Did Gilead Sciences Just Buy Its Next Blockbuster Drug?
Important events don't always seem so important when they first occur. That could be the case with Gilead Sciences' announcement on Tuesday that it was buying Phenex Pharmaceuticals' development program fornonalcoholic steatohepatitis, or NASH. Gilead's shares rose slightly -- but not enough to raise any eyebrows. Could this deal be one that leads to Gilead's next blockbuster drug?
Lessons from the pastGilead stirred things up in the biotech world back in 2011 with the $11 billion buyout of Pharmasset. Many Wall Street observers thought that the price was too high. Gilead, though, saw the potential in Pharmasset's hepatitis C program.
That hepatitis C program that the biotech wanted so badly included the drug now known as Sovaldi, which gained FDA approval in December 2013. In its first nine months on the market, Sovaldi generated revenue topping $8.5 billion. In hindsight, Gilead's purchase was ridiculously smart despite the high price tag.
The cost of Gilead's latest deal is much lower. Privately held Phenex Pharmaceuticals received an undisclosed upfront payment plus the potential for milestone-based payments down the road. Gilead stated that the total package could be worth up to $470 million. That's a lot of money, but only chump change if the NASH program eventually produces a blockbuster drug.
The NASH dashSeveral biotechs are racing to cross the finish line and reach the market first with a drug to treat NASH.Intercept Pharmaceuticals was thought to have captured Gilead's fancy with its NASH drugobeticholic acid, which met its primary endpoint in a phase 2 clinical trial. Gilead's decision to buy Phenex's program resulted in Intercept's shares plunging over 10%.
Raptor Pharmaceuticals lags behind a bit with its drug RP103, which is in a phase 2 trial for treating NASH in children. Results for the phase 2b part of this study are anticipated in the second half of 2015.
Why the big NASH dash? The chronic liver disease affects an estimated 2% to 5% of Americans and is the third leading cause of liver transplants in the U.S. There is no currently approved treatment. NASH represents a market large enough to create the potential for another blockbuster drug.
Gilead's next big thing?Of course, it remains to be seen which company will emerge as the big winner in NASH. With the Phenex deal, Gilead now has its ownFarnesoid X Receptor antagonist with which to battle Intercept. Raptor isn't so far behind that it couldn't make a splash.
This does appear to be yet another smart move by Gilead, though. Adding a NASH drug fits well into the company's broader focus on liver diseases. Making a play for Intercept would have been much more costly than buying Phenex's NASH program. Gilead's recent deal might not lead to its next blockbuster drug -- but then again, it just might.
The article Did Gilead Sciences Just Buy Its Next Blockbuster Drug? originally appeared on Fool.com.
Keith Speights owns shares of Gilead Sciences. The Motley Fool recommends and owns shares of Gilead Sciences. Try any of our Foolish newsletter services free for 30 days. We Fools may not all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. The Motley Fool has a disclosure policy.


text:
Q: Error importing postgres DB dump into empty database I need to import my psql dump into my fresh psql database. When I execute the following command, I get errors.
psql -U user new_database < filename.sql
Error I got:
ERROR:  out of memory
DETAIL:  Cannot enlarge string buffer containing 0 bytes by 1208975751 more bytes.
How do i fix this. Aldo, is there any method to log the import process?
Thanks.

A: I think the most common cause is a "corrupt" SQL file. There's no easy fix. Split the file in half (man split), fix the SQL statements at the tail end of one resulting file and at the head end of the other, and run again. In the past, I seem to recall seeing "Invalid byte sequence for UTF-8", or something like that. I'm sure there are other causes.
PostgreSQL has a lot of logging options; set them in postgresql.conf and restart PostgreSQL. Look at

*

*log_destination

*logging_collector

*client_min_messages

*log_error_verbosity

*log_statement

redpajama_set_name: RedPajamaStackExchange


E-P3 change focus point while recording?
Discussion in 'Olympus Cameras' started by vinay, Apr 6, 2012.
I'm guessing the e-p3 can't change focus points once recording?
No, sadly touch-to-focus isn't available on the E-P3 while recording video. At least, that's what I've read. My E-P3 just arrived yesterday and I didn't really test out video at all.


text:
Page 19, 1 December 1894 — On Tuesday, Mr. Alfred Milner delivered at Toynbee
1 DECEMBER 1894, Page 19
On Tuesday, Mr. Alfred Milner delivered at Toynbee
every interesting lecture on the man from whom that hall has been named,—we mean Mr. Arnold Toynbee, of whom many find a very effective picture in the Sir. Malin of Mrs. Humphry Ward's "Marcella." Toynbee's career was only too short. Born in 1852, be died in 18B3, and yet in the short interval between his coming of age and his death, he Managed to include two lives,—a life of meditation and a life of study. Dr. 'Jciivett, the Master of Balliol; managed to draw him away from
Pembroke to in spite of his not being one of those who read for honours; and even though be took an ordinary degree,--'-thongla the papers he gave in were very remarkable papers,--he was selected to be a Bellied tutor and lecturer, and at once plunged into all the eager life of a social re- former. Nevertheless', his Socialism was always subordinated to the great ideal of conducing to the improvement of the individual, audhe always opposed andcondenmed that kind of Socialism which endeavours to sacrifice the individual to the 'welfare of a class. Arnold Toynbee could not conceive class to be worthy of anygreat sacrifices which did not,owe its distinction, to theligh character and th6 incliVidnid nobility of the members which made it up. Mr. Alfred Milner, who was Toynbee's contemporary at Oxford, could hardly haie placed a higher ideal than Mr. Arnold Toynbee's life before his audience in Toynbee

redpajama_set_name: RedPajamaCommonCrawl


On one of the very last days of July of 2003, we brought Beryl home with us (single mom and 2 kids aged 5 and 3) and she is such a great cat for us! Beryl is gentle and very good with the kids. Somehow my boy and I have come up with the nickname "Minnie Kit" for her. She sleeps on our beds and meows when we come into whatever room she's in. She likes quite a variety of cat toys. I've not been the biggest animal lover in the past partly due to some bad luck with them, but Beryl is really changing that. We are even thinking of getting another cat in the future!


text:
\section{Introduction}
\label{sec:intro}

This paper is devoted to the mathematical analysis of the following
class of parabolic systems:
\begin{align}\label{CH1}
  & u_t - \Delta w = 0,\\
 \label{CH2}
  & w = - a(u) \Delta u - \frac{a'(u)}2 |\nabla u|^2
   + f(u) - \lambda u + \epsi u_t,
\end{align}
on $(0,T) \times \Omega$, $\Omega$ being a bounded
smooth subset of $\RR^3$ and $T>0$ an assigned final
time. The system is coupled with
the initial and boundary conditions
\begin{align}\label{iniz-intro}
  & u|_{t=0} = u_0,
  \quext{in }\,\Omega,\\
 \label{neum-intro}
  & \dn u = \dn w = 0,
  \quext{on }\,\partial\Omega,\
  \quext{for }\,t\in(0,T)
\end{align}
and represents a variant of the Cahn-Hilliard model
for phase separation in binary materials.
Here, $\lambda$ and $\epsi$ are nonnegative parameters,
where the case $\epsi>0$ accounts for a viscosity effect
that may appear the frame of Cahn-Hilliard models
(see, e.g., \cite{Gu}). Moreover,
the function $f$ stands for the derivative
of the so-called logarithmic potential
\begin{equation}\label{logpot}
  F(r) = (1-r)\log(1-r)+(1+r)\log(1+r),
   \quad r\in[-1,1],
\end{equation}
namely,
\begin{equation}\label{f}
  f(r) = \log(1+r)-\log(1-r)= \log \frac{1+r}{1-r},
   \quad r\in(-1,1).
\end{equation}
Finally, we assume that the function $a$
has the form
\begin{equation}\label{defia}
   a(r) = \frac2{1-r^2} = f'(r).
\end{equation}
The initial-boundary value problem given by \eqref{CH1}-\eqref{iniz-intro}
will be noted as Problem~(P) in the sequel.

The above problem is a simplified version (assuming constant mobility and
normalized physical quantities) of the Cahn-Hilliard-de Gennes model
\cite{dG80} of phase separation in binary, incompressible, isothermal
mixtures of polymer molecules of different types $i = 1$ and $i = 2$,
quenched below a critical temperature.

Following the Cahn-Hilliard-de Gennes theory
\cite{dG80}, \cite{dG85} (see also \cite{Bin83}, \cite{NL84}, \cite{Pin81}),
we describe now briefly
the physical basis of the presented model.
Each molecule $i$ in the mixture consists of $N_i$ segments of size $\sigma_i$
(so-called lattice constant) with the quantity
$R^2_{g_i} = \frac{1}{6} N_i \sigma^2_i $
denoting the mean radius of gyration of the i-th polymer molecule.
The variables $u$ and $w$ in Problem (P) have the meaning of the rescaled
order parameter and the exchange chemical potential. For clarity of physical
description let us use just now in place of $u \in [-1,1]$ the original variable
$\chi = \chi_1 \in [0,1]$, denoting the volume fraction of component $i = 1$.
Then, by the incompressibility condition, $\chi_1 + \chi_2 = 1$
everywhere in the sample. The variable
\begin{equation}\label{defiu}
    u:=2\chi - 1 \in [-1,1]
\end{equation}
is introduced for the sake of mathematical convenience.

The model is governed by the Flory-Huggins-de Gennes free energy
functional \cite{dG80}
which in the isothermal case is
\begin{equation}\label{defiFHdG}
\mathcal{F}_{FHdG}(\chi) =\io \Big( F_{FH}(\chi) + \frac{1}{2} a(\chi) |\nabla \chi|^2 \Big).
\end{equation}
The homogeneous (volumetric) free energy $F_{FH}(\chi)$ has the Flory-Huggins form
\begin{equation}\label{defiFH}
  F_{FH}(\chi) = \frac{1}{N_1}\chi \log \chi + \frac{1}{N_2}(1 - \chi) \log(1 - \chi)
      + \lambda \chi(1 - \chi),
\end{equation}
where $\lambda$, called the Flory-Huggins interaction parameter, measures the strength
of interaction between two kinds of species. The second term in the integrand
\eqref{defiFHdG} gives the weighted contribution due to composition gradients.
By de Gennes theory \cite{dG80}, it has the characteristic singular
form
\begin{equation}\label{apolymer}
  \frac{1}{2}a(\chi) = \frac{1}{6}\Big( \frac{R^2_{g_1}}{N_1 \chi} + \frac{R^2_{g_2}}{N_2 (1 - \chi)} \Big)
  = \frac{1}{36}\Big( \frac{\sigma_1^2}{\chi} + \frac{\sigma_2^2}{1 - \chi} \Big)
\end{equation}
which is due to the connectivity of the chains that constitute the polymer molecules.
In the symmetrical case $N_1 = N_2 = N, \; \sigma_1 = \sigma_2 =\sigma,
\; R^2_{g_1} = R^2_{g_2} = R^2_{g} $, the expression \eqref{apolymer} simplifies to
\begin{equation}\label{apolymersymm}
 \frac{1}{2} a(\chi) = \frac{\sigma^2}{36} \frac{1}{\chi (1 - \chi)}.
\end{equation}
Here we note that the singular form of $a(\chi)$  introduces an infinite
energy penalty near the pure phases.
This fact turns out to have an important mathematical consequence
related to the separation property of the solution
(see \eqref{separs} and
Remark~\eqref{mirzel} below).

Under the incompressibility condition,
$\chi_1 + \chi_2 = 1$, the conservation of mass for component $1$ is
given by
\begin{equation}\label{massbalance}
  \chi_t + \nabla \cdot j = 0, \quad j = - \Lambda(\chi) \nabla w.
\end{equation}
Here $j$ is the mass flux defined as the product of the Onsager diffusion coefficient
$\Lambda(\chi)$ (effective mobility) and the gradient of the exchange potential $w$
which is the difference between the chemical potentials of the components,
$w = w_1 - w_2$.
Shortly, we shall refer to $\Lambda$ and $w$ as the mobility
and the chemical potential, respectively.

According to the derivations in \cite{dG80}, \cite{Bin83}, \cite{BJL83},
the expression for $\Lambda$
takes the form
\begin{equation}\label{Onsager}
  \Lambda(\chi) = \frac{\Lambda_1 \Lambda_2}{\Lambda_1  +  \Lambda_2}  = \chi (1 - \chi) \Lambda_0,
\end{equation}
where $\Lambda_i = \chi_i \Lambda_0$ is the Onsager coefficients for the $i$-th component
and $\Lambda_0$ is a positive constant.

We point out that the concentration dependence of the mobility $\Lambda$
is also typical for phase separation of small molecule systems, described by
the classical Cahn-Hilliard equation \cite{Ca}, \cite{CH};
see also \cite{EG}, \cite{GNC00}, \cite {NB89} and the references therein.

The chemical potential $w$ is defined as the first variation of
the functional \eqref{defiFHdG} yielding the following equivalent expressions:
\begin{align}\no
  w & = \frac{\delta \mathcal{F}_{FHdG}}{\delta \chi} \\
  \nonumber
      & =   F^{\prime}_{FH}(\chi) + \frac{1}{2} a^{\prime}(\chi)|\nabla \chi|^2 -
  \dive (a(\chi) \nabla \chi) \\
  \label{chempot}
      & =   F^{\prime}_{FH}(\chi) - \frac{1}{2} a^{\prime}(\chi)|\nabla \chi|^2
  - a(\chi) \Delta \chi \\
      & =   F^{\prime}_{FH}(\chi) - \sqrt{a(\chi)} \dive (\sqrt{a(\chi)} \nabla \chi).
      \nonumber
\end{align}
As in the standard Cahn-Hilliard theory, the chemical potential can
include an additional term $\epsi \chi_t$, with a positive constant $\epsi$,
accounting for possible viscous effects.

The equations \eqref{massbalance}, \eqref{Onsager}, and \eqref{chempot}
with the term $\epsi \chi_t, \;\eps \geq 0$, lead to the following
degenerate singular Cahn-Hilliard-de Gennes polymer system:
\begin{align}\label{CH1p}
  & \chi_t - \Lambda_0 \dive (\chi (1 - \chi) \nabla w)= 0,\\
 \label{CH2p}
  & w = - a(\chi) \Delta \chi - \frac{a'(\chi)}2 |\nabla \chi|^2
   + F^{\prime}_{FH}(\chi)+ \epsi \chi_t
\end{align}
to be considered together with the initial
and appropriate boundart conditions.

As a first step in the study of such system we consider the case
of constant mobility by linearizing the principal part
$\chi (1-\chi)$ in \eqref{CH1p}.
Moreover, for mathematical convenience,
we replace $\chi$ by the variable $u$ defined by \eqref{defiu},
restrict ourselves to the symmetrical mixture \eqref{apolymersymm},
and set all physical constants equal to unity.
This leads to  \eqref{CH1}-\eqref{CH2}.

Apart from the well-known Cahn-Hilliard-de Gennes model described above,
we mention also recently developed two-fluids models for viscoelastic
phase separation in polymer solutions, see \cite{ZZE06}
and the references therein.

It is known that the process of polymer phase separation
is important both for its theoretical aspect and due to
unusual morphology for specific materials applications.
On the contrary to extensive physical and numerical literature
on polymer mixtures (for review see e.g., \cite{Bin83}, \cite{dG85},
\cite{NL84}, \cite{Nos87}, \cite{AP96}),
mathematical aspects of the corresponding models have not been
so far addressed sufficiently.
The structure of steady-state solutions to degenerate singular
polymer system \eqref{CH1p}-\eqref{CH2p} has been analysed by
Mitlin et al. \cite{MiMaE85}, \cite{MiMa90},
and Witelski \cite{Wit98}. The existence of weak solutions to the
Cahn-Hilliard equation with logarithmic potential $F(u)$, degenerate mobility
$\Lambda(u) =1-u^2$, and constant gradient coefficient $a>0$ has been studied
by Elliott and Garcke \cite{EG}; we refer also to \cite{DpGG} and \cite{GNC00}
for further study of degenerate problems.
For the standard Cahn-Hilliard equation with logarithmic potential $F(u)$
and constant coefficient $a>0$, optimal regularity of weak solutions is
analyzed in \cite{MZ}, where in particular the separation property~\eqref{separs}
is obtained in space dimensions 1 and 2.
The existence and uniqueness of weak solutions to the Cahn-Hilliard equation
with logarithmic potential and nonlinear positive, bounded coefficient
$a(u)$ has been recently proved by the authors in \cite{SP11}.

We mention also a closely related sixth order Cahn-Hilliard type problem with
nonlinear coefficient $a(u)$, considered recently in \cite{SP11}
for a singular (e.g., logarithmic) potential, and in \cite{PZ11}
for a polynomial potential. As a special case, in \cite{SP11}
the behavior of the solutions when the sixth order term is let tend to zero
was analysed.

Finally, we point out that the theoretical investigation
of polymer models was initiated by Alt and the second author
in \cite{AP96}, where general nonisothermal phase transition
models with a conserved order parameter
have been derived and, in particular,  polymer free energy models
have been presented along with an extensive list of references.
The authors of \cite{AP96} have obtained also
some partial, unpublished results  \cite{APNotes96} on the existence
of weak solutions to degenerate singular polymer model
\eqref{CH1p}-\eqref{CH2p}
by applying the methods due to Elliott and Garcke \cite{EG},
and Elliott and Luckhaus \cite{EL91}.
This unsolved problem has become the motivation of the present study
which uses a different approach developed previously in \cite{SP11}.

As already mentioned, as a first step of the analysis
we assume that the mobility is constant,
and extend our methods applied in \cite{SP11}
in the case of a nonlinear (but bounded)
coefficient $a(u)$ to singular $a(u)$.

There are two main ideas behind our approach.
The first one, standard in the analysis of Cahn-Hilliard systems,
exploits the characteristic variational structure
of the system \eqref{CH1}-\eqref{CH2}.
The second non-standard one consists in introducing appropriate
changes of variables.

The variational structure becomes evident by (formally)
testing \eqref{CH1} by $w$, \eqref{CH2} by $u_t$,
taking the difference of the obtained relations,
integrating with respect to space variables,
using the {\sl no-flux}\/ conditions \eqref{neum-intro},
and performing suitable integrations
by parts. Then one readily gets the {\sl a-priori}\/ bound
\begin{equation}\label{energyineq}
  \ddt\calE(u) + \| \nabla w \|_{L^2(\Omega)}^2
   + \epsi \| u_t \|_{L^2(\Omega)}^2
   = 0,
\end{equation}
which has the form of an {\sl energy equality}\/ for the
{\sl energy functional}
\begin{equation}\label{defiE}
  \calE(u)=\io \Big( \frac{a(u)}2 |\nabla u|^2
   + F(u) - \lambda \frac{u^2}2 \Big),
\end{equation}
where the interface (gradient) part contains the nonlinear
function $a$. In other words, the system \eqref{CH1}-\eqref{CH2}
arises as the $(H^1)'$-gradient flow problem for the
functional $\calE$.

However, the energy estimate~\eqref{energyineq} is not sufficient
to obtain existence of a solution to \eqref{CH1}-\eqref{CH2}
via approximation-compactness methods. Actually, to apply
this strategy, one also
needs some control of the second space derivatives of $u$ and of the
singular coefficients $a$ and $a'$ in \eqref{CH2}. To obtain this,
two changes of variables will play an important role. The first one,
motivated by the structure of the fourth formula in
\eqref{chempot} and applied also in \cite{SP11}, consists in introducing
the variable $z$ such that $\nabla z = \frac{1}{\sqrt{2}} \sqrt{a(u)} \nabla u$.
Then, the Laplacean of $z$ can be (formally)
estimated simply by testing the equivalent
$z$-formulation of \eqref{CH2} (namely, \eqref{CH2z} below) by
$-\Delta z$. This gives the desired control on second space derivatives.

On the other hand, even in the equivalent formulation \eqref{CH2z},
one has to control a coefficient (namely, $\phi'(u)$), that explodes
polynomially fast as $|u|$ approaches $1$. This is a nontrivial
issue since the nonlinear term $f(u)$ in \eqref{CH2} (or in
\eqref{CH2z}), for which is relatively simple to get
a $L^2$-control, explodes only {\sl logarithmically}\/ fast
(and, hence, an $L^p$-estimate of $f(u)$ would help to control
$\phi'(u)$, or $a(u)$, only for $p=\infty$).
To overcome this further difficulty, a second additional
change of variable comes into help. Namely, we set
$v := f(u)$, which represents
the monotone part of the volumetric
chemical potential. We then see that the formulation
of equation \eqref{CH2} in terms of $v$
(namely, \eqref{CH2v} below) does no longer contain
singular coefficients of polynomial type;
it presents, however, the cubic term $u |\nabla v|^2$.
To control it, we use techniques based on entropy-type
estimates (cf.~\cite{DpGG}), with a rather careful and
{\sl ad-hoc}\/ choice of test functions
(cf.~\eqref{defim} and \eqref{defimp} below).
In this way we can both prove an $L^\infty$-bound
for $v$ (and consequently the ``separation property''
\eqref{separs}), for strictly positive times,
and also control the cubic term $u |\nabla v|^2$
starting from the initial time $t=0$. This is the key
step that permits to get existence of a weak solution
(for the $v$-formulation of the system) for initial
data $u_0$ that have only the natural energy regularity
(i.e.,~such that $\calE(u_0)<+\infty$).

Regarding additional properties of solutions, we remark
that, as already noticed in \cite{DNS}, the energy
\eqref{defiE} with the coefficient $a$ given by
\eqref{defia} is convex with respect to~$u$
(up to the lower order $\lambda$-perturbation). This
basic property permits to prove parabolic time-regularization
properties of weak solutions, as well as uniqueness,
in a relatively standard way.

The plan of the paper is as follows.
In the next Section~\ref{sec:not} we will present
the main assumptions and give the statement of
our main results. In Section~\ref{sec:strong}, we will
prove local existence and uniqueness of a strong (i.e.,
lying in a suitable H\"older class) solution.
The main a-priori estimates needed in the proof
of global existence will be detailed in the subsequent
Section~\ref{sec:apriori}. On the basis of these
estimates, in the final Section~\ref{sec:weak}
we shall show global existence, uniqueness, and
time-regularization properties of weak solutions.

\medskip
\noindent
{\bf Acknowledgment.}~~The second author has greatly benefited
from the cooperation on phase transition models with Prof.~H.W.~Alt
during her stay at the Institute of Applied Mathematics,
University of Bonn. The support of SFB~256
``Nichtlineare partielle Differentialgleichungen''
is greatly acknowledged.

\section{Notation and main results}
\label{sec:not}

Let $\Omega$ be a smooth bounded domain of $\RR^3$
of boundary $\Gamma$, $T>0$ a given final time, and
let $Q:=(0,T)\times\Omega$. Let $H:=L^2(\Omega)$,
endowed with the standard scalar product $(\cdot,\cdot)$
and the norm $\| \cdot \|$. Let also $V:=H^1(\Omega)$.
We identify $H$ with $H'$ so that the chain of continuous
embeddings $V\subset H \subset V'$ holds.
We indicate by $\duav{\cdot,\cdot}$ the duality between
$V'$ and $V$ and by $\|\cdot\|_X$ the norm in
the generic Banach space $X$. We note as $A$ the
weak Laplace operator with no-flux boundary conditions,
namely
\begin{equation} \label{defiA}
   A: V \to V', \qquad
    \duav{Av,z} := \io \nabla v \cdot \nabla z,
      \quad \perogni v, z \in V.
\end{equation}
We also set
\begin{equation} \label{defiW}
  W := \big\{ z \in V:~Az \in H \big\}
   = \big\{ z \in H^2(\Omega):~\dn z = 0~\text{on }\Gamma \big\},
\end{equation}
which is a closed subspace of $H^2(\Omega)$.
In all what follows we shall assume that $F$, $f$ and
$a$ are given, respectively, by \eqref{logpot},
\eqref{f} and \eqref{defia}. Moreover, we will
assume that $u_0$ is an initial datum having
{\sl finite energy}\, $\calE$ (cf.~\eqref{defiE}),
namely
\begin{equation} \label{EE0}
  \EE_0:= \calE(u_0) < +\infty.
\end{equation}
It is worth noting that, since $F$ has
the expression \eqref{logpot},
the above is equivalent to asking
\begin{equation} \label{regou0}
  -1 \le u_0 \le 1~~\text{a.e.~in }\,\Omega, \qquad
  \qquad  a^{1/2}(u_0) \nabla u_0  \in H.
\end{equation}
Moreover, due to \eqref{defia}, if $u_0$ satisfies \eqref{regou0}
then $u_0\in V$, in particular. Letting, for a generic
summable function $v:\Omega\to \RR$, $v\OO$ denote its
spatial mean value, we will also assume that
\begin{equation} \label{regou02}
  m := (u_0)\OO = \frac1{|\Omega|} \io u_0
   \in (-1,1).
\end{equation}
In other words, we cannot admit the case when
$u_0$ coincides with $+1$ (or with $-1$) almost everywhere
in~$\Omega$. This is a standard assumption when dealing
with Cahn-Hilliard systems containing constraint terms
(cf, e.g., \cite{KNP} for more details).

Let us note also that, substituting the expression
\eqref{defia} for $a$, \eqref{CH2} can be rewritten as
\begin{equation}\label{CH2a}
  w = - \frac2{1-u^2} \Delta u
   - \frac{2u}{(1-u^2)^2} |\nabla u|^2
   + f(u) - \lambda u + \epsi u_t.
\end{equation}
It is now convenient to introduce
a couple of additional variables which permit to give
alternative formulations of \eqref{CH2a}.
To start with, we compute some derivatives of $a$.
From \eqref{logpot} and \eqref{defia}, we have
\begin{equation}\label{aprimo}
   a'(r) = f''(r) = \frac{4r}{(1-r^2)^2},
\end{equation}
as well as
\begin{equation}\label{asecondo}
   a''(r) = f'''(r) = \frac{4(1+3r^2)}{(1-r^2)^3}.
\end{equation}
In the sequel, for a generic locally integrable
real-valued function $\psi$ defined in an open
neighbourhood of $0$, we will write
\begin{equation}\label{deficiapo}
  \widehat{\psi}(r):=\int_0^r \psi(s)\,\dis.
\end{equation}
Then, of course, we have
\begin{equation}\label{aciapo}
   \aciapo(r) = \int_0^r a(s)\,\dis
    = f(r) = \log (1+r) - \log (1-r).
\end{equation}
Next, we introduce
\begin{equation}\label{defiphi}
  \phi(r) := \frac{\sqrt2}2\int_0^r a^{1/2}(s)\,\dis
    = \int_0^r \frac{ \dis }{ (1-s^2)^{1/2} }
    = \arcsin r.
\end{equation}
Then, we notice that, setting
\begin{equation} \label{defiz}
  z := \phi(u) = \arcsin u,
\end{equation}
equation \eqref{CH2} can be rewritten in the equivalent
form
\begin{equation}\label{CH2z}
  w = - 2 \phi'(u) \Delta z + f(u)
   - \lambda u + \epsi u_t.
\end{equation}
Next, we put $v:= f(u)$. Then, a simple computation gives
\begin{equation}\label{co61}
  u = f^{-1}(v) = \frac{e^v - 1}{e^v + 1} =: j(v).
\end{equation}
Moreover, \eqref{CH2} can be rewritten as
\begin{equation}\label{CH2v0}
  w = - \Delta v + v + \frac{a'(u)}2 | \nabla u |^2
   - \lambda u + \epsi u_t.
\end{equation}
Noting that
\begin{equation}\label{co62}
  \frac{a'(u)}2 | \nabla u |^2
   = \frac{2u}{(1-u^2)^2} | \nabla u |^2
   = \frac{u}2 \big| f'(u)\nabla u \big|^2,
\end{equation}
we finally obtain from \eqref{co61}
\begin{equation}\label{CH2v}
  w = - \Delta v + v + \frac{j(v)}2 | \nabla v |^2  - \lambda j(v) + \epsi u_t
   = - \Delta v + v + \frac{1}2 \frac{e^v - 1}{e^v + 1} | \nabla v |^2
      - \lambda j(v) + \epsi u_t.
\end{equation}

In the sequel, we shall indicate by $c$ a generic positive
constant, whose value may vary on occurrence,
allowed to depend on the parameters of the system
(more precisely, on the functions $a$ and $f$, on $\lambda$
and on $\Omega$), and, in particular, not on approximating
parameters. Moreover, the constants $c$ will not be allowed
to depend on the choice of initial data. However,
they may depend on the prescribed mean value $m$.
The notation $\kappa$ will be used for positive constants
(depending on the same quantities as $c$) appearing in
estimates from below. We will also use the notation
$Q(\cdot)$ (or $Q(\cdot,\cdot)$),
with $Q$ indicating a computable function with values in
$[0,+\infty)$, increasingly monotone in each of its argument,
whose expression can depend on the same quantities as $c$.
For instance, the expression $Q(\EE_0)$ will stand for a
monotone function of the ``initial energy'' $\EE_0$.

We can now introduce the concepts of ``classical'' and of ``weak''
solution needed in the subsequent analysis.
\bede\label{def:classsol}

 A ``strong'', or ``classical'', solution to~{\rm Problem~(P)}
 over the time interval $(0,T)$
 is a couple $(u,w)$ with the regularity

 \begin{align}\label{regoust}
   & u\in W^{1,\infty}(0,T;V') \cap H^1(0,T;V)
      \cap L^\infty(0,T;H^2(\Omega)),
    \qquad \epsi u\in W^{1,\infty}(0,T;H),\\
  \label{regowst}
   & w\in L^\infty(0,T;V) \cap L^2(0,T;H^3(\Omega)), \qquad
    \epsi w\in L^{\infty}(0,T;H^2(\Omega)),
 \end{align}

 satisfying, a.e.~in~$(0,T)$, the equations

 \begin{align}\label{CH1s}
   & u_t - \Delta w = 0, \quext{a.e.~in }\,\Omega,\\
  \label{CH2s}
   & w = - a(u) \Delta u - \frac{a'(u)}2 |\nabla u|^2
   + f(u) - \lambda u + \epsi u_t,
    \quext{a.e.~in }\,\Omega,\\
  \label{neums}
   & \dn u = \dn w = 0, \quext{a.e.~on }\,\Gamma,
\end{align}
together with the initial condition $u|_{t=0} = u_0$
and, for all $(t,x)\in [0,T]\times \barO$,
the\/ {\rm separation property}
\begin{equation} \label{separs}
  -1 + \epsilon \le u(t,x) \le 1 - \epsilon,
   \quext{for some }\,\epsilon>0.
\end{equation}
\edde
\beos\label{morereg}

 Thanks to \eqref{separs}, the component $u$ of
 any ``classical'' solution to Problem~(P) is uniformly
 separated from the singular values $\pm1$ of $a$ and $f$.
 Hence, by applying the standard theory of quasilinear
 parabolic equation and a bootstrap argument,
 we can see that $(u,w)$ is in fact smoother, with
 its regularity being limited only by the regularity of the
 initial datum. In other words, at
 least for times $t>0$, a classical solution
 can be thought to be arbitrarily regular.

\eddos
\bede\label{def:weakssol}

 A ``weak'', or ``energy'', solution to~{\rm Problem~(P)}
 over the time interval $(0,T)$
 is a couple $(u,w)$ with the regularity

 \begin{align}\label{regou4}
   & u\in \HUVp\cap L^\infty(0,T;V) \cap L^\infty((0,T)\times\Omega),
    \qquad \epsi u\in \HUH,\\
   \label{regoFu4}
   & F(u) \in L^\infty(0,T;L^1(\Omega)),\\
   \label{regofu4}
   & v = f(u) \in L^2(0,T;V),
    \qquad \nabla v\in L^p((0,T)\times\Omega)~~
     \text{for some }\,p>2,\\
  \label{regow4}
   & w\in L^2(0,T;V),
 \end{align}

 satisfying, a.e.~in~$(0,T)$, the equations

 \begin{align}\label{CH1w}
   & u_t + A w = 0, \quext{in }\,V',\\
  \label{CH2w}
   & w = - \Delta v + v + \frac{u}2 | \nabla v |^2  - \lambda u + \epsi u_t,
    \quext{a.e.~in }\,\Omega,\\
  \label{uv}
   & v = f(u), \quext{a.e.~in }\,\Omega,\\
  \label{neumw}
   & \dn v = 0, \quext{a.e.~on }\,\Gamma.
 \end{align}

 together with the initial condition~$u|_{t=0} = u_0$.

\edde
\noinden

We can now state our main result, regarding existence,
uniqueness, and regularization properties of
weak solutions to~Problem~(P):
\bete\label{teo:main}

 Let $f$ and $a$ be given by \eqref{f}, \eqref{defia}.
 Let $u_0$ satisfy\/ \eqref{EE0} and~\eqref{regou02}.
 Finally, let $\Omega$ be\/ {\rm convex}.
 Then, for any $T>0$, {\rm Problem~(P)} admits at least
 a weak solution  $(u,w)$ defined over $(0,T)$.
 Moreover, for any $\tau>0$, $(u, w)$ is a ``classical'' solution
 over $(\tau,T)$. In particular, the separation
 property~\eqref{separs} holds on $(\tau,T)$
 with

 \begin{equation}\label{separepsi}
   \epsilon^{-1} = Q(\EE_0,\tau^{-1}).
 \end{equation}

 Finally, uniqueness holds in the class of weak
 solutions that are classical for strictly positive times.

\ente
\noinden
For strictly positive times, \eqref{CH2} can be interpreted
in any of the equivalent formulations \eqref{CH2a}, \eqref{CH2z},
or \eqref{CH2v}; actually, $u$ is a classical solution
for $t>0$. On the other hand, when looking at the behavior
near $t=0$, it is crucial to view \eqref{CH2} in
the form~\eqref{CH2w} which appears to be the only
formulation permitting to take the approximation limit
$\delta\searrow 0$ starting from the initial time.
\beos\label{morereg2}

 If we have in addition that the initial datum
 satisfies $u_0\in H^2(\Omega)$ with
 $u_0(x)\in [-1+\epsilon_0,1-\epsilon_0]$ for all
 $x\in \barO$ and some $\epsilon_0\in(0,1)$, there are no
 complications due to the boundary layer $t\searrow 0$,
 and so $u$ can be seen as
 a ``classical'' solution over the whole $(0,T)$. This can be deduced
 simply by using the classical Gronwall lemma (instead of the
 uniform Gronwall lemma) in the a-priori estimates.
 As a consequence, estimates \eqref{stz1}, \eqref{st72} and
 \eqref{st72c} below hold in fact with $\tau=0$ in this case.

\eddos
\beos\label{mirzel}

 It is worth stressing that the separation property~\eqref{separs}
 holding for our model is instead an open issue in the frame of the
 standard Cahn-Hilliard system with logarithmic nonlinearity
 $f(u)$, at least in the three-dimensional setting
 (cf.~\cite{MZ} for further remarks).

\eddos

\section{Local strong solutions}
\label{sec:strong}

In this section, we will prove existence
of at least one {\sl local in time}\/ classical
solution to Problem~(P). With this aim, we first
introduce a regularization of the
initial datum $u_0$. This is the object of the following
\bele\label{lemma:u0}

 Let $u_0$ satisfy~\eqref{EE0} and~\eqref{regou02}.
 Then, for any $\delta\in(0,1/6)$, there exists
 $u\zzd$ such that $u\zdd \in C^{0,a}(\barO)$
 for any $a\in(0,1/2)$. Moreover,

 \begin{equation} \label{uzzd0}
   -1+3\delta \le u\zzd(x) \le 1-3\delta \quad \perogni
   x\in \barO.
 \end{equation}

 Finally, we have that

 \begin{align}\label{uzzd1}
   & u\zzd \to u_0 \quext{weakly in }\,V,\\
  \label{uzzd2}
   & \calE(u\zzd) \le c \big( 1 + \calE(u_0) \big) \quad \perogni \delta\in(0,1/6).
 \end{align}
\enle
\begin{proof}
First of all, we set
\begin{equation} \label{uzzd3}
  u\zzd^{(1)}:= \min\big\{ 1 - 3\delta, \max\{ u_0, -1 + 3\delta \} \big\},
\end{equation}
so that $u\zzd^{(1)}$ satisfies a.e.~in $\Omega$ the equivalent of
\eqref{uzzd0}. Moreover, it is clear that
$\calE(u\zzd^{(1)})\le \calE(u_0)$ for all $\delta$.
Next, we define $z\zzd^{(1)}:=\phi(u\zzd^{(1)})=\arcsin u\zzd^{(1)}$.
Then, we proceed by singular perturbation, defining $z\zzd$ as the
unique solution of the elliptic problem
\begin{equation} \label{uzzd4}
  z\zzd + \delta A z\zzd = z\zzd^{(1)}.
\end{equation}
Being $z\zzd^{(1)}\in V$, then, by elliptic regularity
(recall that $\Omega$ is a smooth domain),
$z\zzd\in H^3(\Omega)$ for all $\delta$. Setting
$u\zzd:= \sin z\zzd$, a direct check permits to verify
that (at least) $u\zzd\in H^2(\Omega)$ for all $\delta$.
Hence, $u\zzd$ is H\"older continuous, as desired.
Moreover, by monotonicity of $\phi$ and a standard
maximum principle argument, it is clear that
\eqref{uzzd0} holds.

The key step consists in proving \eqref{uzzd2}.
Actually, it is obvious that $\| F(u\zzd) \|_{L^1(\Omega)} \le c$
(cf.~\eqref{regou0}).
To control the gradient term of $\calE$,
we test \eqref{uzzd4} by $A z\zzd$. We obtain
\begin{align} \no
  \frac12 \io a(u\zzd) |\nabla u\zzd|^2 &
   = \| \nabla z\zzd \|^2
   \le \| \nabla z\zzd^{(1)} \|^2\\
 \label{uzzd5}
  & = \frac12 \io a(u\zzd^{(1)}) |\nabla u\zzd^{(1)}|^2
   \le \frac12 \io a(u_0) |\nabla u_0|^2
   \le \calE(u_0) + c,
\end{align}
as desired. Hence, we have \eqref{uzzd2}. To conclude, we
have to prove \eqref{uzzd1}, which is however an immediate
consequence of standard weak compactness arguments.
The proof is complete.
\end{proof}
\noinden
As a next step, we also provide a modification of
the function $f$ given by~\eqref{f}.
Namely, for all $\delta\in(0,1/6)$,
we take $f_\delta\in C^1(-1+\delta,1-\delta)\to \RR$ such
that $f_\delta$ is monotone, $|f_\delta(r)| \ge |f(r)|$
for all $r\in(-1+\delta,1-\delta)$, and
\begin{align} \label{defifd}
  & f_\delta(r) = f(r) \quext{if }\,r\in[-1+2\delta,1-2\delta],\\
 \label{defifd2}
  & \lim_{|r|\to 1-\delta} f_\delta(r)\sign r = +\infty.
\end{align}
It is obvious that, for any $\delta\in(0,1/6)$, a function
$f\dd$ with the above properties exists.

Finally, we modify $a$ by taking $a\dd\in C^2(\RR;\RR)$ such
that
\begin{equation} \label{defiad0}
  a\dd(r) = a(r) \quad\perogni r\in[-1+\delta, 1-\delta].
\end{equation}
In particular,
\begin{equation} \label{defiad}
  a\dd''(r)\ge 0, \quad \Big(\frac1{a\dd}\Big)''(r) = -1
   \quad \perogni r\in [-1+\delta,1-\delta].
\end{equation}
Outside $(-1,1)$, $a\dd$ is taken as a constant
$K_\delta>0$ (exploding as $\delta\searrow 0$),
whereas for $|r|\in (1-\delta,1)$, $a\dd$ is chosen
in such a way to have
\begin{equation} \label{defiad2}
  1 \le a\dd(r) \le K_\delta
   \quad \perogni r \in \RR.
\end{equation}
Then, for $\delta\in(0,1/6)$, we can consider the system
\begin{align}\label{CH1d}
  & u_{\delta,t} - \Delta w\dd = 0,
   \quext{in }\,(0,T)\times\Omega,\\
 \label{CH2d}
  & w\dd = - a\dd(u\dd) \Delta u\dd - \frac{a\dd'(u\dd)}2 |\nabla u\dd|^2
   + f\dd(u\dd) - \lambda u\dd + \epsi u_{\delta,t},
   \quext{in }\,(0,T)\times\Omega,\\
 \label{inizd}
  & u\dd|_{t=0} = u\zzd,
  \quext{in }\,\Omega,\\
 \label{neumd}
  & \dn u\dd = \dn w\dd = 0,
   \quext{on }\, \Gamma.
\end{align}
We then have:
\bete\label{teo:loc:delta}

 Let\/ $f$ and\/ $a$ be given respectively by\/ \eqref{f},
 \eqref{defia},
 and let\/ $u_0$ satisfy~\eqref{EE0} and~\eqref{regou02}.
 For $\delta\in(0,1/6)$, let $u\zzd$ be defined by\/
 {\rm Lemma~\ref{lemma:u0}} and $f\dd$, $a\dd$ be given by\/
 \eqref{defifd}-\eqref{defiad2}.
 Then, there exists one and only one
 solution $(u\dd,w\dd)$ to system \eqref{CH1d}-\eqref{neumd}
 with the regularity

 \begin{align}\label{regoudd}
   & u\dd \in W^{1,\infty}(0,T;V') \cap H^1(0,T;V) \cap L^\infty(0,T;H^2(\Omega)),
    \quad \epsi u\dd \in W^{1,\infty}(0,T;H),\\
  \label{regowdd}
   & w\dd \in L^\infty(0,T;V)\cap L^2(0,T;H^3(\Omega)), \quad
    \epsi w\dd \in L^\infty(0,T;H^2(\Omega)).
 \end{align}

\ente
\begin{proof}

We claim that this result is essentially a consequence of the
results of \cite{SP11}. Actually, we see that, for any
$\delta\in (0,1/6)$, $a\dd$ satisfies the assumptions
\cite[(2.1)-(2.2)]{SP11} (where $-1$ and $1$ replace
$-2$ and $2$ in \cite[(2.2)]{SP11}) and
\cite[(6.1)]{SP11} (where $[-1,1]$ is replaced by
$[-1+\delta,1-\delta]$, cf.~\eqref{defiad}).

Moreover, $f\dd$ satisfies \cite[(2.3)-(2.4)]{SP11},
with $(-1+\delta,1-\delta)$ replacing $(-1,1)$.
Thus, we can apply \cite[Theorems~5.1, 6.1, 6.2]{SP11}
which give the existence and uniqueness of a weak solution
$(u\dd,w\dd)$ to \eqref{CH1d}-\eqref{neumd}. The regularity
of this solution is specified
by~\cite[(4.3)-(4.4) and (6.14)]{SP11}. More precisely,
since the initial datum $u\zzd$ is smooth and
separated in the uniform norm from the singular values
$\pm(1-\delta)$ of $f\dd$ due to \eqref{uzzd0},
we have here that \cite[(6.14)]{SP11}
holds starting from the initial
time, i.e., with $\tau=0$. Moreover,
a closer inspection of \cite[Proof of Theorem~6.2]{SP11}
(see in particular estimates~(6.19)-(6.20) therein)
permits to see that the additional regularity
for $u_{\delta,t}$ stated in \eqref{regoudd} holds
over $(0,T)$.
The $L^2(0,T;H^3(\Omega))$ regularity of $w\dd$ follows
from the $L^2(0,T;V)$ regularity of $u_{\delta,t}$ and elliptic
regularity estimates applied to \eqref{CH1d}.

Hence, collecting all the information coming from
the results of \cite{SP11}, we obtain exactly
\eqref{regoudd}-\eqref{regowdd}. This concludes the
proof.
\end{proof}
\noinden
Notice now that, as a consequence of \eqref{regoudd}-\eqref{regowdd}
and of the arguments in \cite{SP11}, we have, more precisely,
the a-priori estimate
\begin{equation} \label{std11}
  \| u\dd \|_{H^1(0,T;V')} + \| u\dd \|_{L^\infty(0,T;H^2(\Omega))}
   \le Q ( \| u\zzd \|_{H^2(\Omega)}, \delta^{-1}).
\end{equation}
By interpolation and embedding properties of Sobolev spaces,
we then obtain
\begin{equation} \label{std12}
  \| u\dd \|_{C^{0,b}([0,T]\times \barO)}
   \le Q ( \| u\zzd \|_{H^2(\Omega)}, \delta^{-1}),
   \quext{for some }\,b>0.
\end{equation}
Thus, $u\dd$ is globally H\"older continuous.
Since $u\zzd$ satisfies
 \eqref{uzzd0}, we can deduce that,
once the initial datum $u_0$ is assigned, then
for all $\delta\in(0,1/6)$ there exists
a (computable) time $T_\delta>0$
depending on $\delta$ and $u_0$ such that
\begin{equation} \label{std13}
  -1 + 2\delta \le u\dd(t,x) \le 1 - 2\delta
   \quext{for all }\,(t,x) \in [0,T_\delta]\times \barO.
\end{equation}
However, over $[-1 + 2\delta,1 - 2\delta]$, $a\dd$ coincides
with $a$ by \eqref{defiad0} and $f\dd$ coincides with $f$
by \eqref{defifd}. Hence, we have obtained the
\beco\label{cor:delta}

 Let\/ $f$ and\/ $a$ be given by\/ \eqref{f}, \eqref{defia},
 and let\/ $u_0$ satisfy~\eqref{EE0} and~\eqref{regou02}.
 Let $\delta\in(0,1/6)$ and let $u\zzd$ be given by\/
 {\rm Lemma~\ref{lemma:u0}}.
 Then, there exist a time $T_\delta>0$ depending on $u_0$ and $\delta$,
 and a ``classical'' solution $(u\dd,w\dd)$ to~{\rm Problem~(P)},
 with initial datum $u\zzd$, over the time interval $(0,T\dd)$.

\enco

\section{A priori estimates}
\label{sec:apriori}

In this section, we derive a number of a-priori estimates
for the solutions of our system. We stress that
the procedure leading to these estimates can be rigorously
justified at least for ``classical'' solutions
to Problem~(P) in the sense of Definition~\ref{def:classsol}.
Indeed, owing to~Remark~\ref{morereg}, these solutions can
be thought to be as smooth as we need (possibly paying the
price of additionally regularizing $u_0$).

In particular, the estimates proved below will
hold for local strong solutions given by
Theorem~\ref{teo:loc:delta}. On the other hand, for weaker
notions of solutions the procedure below may just
have a formal character due to insufficient regularity
of test functions. We will clarify this point
in Section~\ref{sec:weak} below.
 Here, we will proceed
assuming that everything is regular enough for our purposes.
Moreover, since the estimates we are going to derive
will have a global-in time character, with some abuse
of notation we will directly work on the time interval
$[0,T]$. The underlying extension argument will be
also detailed in the next section.

We can now start detailing our estimates.

\smallskip

\noinden
{\bf Energy estimate.}~
We test \eqref{CH1} by $w$ and \eqref{CH2} by $u_t$.
This gives rise to the energy equality \eqref{energyineq},
whence we obtain the estimate
\begin{align}\label{st11}
  & \| u_t \|_{L^2(0,T;V')}^2
   + \epsi \| u_t \|_{L^2(0,T;H)}^2
   + \| \nabla w \|_{L^2(0,T;H)}^2 \le Q(\EE_0),\\
 \label{st12}
  & \| F(u) \|_{L^\infty(0,T;L^1(\Omega))}
   \le Q(\EE_0).
\end{align}
Thus, by definition \eqref{logpot} of $F$,
we infer in particular that
\begin{equation}\label{st13}
  -1 \le u \le 1 \quext{a.e.~in }\,
   (0,T)\times\Omega.
\end{equation}
Moreover, recalling also \eqref{defiz}, we obtain
\begin{equation}\label{st14}
  \| u \|_{L^\infty(0,T;V)}
   + \| z \|_{L^\infty(0,T;V)}  \le Q(\EE_0).
\end{equation}

\smallskip

\noinden
{\bf Estimate on time derivatives.}~
Following the lines of \cite[Theorem~6.1]{SP11}, we
indicate by $J$ the gradient part of the energy, i.e.,
\begin{equation} \label{defiJ}
  J:V\to [0,+\infty), \qquad
   J(u):=\io \frac{a(u)}2 | \nabla u |^2.
\end{equation}
Then, we can (formally) compute the first
derivative of $J$, given by
\begin{equation} \label{Jprime}
  \duavg{J'(u),v} =
    \io \Big( a(u) \nabla u \cdot \nabla v
    + \frac{a'(u)}2 |\nabla u|^2 v \Big),
\end{equation}
as well as the second derivative
\begin{equation}\label{Jsecond}
  \duavg{J''(u)v,z} =
  \io \Big( \frac{a''(u)|\nabla u|^2 vz}2
   + a'(u) v \nabla u\cdot\nabla z
   + a'(u) z \nabla u\cdot\nabla v
   + a(u) \nabla v\cdot\nabla z \Big).
\end{equation}
To be more precise, if $u$ is a smooth solution
(and in particular it is separated from singularities,
i.e., it satisfies \eqref{separs}),
formulas \eqref{Jprime} and \eqref{Jsecond}
make sense for $u = u(t)$ at any time $t\in[0,T]$
and, indeed, we have that $J'(u)\in V'$
and $J''(u)\in \calL(V,V')$.

From \eqref{Jsecond}, we then obtain in particular
\begin{align} \no
  \duavg{J''(u)v,v}
  & = \io \Big( \frac{a''(u)|\nabla u|^2 v^2}2
    + 2 a'(u) v \nabla u\cdot\nabla v
    + a(u) | \nabla v |^2 \Big)\\
 \label{Jvv}
  & \ge \io \Big( a(u) - \frac{2 a'(u)^2}{a''(u)} \Big)
   | \nabla v|^2.
\end{align}
We can now test \eqref{CH1} by $t w_t$ and add the time
derivative of \eqref{CH2} tested by $t u_t$.
This leads to
\begin{equation}\label{Irena11}
  \frac{t}{2} \ddt \| \nabla w \|^2
   + \frac{t \epsi}{2} \ddt \| u_t \|^2
   + t \duavg{J''(u)u_t,u_t}
   + t \io f^{\prime}(u) u_t^2 = \lambda t \| u_t \|^2.
\end{equation}
Hence, using \eqref{Jvv}, noting that
\begin{equation}\label{co21}
  a(u)-\frac{2a'(u)^2}{a''(u)}
   = \frac{2}{1+3u^2},
\end{equation}
and using \eqref{st13},
we arrive at the inequality
\begin{align} \no
  & \ddt\Big(
   \frac{t}2 \| \nabla w \|^2
    + \frac{\epsi t}2 \| u_t \|^2 \Big)
   + t \io \Big( \frac{2}{1+3u^2}
   | \nabla u_t |^2 \Big)
   + t \io f'(u) u_t^2\\
  \label{new11}
  & \mbox{}~~~~~
   \le \lambda t \| u_t \|^2
    + \frac12 \| \nabla w \|^2
    + \frac{\epsi}2 \| u_t \|^2
  \le \frac{t}4 \| \nabla u_t \|^2
    + c (1+t) \| \nabla w \|^2
    + \frac{\epsi}2 \| u_t \|^2.
\end{align}
Recalling \eqref{st11} and
noting that $2/(1+3u^2)\ge 1/2$ since
$u$ takes values in $[-1,1]$,
integration in time of \eqref{new11} gives,
for any $\tau \in (0,T)$,
\begin{align}\label{st21}
  & \| u_t \|_{L^\infty(\tau,T;V')}
   + \| \nabla w \|_{L^\infty(\tau,T;H)} \le Q(\EE_0,\tau^{-1}),\\
 \label{st22}
  & \epsi \| u_t \|_{L^\infty(\tau,T;H)}
    + \| u_t \|_{L^2(\tau,T;V)} \le Q(\EE_0,\tau^{-1}).
\end{align}

\smallskip

\noinden
{\bf Estimate of $f(u)$.}~
We test \eqref{CH2} by $u-m$. Integrating
by parts the terms depending on $a$, we obtain
\begin{equation}\label{conto51}
  \io \Big( a(u) + \frac{a'(u)}2 ( u - m ) \Big) | \nabla u |^2
   + \io f(u) ( u - m )
    =\big( w + \lambda u - \epsi u_t, u - m \big).
\end{equation}
We have to estimate some terms. Firstly,
proceeding as in~\cite[Appendix]{MZ}, it
is not difficult to prove that
\begin{equation}\label{conto52}
  \io f(u) ( u - m )
   \ge \frac12 \| f(u) \|_{L^1(\Omega)} - c_m.
\end{equation}
We notice that assumption~\eqref{regou02} is used here.

We now observe that
\begin{equation}\label{co31}
  \io \Big( a(u) + \frac{a'(u)}2 ( u - m ) \Big) | \nabla u |^2
  = \io \Big( \frac{2u(u-m)}{(1-u^2)^2} + \frac2{1-u^2} \Big)
   |\nabla u|^2.
\end{equation}
Moreover, there exist constants $\kappa_m>0$, $c_m\ge 0$
such that
\begin{equation}\label{co32}
  \frac{2u(u-m)}{(1-u^2)^2}
   \ge \kappa_m \frac{1}{(1-u^2)^2}
   - c_m
   \ge \kappa_m \frac{u^2}{(1-u^2)^2}
   - c_m.
\end{equation}
Thus, recalling \eqref{defia}, we get
\begin{equation}\label{co31x}
  \io \Big( a(u) + \frac{a'(u)}2 ( u - m ) \Big) | \nabla u |^2
   \ge \kappa_m \| \nabla f(u) \|^2 - c_m \| \nabla u \|^2.
\end{equation}
Finally, noting that $u$ takes values in $[-1,1]$,
using estimate \eqref{st14}, and applying the
Poincar\'e-Wirtinger inequality, we have
\begin{equation}\label{conto54}
  \big( w + \lambda u - \epsi u_t, u - m \big)
   = \big( w - w\OO + \lambda u - \epsi u_t, u - m \big)
    \le c \big( \| \nabla w \| + 1 + \epsi \| u_t \| \big).
\end{equation}
Thus, collecting the above considerations, \eqref{conto51} gives
\begin{equation}\label{conto51b}
  \| \nabla f(u) \|^2
   + \| f(u) \|_{L^1(\Omega)}
    \le c \big( \| \nabla u \|^2 + \| \nabla w \|
     + 1 + \epsi \| u_t \| \big).
\end{equation}
Squaring \eqref{conto51b}, using
\eqref{conto52}-\eqref{conto54}, and integrating
in time, we arrive at
\begin{equation}\label{st31}
  \| \nabla f(u) \|_{L^4(0,T;H)}
   + \| f(u) \|_{L^2(0,T;L^1(\Omega))}
   \le Q(\EE_0),
\end{equation}
whence, more precisely,
\begin{equation}\label{st32}
  \| f(u) \|_{L^2(0,T;V)} \le Q(\EE_0).
\end{equation}
Taking instead the essential supremum of~\eqref{conto51b}
as $t$ ranges in $(\tau,T)$, and using \eqref{st21}-\eqref{st22},
a straighforward modification of the above procedure
leads to
\begin{equation}\label{st35}
  \| f(u) \|_{L^\infty(\tau,T;V)} \le Q(\EE_0,\tau^{-1})
   \quad \perogni \tau\in(0,T).
\end{equation}
Next, integrating \eqref{CH2} in space,
using \eqref{aprimo}, \eqref{st13}, \eqref{defia},
and noting that $(u_t)\OO\equiv 0$, we get
\begin{align}\no
  | w\OO |
   & \le c \Big( \io \frac{a'(u)}2 | \nabla u |^2
    + \| f(u) \|_{L^1(\Omega)}
     + 1  \Big)\\
 \label{conto51c}
  & \le c \Big( \| \nabla f(u) \|^2
    + \| f(u) \|_{L^1(\Omega)}
     + 1   \Big).
\end{align}
Thus, squaring, integrating in time, using \eqref{st31},
and recalling the last~\eqref{st11}, we infer
\begin{equation}\label{st34}
  \| w \|_{L^2(0,T;V)} \le Q(\EE_0).
\end{equation}
Taking the essential supremum in \eqref{conto51c} as $t\in(\tau,T)$,
and recalling \eqref{st21} and \eqref{st35}, we also get
\begin{equation}\label{st36}
  \| w \|_{L^\infty(\tau,T;V)} \le Q(\EE_0,\tau^{-1})
   \quad \perogni \tau\in(0,T).
\end{equation}

\smallskip

\noinden
{\bf Estimate of $z$.~~
We consider the equivalent formulation
\eqref{CH2z} and test it by $-\Delta z$. This gives
\begin{equation}\label{coz1}
  \io 2 \phi'(u) | \Delta z |^2
   + \big( f'(u) \nabla u,\nabla z \big)
   = \big( w + \lambda u - \epsi u_t, - \Delta z \big).
\end{equation}
Thus, using the monotonicity of $f$ and $\phi$,
noting that $\phi'(u)\ge 1/2$ for all $u\in(-1,1)$,
and that
$\big( f'(u) \nabla u,\nabla z \big) \geq 2 \| \nabla z \|^2 $,
we can control the \rhs\ this way:
\begin{equation}\label{coz2}
  \big( w + \lambda u - \epsi u_t, - \Delta z \big)
   \le \frac12 \| \Delta z \|^2
    + c \big( \| w \|^2 + 1 + \epsi^2 \| u_t \|^2 \big).
\end{equation}
Hence, integrating \eqref{coz1} in time
and recalling~\eqref{st11}, \eqref{st14}
and \eqref{st34}, we arrive at
\begin{equation}\label{stz2}
  \| z \|_{L^2(0,T;H^2(\Omega))} \le Q(\EE_0).
\end{equation}
Taking instead the (essential) supremum of \eqref{coz1} as $t$
ranges in $(\tau,T)$ for $\tau>0$ and using
\eqref{st21}-\eqref{st22} and \eqref{st36}, we obtain
\begin{equation}\label{stz1}
  \| z \|_{L^\infty(\tau,T;H^2(\Omega))} \le Q(\EE_0,\tau^{-1}).
\end{equation}
The above relations permit to improve also the bounds on
$u$. Actually, computing directly the Laplacean
of $u = \sin z$ and using \eqref{st13} together with
the Gagliardo-Nirenberg
inequality (cf., e.g., \cite[Theorem p.~125]{Ni})
\begin{equation}\label{ineq:gn}
  \| \nabla y \|_{L^4(\Omega)} \le c\OO \| y \|_{H^2(\Omega)}^{1/2}
   \| y \|_{L^\infty(\Omega)}^{1/2}
   + \| y \| \qquad \perogni y \in H^2(\Omega),
\end{equation}
it is not difficult to arrive at
\begin{align}\label{stnewu1}
  & \| u \|_{L^2(0,T;H^2(\Omega))} \le Q(\EE_0),\\
 \label{stnewu2}
  & \| u \|_{L^\infty(\tau,T;H^2(\Omega))} \le Q(\EE_0,\tau^{-1}).
\end{align}

\smallskip

\noinden
{\bf First entropy estimate and separation property.}~
The estimates obtained up to this moment yield a control
of the functions $z$ and $u$ up to their second space derivatives
(cf.~\eqref{stz2}-\eqref{stz1}
and \eqref{stnewu1}-\eqref{stnewu2}), and of the nonlinear
term $f(u)$ up to its first space derivatives
(cf.~\eqref{st32} and \eqref{st35}). However,
this still seems not sufficient to pass to the limit
in the equation \eqref{CH2}, even if its equivalent
formulation \eqref{CH2z} is considered. Indeed,
from \eqref{st32} and \eqref{st35} we get
a control of the term $f(u)$, that explodes logarithmically
fast as $|u|\nearrow 1$. On the other hand,
even in  formulation \eqref{CH2z} one faces the term
$\phi'(u)$ which is much more singular since it
explodes as a negative power of $(1-u^2)$.
To control it in some $L^p$-norm we need more refined
estimates of the so-called {\sl entropy}\/ type and,
in particular, we need to refer to the formulation
\eqref{CH2v} (we recall that all formulations
are equivalent, at least for sufficiently smooth solutions).
Usage of this technique requires the convexity assumption
on $\Omega$ asked in the statement of Theorem~\ref{teo:main}.

The basic tool we need consists in an
integration by parts formula due to Dal Passo,
Garcke and Gr\"un (\cite[Lemma 2.3]{DpGG}):
\bele\label{lemma:dpgg}

 Let $h\in W^{2,\infty}(\RR)$ and $z\in W$. Then,

 \begin{align} \no
   & \io h'(z) |\nabla z|^2 \Delta z
    = -\frac13 \io h''(z) |\nabla z|^4\\
  \label{byparts}
   & \mbox{}~~~~~
    + \frac23 \io h(z) \big( |D^2 z|^2 - | \Delta z|^2 \big)
    + \frac23 \iga h(z) II( \nabla z ),
 \end{align}

 where $II(\cdot)$ denotes the second fundamental form
 of $\Gamma$.

\enle
\noinden
Then, we test \eqref{CH2v} by
\begin{equation}\label{co63}
  - \Delta\mciapo(v) = -\dive (m(v) \nabla v)
   = - m(v) \Delta v - m'(v) | \nabla v|^2,
\end{equation}
with the function $m$ to be chosen later. This gives
\begin{align} \no
  & \io m(v) | \nabla v |^2
   + \io \bigg(  m(v) |\Delta v|^2
   - \frac{m'(v)j(v)}2 |\nabla v|^4
   + \Big( m'(v) - \frac{m(v)j(v)}2 \Big)
            \Delta v | \nabla v|^2 \bigg)\\
 \label{co64}
  & \mbox{}~~~~~
   = \big( m(v) \nabla v, \nabla w + \lambda \nabla u \big)
    + \big( \epsi u_t, m(v) \Delta v + m'(v) | \nabla v|^2 \big).
\end{align}
Now, applying Lemma~\ref{lemma:dpgg} to
the last integral on the \lhs\ of \eqref{co64} we infer
\begin{align}\no
  & \io \Big( m'(v) - \frac{m(v)j(v)}2 \Big) \Delta v | \nabla v|^2
   = - \frac13 \io \Big( m''(v) - \frac{m'(v)j(v)}2 - \frac{m(v)j'(v)}2 \Big) | \nabla v|^4\\
 \label{co65}
  & + \frac23 \io \Big( m(v) - \frac{\mjciapo(v)}2 + K \Big)\big( | D^2 v|^2 - | \Delta v |^2\big)
   + \frac23 \iga \Big( m(v) - \frac{\mjciapo(v)}2 + K \Big) II(\nabla v),
\end{align}
where $K>0$ is an integration constant that will be chosen
later on, and the notation \eqref{deficiapo} is used.

Substituting \eqref{co65} into \eqref{co64}, we get on
the \lhs\ the following ``hopefully good''
terms
\begin{equation}\label{co66}
  \io m(v) |\nabla v|^2 + \io m(v) |\Delta v|^2
   + \frac13 \io \Big( - m''(v) - m'(v) j(v) + \frac{m(v)j'(v)}2 \Big) | \nabla v|^4,
\end{equation}
where we notice that
\begin{equation}\label{co67}
  j'(v) = \frac{2e^v}{(e^v+1)^2}.
\end{equation}
We can now specify our choice of $m$ as
\begin{equation}\label{defim}
  m(v)=\frac1{2(1+v^2)}.
\end{equation}
Actually, this expression arises since we need $m$ to
decay not too fast at infinity (otherwise we
do not get enough information from it), but at the same time
we need it to be summable (cf.~\eqref{co75} below).
The above choice gives
\begin{equation}\label{mprim}
  \mciapo(v) =\frac12 \arctan v,

\end{equation}
as well as
\begin{equation}\label{mder}
  m'(v)=-\frac{v}{(1+v^2)^2}, \qquad
   m''(v)=\frac{3v^2-1}{(1+v^2)^3}.
\end{equation}
Now, noting that $mj'\ge 0$, we can easily observe that
\begin{equation}\label{co68}
  - m''(v) - m'(v) j(v) + \frac{m(v)j'(v)}2
  \ge \frac{(v + v^3)j(v) + 1 - 3 v^2}{(1+v^2)^3}.
\end{equation}
Now, recalling \eqref{co61}, for a suitable $M>0$
(e.g., we can take $M=12$ here), a direct computation shows
that
\begin{equation}\label{co68bis}
  \frac{(v + v^3)j(v) + 1 - 3 v^2}{(1+v^2)^3}
   \ge \frac14 \frac{1 + |v|^3}{(1+v^2)^3}
   \quad\perogni |v|\ge M.
\end{equation}
On the other hand,
\begin{equation}\label{co68ter}
  \frac{(v + v^3)j(v) + 1 - 3 v^2}{(1+v^2)^3}
   \ge \frac1{(1+v^2)^3} - \frac{3 v^2}{(1+v^2)^3}
   \ge \kappa_M \frac{1 + |v|^3}{(1+v^2)^3}
    - c_M \frac{|v|}{(1+v^2)^3}
   \quad\perogni |v| \le M.
\end{equation}
Summarizing, we have
\begin{equation}\label{co68fine}
  - m''(v) - m'(v) j(v) + \frac{m(v)j'(v)}2
   \ge \kappa \frac{1+|v|^3}{(1+v^2)^3}
    - c \frac{|v|}{(1+v^2)^3} \chi_{\{|v|\le M\}},
\end{equation}
where $\chi$ denotes the characteristic function.

Now, we can choose $K$ so large that the function
\begin{equation}\label{co75}
  \Big( m(v) - \frac{\mjciapo(v)}2 + K \Big)
\end{equation}
is strictly positive (and bounded, of course).
Then, thanks the {\sl convexity}\/ assumption on $\Omega$
(that entails positive definiteness of the second
fundamental form), the latter two terms in \eqref{co65} are
positive.

Collecting these observations, we can deduce from
\eqref{co64} the estimate
\begin{equation}\label{co64b}
  \io m(v) | \nabla v |^2
   + \io m(v) |\Delta v|^2

  + \kappa \io \frac{1+|v|^3}{(1+v^2)^3} | \nabla v|^4
  \le \sum_{i=1}^4 T_i,
\end{equation}
and we have to control the ``bad'' terms on the \rhs:
\begin{align}\label{co72}
  & T_1:= c \int_{\{|v|\le M\}} \frac{|v|}{(1+v^2)^3} | \nabla v|^4,\\
 \label{co73}
  & T_2:= \io m(v) \nabla v \cdot \nabla w,\\
 \label{co74}
  & T_3:= \lambda \io m(v) \nabla v \cdot \nabla u,\\
 \label{co74x}
  & T_4:= \big( \epsi u_t, m(v) \Delta v + m'(v) | \nabla v|^2 \big).
\end{align}
To do this, we first notice that
\begin{equation}\label{co74b}
  | T_2 + T_3 |
   \le \frac14 \io m(v) | \nabla v |^2
    + c \big( \| \nabla w \|^2 + \| \nabla u \|^2 \big).
\end{equation}
Next, we have to control $T_1$. Then, we can note
that there exists $\delta\in(0,1)$, depending only on $M$,
such that the restriction
$f:[-1+\delta,1-\delta]\to[-M,M]$ is bijective
and Lipschitz continuous together with its
inverse $j$.
Thus, using \eqref{st13} and
inequality \eqref{ineq:gn}, we deduce
\begin{align}\no
  | T_1 | & \le c \int_{\{|v|\le M\}} | \nabla v |^4
    \le c \int_{\{|v|\le M\}} f'(u)^4 | \nabla u |^4\\
 \no
   & \le c \int_{\{|u|\le 1-\delta\}} \frac1{(1-u^2)^4} | \nabla u|^4
     \le c_\delta \int_{\{|u|\le 1-\delta\}} | \nabla u |^4\\
 \label{co72b}
  & \le c_\delta \io | \nabla u |^4
   \le c_\delta \| u \|_{L^\infty(\Omega)}^2
     \big( 1 + \| u \|_{H^2(\Omega)}^2 \big)
     \le c_\delta \big( 1 + \| u \|_{H^2(\Omega)}^2 \big).
\end{align}
Finally, we have to control $T_4$. We have
\begin{equation}\label{co74d}
  | T_4 |  \le \frac14 \io m(v) | \Delta v |^2
    + c_\sigma \epsi^2 \| u_t \|^2
    + \sigma \io m'(v)^2 |\nabla v|^4,
\end{equation}
for small $\sigma>0$ to be chosen later. Then,
we can go on as follows:
\begin{equation}\label{co74d2}
   \sigma \io m'(v)^2 |\nabla v|^4
    = \sigma \io \frac{ v^2 }{ (1 + v^2)^4 }|\nabla v|^4
    \le \sigma \io \frac{ 1 + |v|^3 }{ (1 + v^2)^3 }|\nabla v|^4.
\end{equation}
Hence, for $\sigma$ sufficiently small, the last
integral is controlled by the last term on
the \lhs\ of \eqref{co64b}. On the other hand, the
first term on the \rhs\ of \eqref{co74d} is
controlled by the corresponding one on the
\lhs\ of \eqref{co64b}.

Now, a direct computation permits to see that
\begin{equation}\label{co71}
  \io \frac{1+|v|^3}{(1+v^2)^3} | \nabla v|^4
  \ge \io \frac{|v|^3}{(1 + v^2)^3} \frac{|v|}{(1+ v^2)^{1/2}} | \nabla v|^4
   \ge \kappa \io \big| \nabla (1+v^2)^{1/8} \big|^4.
\end{equation}
Thus, collecting \eqref{co74b}-\eqref{co71},
\eqref{co64b} gives
\begin{align}\no
  & \io m(v) | \nabla v |^2
   + \io m(v) | \Delta v |^2
   + \io \big| \nabla (1+v^2)^{1/8} \big|^4\\
 \label{co64b2}
  & \mbox{}~~~~~
   \le c \big( \| \nabla w \|^2 + \| \nabla u \|^2
    + 1 + \| u \|_{H^2(\Omega)}^2
    + \epsi^2 \| u_t \|^2 \big).
\end{align}
Hence, integrating in time, and using \eqref{st11}, \eqref{st14}
and \eqref{stnewu1}, we obtain
\begin{equation}\label{st71}
  \big\| m^{1/2}(v) \Delta v \big\|_{\LDH}
   + \big| (1+v^2)^{1/8} \big|_{L^4(0,T;W^{1,4}(\Omega))}
  \le Q(\EE_0).
\end{equation}
On the other hand, taking the essential supremum
in \eqref{co64b2} as $t$ ranges in $(\tau,T)$ for
$\tau>0$, and using \eqref{st21}, \eqref{st22} and
\eqref{stnewu2}, we arrive at
\begin{equation}\label{st72}
  \big\| m^{1/2}(v) \Delta v \big\|_{L^\infty(\tau,T;H)}
   + \big| (1+v^2)^{1/8} \big|_{L^\infty(\tau,T;W^{1,4}(\Omega))}
  \le Q(\EE_0,\tau^{-1}).
\end{equation}
By the continuous embedding
$W^{1,4}(\Omega)\subset L^\infty(\Omega)$ we
have in particular
\begin{equation}\label{st72b}
  \big| (1+v^2)^{1/8} \big|_{L^\infty((\tau,T)\times \Omega)} \le Q(\EE_0,\tau^{-1}).
\end{equation}
In terms of $u$ the above estimate gives rise to
the separation property
\begin{equation}\label{st72c}
  -1 + \epsilon \le u(x,t) \le 1 - \epsilon
   \quext{a.e.~in }\,(\tau,T)\times \Omega,
\end{equation}
for all $\tau>0$, with $\epsilon>0$ depending on $\tau$.

\smallskip

\noinden
{\bf Refined entropy estimate.}~
We repeat the entropy estimate of before taking now,
in place of \eqref{defim},
\begin{equation}\label{defimp}
  m(v)=\frac1{2(1+v^2)^p},
\end{equation}
where the choice of $p\in (1/2,1]$ will be
made precise later on. Then, we have
\begin{equation}\label{mderp}
  m'(v)=-\frac{pv}{(1+v^2)^{p+1}}, \qquad
   m''(v)=\frac{(2p^2+p)v^2-p}{(1+v^2)^{p+2}}.
\end{equation}
Thus, a straighforward modification of
\eqref{co68}-\eqref{co68ter} leads to
\begin{equation}\label{co68finep}
  - m''(v) - m'(v) j(v) + \frac{m(v)j'(v)}2
   \ge \kappa \frac{1+|v|^3}{(1+v^2)^{p+2}}
    - c \frac{|v|}{(1+v^2)^{p+2}} \chi_{\{|v|\le M\}},
\end{equation}
whereas the equivalent of \eqref{co71} gives now rise to
\begin{equation}\label{co71p}
  \io \frac{1+|v|^3}{(1+v^2)^{p+2}} | \nabla v|^4
   \ge \kappa \io \big| \nabla (1+v^2)^{\frac38-\frac{p}4} \big|^4.
\end{equation}
Now, let us observe that, since $p>1/2$, then the function
$\mjciapo$ is bounded. Thus, we can still take $K$ so large,
depending of course on $p$, that the function in
\eqref{co75} is strictly positive.
Moreover, the terms corresponding to $T_j$, $j=1,\dots,4$, can
be controlled similarly as before. Thus, integrating
in time the $p$-analogue of \eqref{co64b}, we obtain
\begin{equation}\label{co81b}
  \iTo \big| \nabla (1+v^2)^{\frac38-\frac{p}4} \big|^4
   \le Q(\EE_0).
\end{equation}
Let us now test \eqref{CH2v} by $v$. This gives
\begin{equation}\label{conew1}
  \| \nabla v \|^2
   + \| v \|^2
   + \io \frac{j(v) v}2 | \nabla v |^2
  \le \big( w + \lambda u - \epsi u_t, v \big).
\end{equation}
Then, integrating in time,
noting that $1 + j(v)v\ge \kappa (1 + v^2)^{1/2}$,
recalling \eqref{st11} and using H\"older's and Young's inequalities
to estimate the \rhs, we arrive at
\begin{equation}\label{co82}
  \iTo (1+v^2)^{1/2} | \nabla v|^2 \le Q(\EE_0).
\end{equation}
Then, let us define
\begin{equation}\label{co83}
  \Omega^+(t):=\big\{x\in\Omega:~|v(x,t)|\ge 1\big\}, \qquad
   \Omega^-(t):=\big\{x\in\Omega:~|v(x,t)|\le 1\big\}.
\end{equation}
Let us now see that \eqref{co81b} and \eqref{co82} permit to
prove higher integrability properties of $\nabla v$.
Firstly, recalling \eqref{st14} and \eqref{stnewu1},
and using inequality \eqref{ineq:gn}, we obtain
\begin{equation}\label{co84-}
  \| u \|_{L^4(0,T;W^{1,4}(\Omega))} \le Q(\EE_0).
\end{equation}
Hence, noting that $j$ is locally Lipschitz continuous
with its inverse, we get an analogous information for $v$ in
the space-time set where it is small:
\begin{equation}\label{co84}
  \iTT \int_{\Omega^-(t)} | \nabla v |^4 \le Q(\EE_0).
\end{equation}
On the other hand, in $\Omega^+(t)$ we can write, for $\eta>0$,
\begin{align}\no
  | \nabla v |^2 & = \big( v ( 1 + v^2 )^{-\eta} \nabla v \big) \cdot
   \Big( \frac{( 1 + v^2 )^{\eta}} v \nabla v \Big)\\
 \label{co85}
   & = c_\eta \nabla ( 1 + v^2 )^{1-\eta} \cdot
   \Big( \frac{( 1 + v^2 )^{\eta}} v \nabla v \Big).
\end{align}
Thus, choosing $\eta=\frac58+\frac{p}4$, we have
$1-\eta=\frac38-\frac{p}4$. Then, we can take $p=\frac12+\epsilon$,
with $\epsilon>0$ as small as we want. We then obtain that
$\eta-\frac14=\frac12+\frac\epsilon4$.
Thus, recalling \eqref{co81b} and \eqref{co82}, we arrive at
\begin{equation}\label{co86}
  | \nabla v |^2 \le c_\eta \bigg| \underbrace{\nabla ( 1 + v^2 )^{1-\eta}}_{L^4}
    \cdot
   \underbrace{\big( ( 1 + v^2 )^{1/4} \nabla v \big)}_{L^2}
   \underbrace{\frac{( 1 + v^2 )^{\eta - \frac14}}{v}}_{L^r} \bigg|,
\end{equation}
where we can take $r$ strictly greater than $4$ since
for~$|v|\ge 1$ we have
\begin{equation}\label{co87}
  \frac{( 1 + v^2 )^{\eta - \frac14}}{|v|}
  = \frac{( 1 + v^2 )^{\frac12+\frac\epsilon4}}{|v|}
  \sim |v|^{\frac{\epsilon}2},
\end{equation}
and we know from \eqref{st32} that $v$ is controlled
in $L^2(0,T;H)$. Thus, estimating the
\rhs\ of \eqref{co86} by Young's inequality,
integrating first over $\Omega^+(t)$ and
then for $t\in(0,T)$, we obtain that
\begin{equation}\label{co88}
  \iTT \int_{\Omega^+(t)} | \nabla v |^q \le Q(\EE_0),
   \quext{for some }\,q~\text{{\bf strictly} larger than }\,2,
\end{equation}
and, of course, combining with \eqref{co84},
\begin{equation}\label{co88b}
  \iTo | \nabla v |^q \le Q(\EE_0),
   \quext{for some }\,q~\text{{\bf strictly} larger than }\,2.
\end{equation}

\section{Existence and uniqueness of weak solutions}
\label{sec:weak}

We detail here the proof of Theorem~\ref{teo:main}, which
is largely based on the estimates derived in the previous
section. As a first step, however, we show uniqueness,
which works similarly to \cite{SP11}. Indeed, the key
assumption \cite[(6.1)]{SP11} is satisfied by our function~$a$
(we have, indeed, that $(1/a)''\equiv -1$).

Then, let us take a couple of weak
solutions $(u_1,w_1)$ and $(u_2,w_2)$
originating from the same initial datum $u_0$,
and assume that both are ``classical'' (and in particular
satisfy the separation property \eqref{separs}),
for strictly positive times.

Setting $(u,w):=(u_1,w_1)-(u_2,w_2)$, we can write both
\eqref{CH1w} and \eqref{CH2w} for the two solutions
and take the difference. Using notation \eqref{defiJ}
we get
\begin{align}\label{CH1di}
  & u_t + A w = 0,\\
 \label{CH2di}
  & w = J'(u_1) - J'(u_2) + f(u_1) - f(u_2) - \lambda u + \epsi u_t.
\end{align}
Then, we can test \eqref{CH1di} by
$A^{-1}u$, \eqref{CH2di} by $u$, and take the difference.
Actually, the operator $A$ is invertible as it is restricted
to $0$-mean valued functions (as in the case of $u$ due to
conservation of mass). Noting that
\begin{equation} \label{contod0}
  \big( Aw, A^{-1} u \big)
   = \big( A (w-w\OO), A^{-1} u \big)
   = ( w - w\OO, u )
   = ( w, u ),
\end{equation}
we then obtain
\begin{equation} \label{contod1}
  \frac12 \ddt \Big( \| u \|_{V'}^2 + \epsi \| u \|^2 \Big)
   + \duavg{J'(u_1) - J'(u_2), u}
   + \big( f(u_1) - f(u_2), u \big)
   \le \lambda \| u \|^2.
\end{equation}
Then, recalling \eqref{Jvv} and \eqref{co21}
and using monotonicity of $f$, we arrive~at
\begin{equation} \label{contod2}
  \frac12 \ddt \Big( \| u \|_{V'}^2 + \epsi \| u \|^2 \Big)
   + \frac12 \| \nabla u \|^2
  \le \lambda \| u \|^2.
\end{equation}
Noting that, by the Poincar\'e-Wirtinger inequality,
\begin{equation} \label{contod3}
  \lambda \| u \|^2
   \le \frac14 \| \nabla u \|^2
   + c \| u \|_{V'}^2,
\end{equation}
we can integrate \eqref{contod2}
over $(\tau,T)$ for $\tau>0$. Using Gronwall's lemma,
we obtain
\begin{equation} \label{fromSP11-2}
  \| u_1 - u_2 \|_{L^\infty(\tau,T;V')}^2
   \le c(T) \| u_1(\tau) - u_2(\tau) \|_{V'}^2,
\end{equation}
where $c(T)$ is independent of $\tau$. Then, uniqueness
follows by taking the limit $\tau\searrow 0$ and owing
to continuity of weak solutions with values in $V'$ (which
is an obvious consequence of \eqref{regou4}).

\smallskip

Let us now switch to existence. To start with,
we approximate the initial datum $u_0$ as specified in
Lemma~\ref{lemma:u0}. Then, thanks to Corollary~\ref{cor:delta},
for any $\delta\in (0,1/6)$, there exists a ``classical''
solution $(u\dd,w\dd)$ to Problem~(P)
defined {\sl at least}\/ on the time interval
$(0,T\dd)$, where $T\dd$ depends on $u_0$ and $\delta$.
Actually, in principle, we may have that $T\dd\searrow 0$ as we let
$\delta\searrow 0$. On the other hand, the forthcoming argument
will exclude this eventuality and show that, in fact,
$(u\dd,w\dd)$ can be extended up to the final time $T$.

Indeed, let us denote as $T\dmax$ the maximum time up to
which $(u\dd,w\dd)$ can be extended in the form of
a ``classical'' solution; namely,
\begin{equation}\label{defitdmax}
  T\dmax:=\sup\big\{ S\in(0,T]:~ u\dd~\text{admits a ``classical'' extension over $(0,S)$}\big\}.
\end{equation}
Due to uniqueness proved above, all extensions of
$(u\dd,w\dd)$ can be ``glued'' together. Consequently,
there exists a (unique) maximal classical extension $(u\dmax,w\dmax)$
defined over $(0,T\dmax)$. We claim that $T\dmax = T$, and,
to prove this claim, we proceed as usual by contradiction.
Actually, due to \eqref{regoust}-\eqref{regowst} and \eqref{separs},
for any $S\in (0,T\dmax)$ we have
\begin{align}\label{regoumax}
  & \| u\dmax \|_{H^1(0,S;V) \cap W^{1,\infty}(0,S;V') \cap L^\infty(0,S;H^2(\Omega))}
   \le C(S), \qquad
   \epsi \| u\dmax \|_{W^{1,\infty}(0,S;H)}\le C(S),\\
 \label{regowmax}
  & \| w\dmax \|_{L^\infty(0,S;V) \cap L^2(0,S;H^3(\Omega))} \le C(S), \qquad
   \epsi \| w\dmax \|_{L^\infty(0,S;H^2(\Omega))}\le C(S),\\
 \label{separmax}
  & - 1 + \epsilon(S) \le u\dmax(t,x) \le 1 - \epsilon(S)
  \quad \perogni (t,x) \in [0,S]\times \barO,
\end{align}
where $C(S),\epsilon(S)>0$ and it may be $C(S)\nearrow\infty$ and
$\epsilon(S)\searrow 0$ as $S\nearrow T\dmax$.
On the other hand, since $(u\dmax,w\dmax)$ is a ``classical''
solution, it satisfies the a-priori estimates of the previous section
on the time interval $(0,T\dmax)$. Then, thanks to \eqref{st72c},
we have that
\begin{equation}\label{separunif}
   - 1 + \bar\epsilon \le u\dmax(t,x) \le 1 - \bar\epsilon
  \quad \perogni (t,x) \in [\tau,S]\times \barO,
\end{equation}
and for all $0<\tau<S<T\dmax$, with $\bar\epsilon$ independent
both of $S$ and of $\delta$.
To be more precise, we have that
\begin{equation}\label{separunif2}
  \bar\epsilon^{-1} = Q(\calE(u\zzd), \tau^{-1})
   \le Q(\EE_0, \tau^{-1}),
\end{equation}
where the second inequality is a consequence of
\eqref{uzzd2}.

Analogously, we have estimates of the norms in
\eqref{regoumax}-\eqref{regowmax} over the time interval $(\tau,S)$
by a constant $C$ independent of $\delta$ and $S$.
Consequently, we obtain
\begin{equation}\label{extens}
  \esiste \baru = \lim_{t\nearrow T\dmax} u\dmax(t,\cdot).
\end{equation}
To be more precise, this limit is
reached in the weak topology of $H^2(\Omega)$. Indeed, it is
a consequence of \eqref{regoumax} that
$u\dmax\in C_w([0,T\dmax];H^2(\Omega))$.
Thus, $- 1 + \bar\epsilon \le \baru(x) \le 1 - \bar\epsilon$ for all
$x\in \barO$ and we can use $\baru$ as a new
``initial'' datum and extend the solution $(u\dmax,w\dmax)$
beyond the time $T\dmax$. Moreover, the extension is still
a classical solution since $\baru$ is H\"older continuous
and uniformly separated from $-1$ and $1$. This contradicts
the maximality of $T\dmax$ and of $(u\dmax,w\dmax)$.
Hence, we necessarily have that $T\dmax=T$.

To conclude the proof, we need to show that we can take
the limit $\delta\searrow 0$ and obtain a weak solution to
Problem~(P). With this purpose we rename simply as $(u\dd,w\dd)$
the maximal solution obtained in the previous part
(which is now defined in the whole $(0,T)$), and observe
that, thanks to estimates \eqref{st11}, \eqref{st14},
\eqref{stz2}, and \eqref{st32}, there hold the following
convergence relations:
\begin{align}\label{conv11}
  & u\dd \to u \quext{weakly star in }\,\HUVp \cap \LIV \cap L^\infty((0,T)\times\Omega),\\
 \label{conv11b}
  & \epsi u\dd \to \epsi u \quext{weakly in }\,\HUH,\\
 \label{conv12}
  & w\dd \to w \quext{weakly in }\,\LDV,\\
 \label{conv13}
  & z\dd \to z \quext{weakly star in }\,\LIV \cap \LDHD,\\
 \label{conv14}
  & v\dd=f(u\dd) \to v \quext{weakly in }\,\LDV,
\end{align}
for suitable limit functions $u,w,z,v$. The above properties,
as well as the ones that will follow, are to be intended
up to the extraction of (non-relabelled) subsequences
of $\delta\searrow 0$. We then immediately see that relation
\eqref{CH1} passes to the limit. However, since it is only
$w\in \LDV$, the Laplace operator with the boundary
condition $\dn w=0$ have to be interpreted in the weak
form through the operator $A$ (cf.~\eqref{defiA}).

Next, applying the Aubin-Lions lemma, \eqref{conv11} gives
\begin{equation}\label{conv21}
  u\dd \to u \quext{strongly in }\, L^p((0,T)\times\Omega),
   \quad \perogni p\in[1,\infty).
\end{equation}
Thus, a standard monotonicity argument (see, e.g.,
\cite[Prop.~1.1, p.~42]{barbu})
permits to infer from \eqref{conv14}
that $v=f(u)$ almost everywhere. Moreover, by
the generalized Lebesgue's theorem, we get more
precisely
\begin{equation}\label{conv22}
  v\dd=f(u\dd) \to v=f(u) \quext{strongly in }\, L^p((0,T)\times\Omega),
   \quad \perogni p\in[1,2).
\end{equation}
Now, as a consequence of estimate \eqref{co88b}, we obtain
\begin{equation}\label{conv23}
  \nabla v\dd \to \nabla v \quext{weakly in }\, L^p((0,T)\times\Omega),
   \quext{for some }\, p>2.
\end{equation}
Collecting \eqref{conv21} and \eqref{conv23}, we infer
\begin{equation}\label{conv24}
  \frac{u\dd}2 |\nabla v\dd|^2 \to \Phi \quext{weakly in }\, L^p((0,T)\times\Omega),
   \quext{for some }\, p>1,
\end{equation}
and for a suitable limit function $\Phi$.

We can now write equation \eqref{CH2v} for the solution
$(u\dd,w\dd)$ and see that, thanks to the above
convergence relations, all terms pass to the limit.
Actually, what we obtain for $\delta\searrow 0$ is
\begin{equation}\label{CH2vlim}
  w = - \Delta v + v + \Phi  - \lambda u + \epsi u_t,
\end{equation}
at least in the distributional sense.
To conclude the proof, we need to identify
the function $\Phi$. To this aim, we notice that,
by \eqref{conv11}, \eqref{conv12}, \eqref{conv24}, and
a comparison of terms in \eqref{CH2w},
\begin{equation}\label{conv31}
  \| -\Delta v\dd \|_{L^p((0,T)\times\Omega)} \le c,
   \quext{for some }\, p>1.
\end{equation}
Thus, we get that, also in the limit,
$-\Delta v$ (the distributional Laplacean)
lies in $L^p((0,T)\times\Omega)$ for some $p>1$.
More precisely, thanks to the no-flux condition and
to elliptic regularity, we deduce
\begin{equation}\label{conv32}
  v\dd \to v \quext{weakly in }\, L^p(0,T;W^{2,p}(\Omega)),
   \quext{for some }\, p>1,
\end{equation}
and $\dn v=0$ on $(0,T)\times \Gamma$ in the sense of traces.
Coupling \eqref{conv22} and \eqref{conv32} we obtain
strong convergence of $\nabla v\dd$ by interpolation.
Indeed, we can use (for example) the Gagliardo-Nirenberg
inequality (see again \cite{Ni}) in the form
\begin{equation}\label{conv33}
  \big\| \nabla (v\dd - v) \big\|_{L^{24/19}(\Omega)}
   \le c \big\| D^2 (v\dd - v) \big\|_{L^1(\Omega)}^{1/2}
              \| v\dd - v \|_{L^{12/7}(\Omega)}^{1/2}
  + \| v\dd - v \big\|_{L^1(\Omega)}.
\end{equation}
Then, integrating in time, and using the boundedness of
$W^{2,p}$-norms resulting from \eqref{conv32}, we infer
\begin{align} \no
  & \big\| \nabla (v\dd - v) \big\|_{L^1(0,T;L^{24/19}(\Omega))}
   \le c \big\| D^2 (v\dd - v) \big\|_{L^1((0,T)\times\Omega)}^{1/2}
              \| v\dd - v \|_{L^1(0,T;L^{12/7}(\Omega))}^{1/2}
   + \| v\dd - v \big\|_{L^1((0,T)\times \Omega)}\\
 \no
  & \mbox{}~~~~~
   \le c \big( \| D^2 v\dd \|_{L^1((0,T)\times\Omega)}^{1/2}
   + \| D^2 v \|_{L^1((0,T)\times\Omega)}^{1/2} \big)
              \| v\dd - v \|_{L^1(0,T;L^{12/7}(\Omega))}^{1/2}
  + \| v\dd - v \big\|_{L^1((0,T)\times \Omega)}\\
  \label{conv34}
  & \mbox{}~~~~~
    \le c \| v\dd - v \|_{L^1(0,T;L^{12/7}(\Omega))}^{1/2}
    + \| v\dd - v \big\|_{L^1((0,T)\times \Omega)}.
\end{align}
Owing to \eqref{conv22}, we then obtain that
$\nabla v\dd$ tends to $\nabla v$, say, strongly
in $L^1((0,T)\times \Omega)$, and consequently
almost everywhere. Thus, recalling \eqref{conv23} and using
once more the generalized Lebesgue theorem, we have
\begin{equation}\label{conv23b}
  \nabla v\dd \to \nabla v \quext{strongly in }\, L^p((0,T)\times\Omega),
   \quext{for some }\, p>2.
\end{equation}
whence, by \eqref{conv21}, \eqref{conv24} is improved up to
\begin{equation}\label{conv24b}
  \frac{u\dd}2 |\nabla v\dd|^2 \to \frac{u}2 |\nabla v|^2
   \quext{strongly in }\, L^p((0,T)\times\Omega),
   \quext{for some }\, p>1.
\end{equation}
Consequently, the function $\Phi$ in \eqref{CH2vlim}
is identified to its expected limit. Hence, we get
\eqref{CH2w} (holding as a relation in $L^p(\Omega)$ for some
$p>1$, hence almost everywhere) and the boundary condition
\eqref{neumw}. Finally, we notice that the limit $\delta\searrow 0$
can be taken trivially in the initial condition \eqref{iniz-intro}.
This concludes the proof.
\beos\label{rem:uniq}

 Theorem~\ref{teo:main} states that uniqueness
 holds for weak solutions that are classical on
 all intervals $(\tau,T)$, $\tau>0$. This is, in fact,
 the same regularity class where we are able to
 prove existence. However, we cannot exclude
 that uniqueness might instead fail as one considers
 the (larger) class of {\sl all}\/ weak solutions,
 which in particular may contain some trajectory that
 does not achieve the ``classical'' regularity for strictly positive
 times. Actually, some of the calculations given
 in the proof (in particular, those related to
 the gradient terms) seem not be justified under
 the sole regularity conditions proper of weak solutions
 (which, for instance, may not be ``separated'' from
 the singular values $\pm1$).

\eddos

redpajama_set_name: RedPajamaArXiv


We support blind and visually impaired children, we provide free events during school holidays and free short breaks in our respite facility at Chloe's Den, a wheelchair accessible (with wet room and facilities for severely disabled children) caravan in a five star facility, at Amble Links in Northumberland. Most of the children we support are severely disabled.
We seek to include the children and families in community, caring for a child with disability often brings isolation.
We seek to overcome financial hardship, caring for a child with severe disability often means one or both of the parents are unable to work to meet the medical needs of the child.
We are a group with similar needs, we share experiences, informally support each other and make friendships.
We are unique in so far as no one else in the North East provides the same services and offers free holidays.
We seek to include the whole of the family, including siblings in all of events. Often siblings miss out too, because the needs of the disabled child can often take prioirty.
We seek to make life-long memories for the families we support, caring for children with "life-limiting" illness, can be so upsetting and stressful, so we aim to make the best memories we can - hence we offer our bespoke caravan (Chloe's Den) for respite - everything is provided free.
Chloe's Den, is named after one of the children that passed away aged 9 years.
It is so important to us all to be spirited, be included, make friendships and have some fun.
There are very limited places we can include our children in the community, so this Charity has grown because there is a need to support families caring for children with severe disability.
We aim to continue provide free events during school holiday periods to the disabled child, their siblings and families (as a group to meet their needs). Promoting friendships and inclusion.
To continue to provide free short-break holidays at Chloe's Den a 5 star caravan at Amble Links: relieving stress; recharging batteries; make life-long memories. Providing bespoke accommodation.
Generally we aim to include our children and families in the community despite financial hardship, seeking to prevent isolation, promote friendships with other families in is similar circumstances.


Ame had us all fooled.
What's the name of the sauce? I must know for research purposes!
Onoes, I forgot to sensor out the eyes.
Onoes, I used the wrong 'sensor'.
I thought I was going to be trolled but I wasn't, so I searched on the webernet for Mecha-Con! You know for >insert excuse here< and then I forgot I left this tab open for quite a while before closing it as calmly as possible when I heard noises coming from behind the door in the room xD.


Get dolled up in your nifty glam rags and head to the juice joint for some fun!
Our silk Flapper Dresses are sure to make you the Cat's Meow.
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On the level these great Flapper Dresses and accessories won't last forever, so get them before they skedaddle.


text:
FC Namys Almaty (, Namys fýtbol klýby)  are a Kazakhstani football club based in Almaty, Kazakhstan. The club played in the Kazakhstan Super League in season 1993. They were members of the Kazakhstan First Division from 1994.

Name history
1992 : Founded as Namys Almaty based in Almaty.

References

Football clubs in Almaty
Football clubs in Kazakhstan
1992 establishments in Kazakhstan

redpajama_set_name: RedPajamaWikipedia


!Kheis Local Municipality is part of ZF Mgcawu District Municipality.
Description: The !Kheis Local Municipality is a Category B municipality situated within the ZF Mgcawu District in the Northern Cape Province. It is one of the five municipalities that make up the district, accounting for 11% of its geographical area.
It was established from the former Groblershoop Municipality, from settlements that were previously part of the ZF Mgcawu and Karoo District Municipalities. These municipalities administrated these settlements and provided them with services up until the demarcation in November 2000. From December 2000, !Kheis Municipality took over services and personnel, and total service provision took place from 1 July 2001.
The !Kheis municipal area was initially inhabited by the Khoisan people, who were also the first permanent inhabitants of South Africa. The San, who lived a nomadic life, migrated through the area. !Kheis is a Khoi name meaning 'a place where you live' or 'your home'.
The existing roads in the !Kheis municipal area are mainly gravel. The main road (N10) from Upington to Groblershoop is a tar road. Groblershoop is also connected by the N8 to Griekwastad and by the N10 to Marydale.
Due to the climate of the area, there is huge potential to utilise solar energy more widely, especially in the remote areas of the district. The Orange River stretches through the area, from the south-east to the north-west. The presence of perennial river water contributes to the establishment of table grapes, which are marketed and exported to Western Europe.


text:
W&L Alum Champions Fix for Healthcare.gov
By Julie Campbell
From his desk in Hong Kong, Sam Reed, a JavaScript developer, is pounding away at his keyboard as part of the open-source community trying to help federal contractors fix the healthcare.gov website.
A member of Washington and Lee's class of 2010, Sam is putting his B.S. in computer science to good use. He got to work after the troubled Oct. 1 launch of the health-care website: downloading parts of the code, sharing it on GitHub, and buckling down to analyze the problems.
Sam thinks it can be repaired and that "it's not the worst code I've ever seen," as he told "NBC News" in a story about the efforts that he and other programmers are undertaking.
"If we can get the open-source community involved in the issue, people in the trenches (working for the government) can see the fixes," he told the reporter.
Related //Computer Science, The College
Tagged //Old Wordpress, science


Published: Jan 15 at 11:05 a.m.
YARMOUTH, NS – The Par-en-Bas Sharks brought home the 2019 Danny Dorrington High School Hockey tournament championship banner this past weekend.
The seven-team high school hockey tournament was held in New Glasgow Jan. 11-13.
The PEB Sharks defeated the host Northumberland Regional High School team 2-0 in Sunday's championship final.
The Sharks had two wins and a tie during the round robin. PEB defeated West Kings 7-4 and had a 3-2 win over the Dr. J H Gillis team from Antigonish. The Sharks tied North Nova Education Centre of New Glasgow 3-3. Par-en-Bas advanced to the championship game following a 1-0 win over North Nova Education Centre in Saturday's semi-final.
This weekend the Par-en-Bas Sharks return to Valley High School Hockey League action with a Jan. 18 road trip to Berwick to play the Central Kings Gators.
Updated Nov 24, 2018 at 10:53 a.m.
Updated Nov 19, 2018 at 12:07 p.m.
Updated Jan 09, 2019 at 12:17 p.m.
Updated Jan 07, 2019 at 10:57 p.m.
Published Jan 07, 2019 at 10:55 p.m.
Published Jan 15, 2019 at 11:16 a.m.
Published Jan 07, 2019 at 11:05 p.m.
Updated Jan 18, 2019 at 7:33 a.m.


text:
The Suitcase Under The Bed is playing at Theatre Row located at 410 West 42ns Street. It runs two hours and ten minutes with one intermission. The play closes on September 23, 2017.
Teresa Deevy is the playwright. She was a deaf Irish dramatist born on January 21, 1894. She died on January 19, 1963. Teresa started writing in 1919.
Four short plays are performed. They are The King of Spain's Daughter (1935 theatre play), Strange Birth (world premiere stage play 1946), Holiday House (world premiere was a radio play) and In the Cellar of My Friends (world premiere a radio play).
Strange Birth takes place in the hall of Mrs. Taylor (Cynthia Mace). It is a summer evening. Sara Meade (Ellen Adair) is mopping the hall when the mail is delivered. Mrs. Taylor is thrilled when she opens the letter from her son. He is coming to visit her. Two of the tenets come into the hall looking for the mail. Mrs. Stims (Gina Costigan) is not happy that Sara did not bring her the mail for her. Mr. Bassett (A. J. Shively) is disappointed that he has not received a letter from a certain young lady. The mailman Bill (Aidan Redmond) delivers a bogus letter to the house. He confesses his love for Sara. She gets scared but then calms down.
Entr'acte: "The spiritual Canticle by St. John of the Cross" is told by Sarah Nicole Deaver.
In the Cellar of My Friend takes place in the breakfast room of Grantsthorn residence of Thomas Keane, a July morning. Patricia Keane (Cynthia Mace) is thrilled Belle Dobby (Sara Nicole Deaver) is visiting her. She calls her Aunt Patricia . Belle has brought her roses.
Belle has met her grandson Barney (A. J. Shively) the night before. She has known him her whole life. Barney proposed to her.
Patricia tells her Barney has left. He had a falling out with his father Thomas Keane (Colin Ryan). When Thomas comes down he tells them Barney has left for good. Barney comes in surprised to see Belle. He said he sent a letter explaining everything.
After intermission Holiday House is done. It takes place in the hall of Seafield House in August. The mother (Cynthia Mace) has rented the house. Her daughter Hetty (Sarah Nicole Deaver) is helping her unpack. She is frantic; both of her brothers are coming with their wife's. Her friend Charlie Moore (A. J. Shively) tries to calm her down. Her brother Derek (Colin Ryan) arrives first. His wife Jil (Gina Costigan) comes later. Neil (Aidan Redmond) her other brother arrives with his wife Doris (Ellen Adair). Doris was involved with Derek. Things happen and in rebound she marries Neil.
Things flare up when they get together.
Entr'acte "A Drover by Padraic Colum" is told by Aidan Redmond.
The King of Spain's daughter takes place on a grass road. It is Dinner hour on an April day. Peter Kinsella(Aidan Redmon) and Jim (A. J. Shively) are laborers. Peter is waiting for his daughter Annie (Sarah Nicole Deaver) to bring his dinner. She is late. Annie is flirting with Roddy Mann (Colin Ryan) despite the fact she knows her father is going to be mad.
Jim is in love with Annie.
Mrs. Marks (Cynthia Mace), a neighbor sticks her business in where it doesn't concern her.
This is the basis of the stories, some are twenty minutes the others around forty five.
All the plays were well done and enjoyable.
The sets were well done modest, but delightful. During the Entr'acte the sets are changed.
All of the actors were outstanding. Each actor shines, in some roles even more.
The plays were simple; some people may have like a little more. There was just enough that if you wanted to you could have come to your own conclusion or not as to what could have happened.
To read a candid interview with the cast, scroll down to the left for recent photos. Click on a photo, then click on back to album and finally click on back to gallery for this and other shows.


"Swim" original art for subway, 1937. Tempura water color on tissue paper; artist unknown. Department of Parks General Files, NYC Municipal Archives.
A life-long swimmer, Parks Commissioner Robert Moses vastly expanded access to aquatic facilities for New Yorkers. In 1936, he opened ten new swimming pools and during his long tenure he built and improved public beaches throughout the city.
Recently, the Municipal Archives received an inquiry from a potential patron asking if it was true "…that their [Department of Parks] internal records from 1934-45 are offsite and inaccessible, perhaps rotting away in a barn somewhere in New Jersey, piled up in banker's boxes..." The answer is . . . not true! Although the Parks records did make a brief trip to New Jersey, they are very much accessible in the Municipal Archives and constitute one of the most important collections documenting the built environment of New York City and the decades-long era of Parks Commissioner Robert Moses.
In 1984, the National Endowment for the Humanities (NEH) awarded the Archives a grant to identify and appraise historical records still held in municipal offices. During the course of the project, City archivists visited the Parks storage facility at the Boat Basin and discovered nearly 800 cubic feet of administrative records. Despite their location only a few hundred yards from the Hudson River, directly below a busy traffic interchange, the archivists found the records to be in remarkably good order.
Aerial view of the 79th Street Boat Basin and Henry Hudson Parkway interchange, ca. 1936. NYC Municipal Archives Collection. Moses leveraged federal highway funding to complete an earlier "West Side Improvement" plan and added a marina, known as the 79th Boat Basin in Riverside Park.
Recognizing the importance of this material, City archivists transferred the records to the Municipal Archives, after a detour to a laboratory in New Jersey for mold remediation.
Astoria Pool, Queens, August 20, 1936. Department of Parks Collection, NYC Municipal Archives. Opened July 2, 1936, Astoria Pool is the largest of the eleven pools Moses built with funding from the federal Works Progress Administration program.
And fortunately for generations of historians, Commissioner Moses and his staff were prolific correspondents and meticulous record-keepers. The records extracted from the Boat Basin consisted of two series—740 cubic feet of the Department of Parks General Files, 1934-1966, and 44 cubic feet of the Office of the Parks Commissioner/City Planning Commissioner files, 1940-1956. The material includes carbons or originals of incoming and outgoing correspondence, memoranda, transcripts, reports, contracts, news clippings, maps, blueprints, plans, printed materials, press releases, invitations, and photographs. The records document virtually every component of the Department's administrative actions from 1934 through 1966. The General Files series is further divided into three subseries: Administrative Files, Borough Files and Index Cards.
Sara Delano Roosevelt Park, March 6, 1934. Department of Parks Collection, NYC Municipal Archives. Several blocks of tenements in Manhattan's lower East Side, from Houston to Rivington Streets, were razed for construction of the Sara Delano Roosevelt Park.
The Parks/City Planning Commissioner series comprises the files created by Moses in his capacity as Commissioner of the Department of Parks, Commissioner of the City Planning Department, and after 1946, City Construction Coordinator. (Moses simultaneously held up to twelve official positions.) The material in this series is 100% Robert Moses. With the exception of a few folders labeled "Hazel Tappan," his personal secretary, it does not contain the correspondence of any of his deputies or assistants.
Pelham Bay Park: Concessions Building and comfort station, October 22, 1941. Department of Parks Collection, NYC Municipal Archives. No detail was too small or building too insignificant for Moses and his talented team of architects as illustrated by the handsome design of this comfort station.
Invitation to the Menagerie in Central Park, 1934. Color half-tone on paper; artist unknown. Mayor LaGuardia Parks Department correspondence, NYC Municipal Archives.
Good design was a hallmark of Commissioner Moses' public works as evidenced by this 1934 invitation and during the Great Depression he could draw on a large pool of readily available talented architects and designers.
In 1998 the NEH awarded $64,000 to the Municipal Archives to microfilm the entire Commissioner series, and selected documents from the General Files series.
Robert Moses' career and the physical changes he wrought on New York City, for better or worse, have long been the subject of analysis and debate. Beginning even before his death in 1981, there has been a steady stream of works pertaining to Moses ranging from the Caro biography to innumerable articles, dissertations, papers, conferences, documentary films and at least one opera. Access to the voluminous Parks collection—and its rich content of Moses-related material—will continue to facilitate the ongoing debate about one of the most influential figures of 20th century New York City history.


text:
Robert E. Hughes (born March 10, 1971) is a former Major League Baseball catcher who played for the Milwaukee Brewers in  and .

Hughes attended the University of Southern California. In 1991, he played collegiate summer baseball with the Wareham Gatemen of the Cape Cod Baseball League.

References

External links

1971 births
Living people
Los Angeles Valley Monarchs baseball players
Major League Baseball catchers
Milwaukee Brewers players
Baseball players from California
Sportspeople from Burbank, California
Notre Dame High School (Sherman Oaks, California) alumni
USC Trojans baseball players
Helena Brewers players
Beloit Brewers players
Stockton Ports players
El Paso Diablos players
New Orleans Zephyrs players
Tucson Toros players
Louisville RiverBats players
Buffalo Bisons (minor league) players
Rochester Red Wings players
Wareham Gatemen players
Anchorage Glacier Pilots players
Alaska Goldpanners of Fairbanks players


Off We go. We have heard mention of visitors coming this weekend. As loyal readers know Me and Nellie love having guests to stay. More people to give Us the love and attention that We so richly deserve. Not to mention more tasty treats coming our way. Itai has yet to experience guests so he will be in for a pleasurable learning curve.
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Good to know that the 3 of you will be receiving the love, attention, and treats from some fresh visitors. Maybe there will be a party with some yummy floor pies will drop all over the place. Paws crossed.
We hope the wind is done downing branches!
Just great? You get firewood in the summer? You probably get swim suits as winter presents .
Good thing that branch didn't hit anything or anybody!
Did that wind blow in your guests? We hope they spend all their time giving you attention! Mom is sick of winter here.
Gosh, hope you all stay safe with your weekend visitors. Sad to lose a big branch like that.


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