tokens = line.split(";", 4);
        final @Nonnull Level threshold = Level.valueOf(tokens[0].trim().toUpperCase());
        final @Nullable String callerPrefix = getNonEmpty(tokens, 1);
        final @Nullable String threadPrefix = getNonEmpty(tokens, 2);
        final @Nullable String messageRegex = getNonEmpty(tokens, 3);
        return new LoggingRule(threshold, callerPrefix, threadPrefix, messageRegex);
    }

    /* -------------------------------------------------- Object -------------------------------------------------- */

    @Pure
    @Override
    public boolean equals(@NonCaptured @Unmodified @Nullable Object object) {
        if (object == this) { return true; }
        if (object == null || !(object instanceof LoggingRule)) { return false; }
        final @Nonnull LoggingRule that = (LoggingRule) object;
        return this.threshold == that.threshold && Objects.equals(this.callerPrefix, that.callerPrefix) && Objects.equals(this.threadPrefix, that.threadPrefix) && Objects.equals(this.messageRegex, that.messageRegex);
    }

    @Pure
    @Override
    public int hashCode() {
        int hash = 7;
        hash = 83 * hash + threshold.hashCode();
        hash = 83 * hash + Objects.hashCode(callerPrefix);
        hash = 83 * hash + Objects.hashCode(threadPrefix);
        hash = 83 * hash + Objects.hashCode(messageRegex);
        return hash;
    }

    @Pure
    @Override
    public @Nonnull String toString() {
        return "LoggingRule(threshold: " + threshold + ", callerPrefix: " + Quotes.inCode(callerPrefix) + ", threadPrefix: " + Quotes.inCode(threadPrefix) + ", messageRegex: " + Quotes.inCode(messageRegex) + ")";
    }

}