text:
The Green Blotch: Hurricane Andrew Through A Six-Year-Old's Eyes
By Trina Sargalski & Daniel Rivero • Jul 12, 2012
Danny Rivero and his brother, Alexis, at their uncle's wedding the day before Hurricane Andrew. They were furious about having to wear tuxedoes. Danny was the ring-bearer.
Danny Rivero
WLRN social media editor, Danny Rivero, at six years old.
Danny Rivero is our social media editor here at WLRN. You don't hear his voice on the radio, but Danny's is the voice of WLRN on Twitter, Facebook, and Tumblr.
In 1992, he was six years old and living in Fairfax County, Virginia, but he happened to be visiting Miami during Hurricane Andrew. Around the same time, he'd been listening to tapes of the story, "The War of the Worlds" by Orson Welles. His mom would play them for Danny and his brother, Alexis, before they went to bed at night.
Before she ever played the tapes, Danny's mom explained the back story of the original 1938 broadcast of a dramatization of "The War of the Worlds"–that people thought this was really happening and panicked. Young Danny was fascinated by this back story as he listened and suspended his disbelief, wondering if this alien invasion really could be happening somewhere.
The day before Hurricane Andrew struck, Danny was in Miami for his uncle's wedding. He noticed the adults were transfixed on a green blotch on the TV screen. Six-year-old Danny absorbed that this splotch was scary to adults and meant destruction. However, he had a hard time sorting out whether this was a real broadcast or a dramatization like "War of the Worlds." Listen to Danny's account of Hurricane Andrew through his six-year-old eyes on the player above.Danny's family's house was alright after the storm, except for a large awning that blew down the street. Danny particularly remembers an uprooted banyan tree:
We went to my tia's house in Shanandoah, and they had a big banyan tree or a willow tree with those spreading roots. The wind had just knocked it over. The tree went sideways, and the roots were three times taller that I was. I'm looking up at this huge wall of roots and it's like 10-12 feet high. It was huge to me as a child. I was in awe that yesterday I was walking on these roots, and now they were my ceiling. It was a weird feeling.
You can read or listen to more personal accounts commemorating the 20th anniversary of Hurricane Andrew when you visit our "Remembering Andrew" Storify.
Remembering Andrew


text:
\section{Introduction}
In as early as 1952, A. Turing in his seminal paper \cite{t52} attempted, by using reaction-diffusion systems, to model the phenomenon of {\it morphogenesis}, the regeneration of tissue structures in hydra, an animal of a few millimeters in length made up of approximately 100,000 cells. Further observations on the morphogenesis in hydra led to the assumption of the existence of two chemical substances (morphogens), a slowly diffusing (short-range) activator and a rapidly diffusing (long-range) inhibitor. A. Turing was the first to indicate that although diffusion has a smoothing and trivializing effect on a single chemical, for the case of the interaction of two or more chemicals different diffusion rates could force the uniform steady states of the corresponding reaction-diffusion systems to become unstable and to lead to nonhomogeneous distributions of such reactants. Such a phenomenon is now known as {\it diffusion driven  instability (DDI)}, or {\it Turing instability}.

Exploring Turing's idea further,  A. Gierer and H. Meinhardt, \cite{gm72}, proposed in 1972 the following activator-inhibitor system, known since then as a Gierer-Meinhardt system, to model the regeneration phenomenon of hydra located in a domain $\Om\subset \R^N, N\geq 1$ \bge
&& u_t= \ep^2 \Delta u-u+\displaystyle\frac{u^p}{v^q}, \quad\mbox{in}\quad \Om \times (0,T), \label{gm1}\\
&& \tau v_t = D \Delta v-v+\displaystyle\frac{u^r}{v^s}, \quad\mbox{in}\quad \Om \times (0,T),\label{gm2}\\
&&\displaystyle\frac{\partial u}{\partial \nu}=\displaystyle\frac{\partial v}{\partial \nu}=0,\quad\mbox{on}\quad \pl\Om\times (0,T),\label{gm3}\\
&& u(x,0)=u_0(x)>0,\quad v(x,0)=v_0(x)>0,\quad\mbox{in}\quad \Om, \label{gm4}
\ege
where $\nu$ denotes the unit outer normal vector to $\pl \Om$ whilst $u$ and $v$ stand for the concentrations of the activator and the inhibitor respectively. System \eqref{gm1}-\eqref{gm4} intends to provide a thorough explanation of symmetry breaking as well as of {\it de novo} pattern formation by virtue of the coupling of a local activation and a long-range inhibition process. The inserted nonlinearities describe the fact that the activator promotes the differentiation process and it stimulates its own production, whereas  the inhibitor acts a suppressant against the self-enhancing activator to prevent the unlimited growth.

Here, $\ep^2, D$ represent the diffusing coefficients whereas the exponents satisfying the conditions:
$$p>1,\; q, r,>0,\quad\mbox{and}\quad s>-1,$$ measure the morphogens interactions. In particular, the dynamics of system \eqref{gm1}-\eqref{gm4} can be characterised by two numbers: the {\it net self-activation index} $\rho\equiv (p-1)/r$ and the {\it net cross-inhibition index} $\gamma\equiv q/(s+1).$ Indeed, $\rho$ correlates the strength of self-activation of the activator with the cross-activation of the inhibitor. So, if $\rho$ is large, then the net growth of the activator is large no matter the inhibitor's growth. On the other hand, $\gamma$ measures how strongly the inhibitor suppresses the production of the activator and that of itself. Now if $\gamma$ is large then the production of the activator is strongly suppressed by the inhibitor. Finally, the parameter $\tau$ quantifies the inhibitor's response against the activator's growth.

Guided by biological interpretation as well as by mathematical reasons, it is usually assumed that the parameters $p,q,r,s$ satisfy the following condition
\bgee
\rho\equiv\frac{p-1}{r}<\displaystyle\frac{q}{s+1}\equiv \gamma,
\egee
or equivalently
\begin{equation}\label{turing2}
p-r\gamma <1
\end{equation}
Condition \eqref{turing2} is called a {\it Turing condition} whilst
the reverse inequality
\begin{equation}\label{aturing}
p-r\gamma > 1
\end{equation}
will be referred to as an {\it anti-Turing condition}.

The {\it Turing condition} guarantees, \cite{nst06}, that the spatially homogeneous equilibrium $(u,v)=(1,1)$ of the corresponding kinetic (ODE) system
\begin{equation}
\displaystyle\frac{du}{dt}=-u+\displaystyle\frac{u^p}{v^q}, \quad
\tau \displaystyle\frac{dv}{dt}=-v+\displaystyle\frac{u^r}{v^s}.
 \label{ode1}
\end{equation}
is stable if $\tau<\frac{s+1}{p-1}.$ Nevertheless, once diffusion terms are introduced, with $\ep^2\ll D,$ and under \eqref{turing2} then $(u,v)=(1,1)$ becomes unstable and bifurcation occurs, see also \cite{nst06}. Therefore, {\it diffusion driven instability (DDI)} takes place which leads to pattern formation and then explains the phenomenon of morphogenesis.

Apart from its vital biological importance the system \eqref{gm1}-\eqref{gm4} has also interesting mathematical features and emerging singularities. As such, it has attracted a lot of attention from the field of mathematical analysis. Subjects of interest include the existence of global-in-time solutions, which was first investigated in \cite{r84} and then studied more thoroughly in \cite{msy95, mt87}. The author in \cite{j06} proved that under the condition
$\frac{p-1}{r}<1$, a global-in-time solution exists, which is an almost optimal result, also taking into consideration the results in \cite{nst06}. Furthermore, \cite{ksy13} contains an investigation of  the asymptotic behaviour of the solution of (\ref{gm1})-(\ref{gm4}). In particular the authors showed that if $\tau=\frac{s+1}{p-1}$, $s>0$, and
\bgee\frac{2\sqrt{d_1\,d_2}}{d_1+d_2}\geq \sqrt{\frac{(s+1)(p-1)}{sp}}, \quad d_1=\ep^2, \ d_2=\tau^{-1} D, \egee
then the global-in-time solution of (\ref{gm1})-(\ref{gm4}) is approaching uniformly a spatially homogeneous solution, which is always periodic-in-time unless it is a constant one. The occurrence of finite-time blow-up, which actually means unlimited growth for the activator,  was first established in \cite{msy95} and later in \cite{ksz16, lps17, z15}, whereas the case of nondiffusing activator finite-time blow-up is also investigated in \cite{ksz16}. The existence and stability of spiky stationary solutions is thoroughly studied in the survey paper \cite{wei08}.

As specified above, in the case of the Gierer-Meinhardt system, the inhibitor diffuses much faster compared to the activator, i.e. $\eps^2\ll D,$ and thus the system (\ref{gm1})-(\ref{gm4}) can be fairly approximated by its {\it shadow system} when $D\gg 1.$ The concept of a {\it shadow system} was introduced by Keener, \cite{ke78}, to describe the qualitative behaviour of reaction-diffusion systems when one of the diffusing coefficients is very large. Such a system is formed by a reaction-diffusion equation coupled with an ordinary differential equation (ODE) with non-local effects and it actually contains all the essential dynamics of the original reaction-diffusion system. In particular, if there is a compact attractor for the  {\it shadow system} the original reaction-diffusion system has a compact attractor too, see also \cite{hs89}.

In the following we provide a formal derivation of the {\it shadow system} of the Gierer-Meinhardt system (\ref{gm1})-(\ref{gm4}). A rigorous proof can be found in \cite{mm17, mchks} where it is also shown that the convergence  of the original reaction-diffusion system towards its {\it shadow system} is valid locally in time except for an initial layer. Now, dividing (\ref{gm2}) by $D$ and letting $D\uparrow+\infty$ for any fixed $t\in(0,T),$ then due to the boundary condition \eqref{gm3} $v$  becomes spatial homogeneous, i.e. $v(x,t)=\xi(t).$  Next, integrating the resulting equation over $\Om$  we finally derive that $u(x,t),\xi(t)$ satisfy the {\it shadow system}:
\begin{eqnarray}
& & u_t=\ep^2 \Delta u-u+\displaystyle\frac{u^p}{\xi^q}, \quad\mbox{in}\quad \Om \times (0,T), \label{sd1}\\
& & \tau \xi_t=-\xi+\displaystyle\frac{1}{\xi^s} \displaystyle\avint u^r \,dx \quad\mbox{in}\quad \Om \times (0,T),\label{sd2}\\
& & \displaystyle\frac{\partial u}{\partial \nu}=0, \quad\mbox{on}\quad \partial \Om\times (0,T),\label{sd3}\\
& & u(x,0)=u_0(x)>0,\quad  \xi(0)=\bar{v}_0=\frac{1}{|\Om|}\int_{\Om} v_0,\quad\mbox{in}\quad \Om,
 \label{sd4}
\end{eqnarray}
where
\bgee
\avint u^r \, dx\equiv\frac{1}{|\Om|} \int_{\Om} u^r \, dx.
\egee
Note that \eqref{sd1}-\eqref{sd4} is  non-local due to the presence of the integral term in (\ref{sd2}).

Since the convergence towards \eqref{sd1}-\eqref{sd4} holds only locally in time there might be discrepancies in global-in-time dynamics between of the (\ref{gm1})-(\ref{gm4}) and those of \eqref{sd1}-\eqref{sd4} for some range of the involved parameters $p,q,r,s;$  this has been also indicated in \cite{ln09, ly14}.  On the other hand, there are ranges of the involved parameters, where the two systems have exactly the same long-time behaviour \cite[Theorem 1]{ln09} and thus it is worth investigating the {\it shadow system} \eqref{sd1}-\eqref{sd4}, which is simpler compared to the full system (\ref{gm1})-(\ref{gm4}), so we can capture some of the features of (\ref{gm1})-(\ref{gm4}).

Henceforth, we focus on the case where $\tau=0;$ i.e. when the inhibitor's response rate is quite small against the inhibitor's growth. We will investigate the dynamics of \eqref{sd1}-\eqref{sd4} in a forthcoming paper.  For $\tau=0$ the second equation  \eqref{sd2} is solved as
\[ \xi(t)=\left( \avint u^r(x,t)\,dx\right)^{\frac{1}{s+1}}, \]
and thus the {\it shadow system} reduces to the following non-local problem
\bge
&& u_t= \Delta u -u +\displaystyle\frac{u^p}{\Big(\displaystyle\avint u^r \, dx\Big)^{\gamma}}, \quad \mbox{in}\quad \Omega \times (0, T),\label{bsd1}\\
&& \displaystyle\frac{\partial u}{\partial \nu}=0, \quad\mbox{on}\quad \partial \Om\times (0,T),\label{bsd2}\\
&& u(x,0)=u_0(x)>0,\quad  \mbox{in}\quad \Om,
 \label{bsd3}
\ege
where for simplicity it has been considered $\ep=1.$

The rest of the current work is devoted to the study of problem the \eqref{bsd1}-\eqref{bsd3} whose mathematical structure is intriguing. In particular, due to the presence of the non-local term and the monotonicity of its nonlinearity, then problem  \eqref{bsd1}-\eqref{bsd3} does not admit a maximum principle, \cite{qs07}, and so alternatives to comparison techniques should be employed to investigate its long-time behaviour. Some global-in-time existence and blow-up results for problem \eqref{bsd1}-\eqref{bsd3} were presented in \cite{ly14}, whereas some slow moving spike solutions were constructed in \cite{iw00}.  In the current paper, we provide novel global-in-time and blow-up results, extending further the mathematical analysis provided in \cite{ly14}, as well as describing the form of the destabilization patterns developed due to the phenomenon of {\it DDI}.

In addition, the investigation of the non-local problem \eqref{bsd1}-\eqref{bsd3} is also attractive from the biological point of view. Specifically, it will reveal under which circumstances the dynamics of the interaction of the two morphogens (activator and inhibitor) can be controlled by governing only the dynamics of the activator itself.

The rest of the manuscript is composed of nine sections. In the next section we provide the main notation  as well as some preliminary results used throughout the manuscript. Our main results, which include the existence of global-in-time and  blowing up solutions of \eqref{bsd1}-\eqref{bsd3}, are presented in section \ref{sec:mr}. Section \ref{sec:qi} contains the proof of a lower estimate of any solution of \eqref{bsd1}-\eqref{bsd3} which actually guarantees its well posedness.  In section \ref{sec:2} we treat the special case $r=p+1,$ whence problem \eqref{bsd1}-\eqref{bsd3} has a variational structure, and under the Turing condition we  prove that global-in-time solutions converge towards steady states through Turing patterns. Section \ref{sec:3} contains a global-in-time existence result for \eqref{bsd1}-\eqref{bsd3} analogous to the one in \cite{j06}. In section \ref{sec:4} we derive proper estimates of $L^{\ell}-$norms, $\ell>1,$ of the solution $u(x,t)$ which either lead to global-in-time existence or to finite-time blow-up. A {\it DDI} result, which is actually exhibited in the form of a diffusion-driven blow-up for peaky initial data,  is proven in section \ref{sec:5}. Finally section \ref{sec:7} investigates the blow-up rate as well the blow-up profile of the derived blowing up solutions and section \ref{conc} summarizes the main conclusions of the current work.

\section{Preliminaries}\label{prl}
Throughout the manuscript by $||h||_{\ell}, \;\ell>0,$ we denote the $L^{\ell}-$norm of function $h$ defined as:
\bgee
||h||_{\ell}:=\left(\avint |h|^{\ell}\,dx\right)^{1/\ell}, \quad 0<\ell<\infty,
\egee
and
\bgee
||h||_{\infty}:=\esssup_{\Om} |h|,
\egee
whereas $||h||_{H^1}$ stands for the norm of the Sobolev space $H^1(\Om)$ defined as:
\bgee
||h||_{H^1}:=\left(\avint (|h|^{2}+|\nabla h|^2)\,dx\right)^{1/2}.
\egee
Moreover, if $\Delta$ is the Laplace operator associated with Neumann boundary conditions then by $e^{t\Delta}$ we denote its semigroup. Then the well-known estimate, \cite{r84}, holds
\begin{equation}
\Vert e^{t\Delta}h\Vert_q\leq C\max\{ 1,
t^{-\frac{N}{2}(\frac{1}{\ell}-\frac{1}{q})}\} \Vert h\Vert_\ell, \quad 1\leq \ell\leq q\leq \infty.
 \label{eqn:1.2}
\end{equation}

Note that under condition \eqref{turing2}  the solution of the spatially homogeneous part
\bge\label{ts1}
\frac{du}{dt}=-u+u^{p-r\gamma}, \quad u(0)=\bar{u}_0>0,
\ege
never exhibits blow-up, since the non-linearity is sublinear, and its unique stationary state $u=1$ is asymptotically stable. Below, by using stability analysis we show that condition \eqref{turing2} implies linear instability.

Indeed, the linearized problem of (\ref{bsd1})-(\ref{bsd3}) around $u=1$ is given by
\bgee
&&\phi_t=\Delta \phi+(p-1)\phi-r\gamma \avint \phi, \quad \mbox{in}\quad\Omega \times (0, T),\label{ln1}\\
&&\frac{\partial \phi}{\partial\nu}=0, \quad\mbox{on}\quad\partial\Om, \label{ln2}
\egee
and can be written in the form of an evolution equation in $X=L^2(\Omega)$ as
\[ \frac{d\phi}{dt}=-A\phi. \]
Here the generator $A$ is a self-adjoint operator associated with the bi-linear form (see Kato \cite{kato})
\[ a(\phi,w)=\avint\left(\nabla \phi\cdot \nabla w+(1-p)\phi w \right)\ dx+r\gamma \avint \phi\cdot\avint w, \quad \phi,w\in V=H^1(\Omega). \]
Now for $\phi=w$ we derive
\begin{eqnarray*}
a(\phi,\phi) & = & \Vert \nabla \phi\Vert_2^2+(1-p)\avint \phi^2+r\gamma \left(\avint \phi\right)^2 \\
& = & \sum_{j=1}^\infty\mu_j^2\vert (\phi,\varphi_j)\vert^2+\sum_{j=1}^\infty(1-p)\vert (\phi, \varphi_j)\vert^2+r\gamma\vert (\phi,\varphi_1)\vert^2 \\
& = & (1-p+r\gamma)\vert (\phi, \varphi_1)\vert^2+\sum_{j=2}^\infty(\mu_j^2+1-p)\vert (\phi, \varphi_j)\vert^2,
\end{eqnarray*}
where $0=\mu_1<\mu_2\leq \cdots \rightarrow\infty$ denote the eigenvalues of $-\Delta$ associated with the Neumann boundary condition, and $\varphi_j$ is the corresponding $j$-th eigenfunction normalized by $\Vert \varphi_j\Vert_2=1$.
Note that under the Turing condition (\ref{turing2}), the linearized instability of the steady-state solution $u=1$ arises if and only if $\mu^2_2<p-1.$ The latter suggests that under condition (\ref{turing2}) a Turing instability phenomenon should be anticipated, which in particular  as shown in Theorem \ref{thm:1.7},  this Turing instability is exhibited in the form of a diffusion-driven blow-up.

\section{Main Results}\label{sec:mr}
In the current section we present our main results and we prove them in the following sections.

The first observation regarding the solution of \eqref{bsd1}-\eqref{bsd3} is that it never quenches in finite time. Indeed, the following holds
\begin{proposition}\label{lbd}
Each $T>0$ admits $C_T>0$ such that
\begin{equation} \label{jg0}
u(x,t)\geq C_T\quad \mbox{in $\Om\times [0,T)$}.
\end{equation}
\end{proposition}
\begin{proof}
By maximum principle and comparison theorems we obtain that $u(x,t)>0$ and $u(x,t)\geq \tilde u(t)$, where $\tilde{u}=\tilde{u}(t)$ solves the following
\[ \frac{d\tilde{u}}{dt}=-\tilde{u}\quad \mbox{in $(0, T)$},\quad \tilde{u}(0)=\tilde{u}_0\equiv \inf_{\Om} u_0(x)>0. \]
Therefore we obtain (\ref{jg0}) with $C_T=\tilde{u}_0e^{-T}$.
\end{proof}
Due to Proposition \ref{lbd} the following alternatives are left; blow-up in finite time indicated by $T<+\infty$, blow-up in infinite time, quenching in infinite time, and global-in-time compact orbit in $C(\overline{\Omega})$. In fact, by (\ref{jg0}) and the parabolic regularity the existence time of the classical solution $u=u(\cdot, t)$ of \eqref{bsd1}-\eqref{bsd3} in $0\leq t \leq \tilde T$ is estimated below by $\tilde T$ and $\Vert u_0\Vert_\infty$ (see, e.g., \cite{r84}). Then there holds
\begin{equation}
T<+\infty \ \Rightarrow \ \lim_{t\uparrow T}\Vert u(\cdot,t)\Vert_\infty=+\infty
  \label{eqn:1.13h}
\end{equation}
as in (8.25) of \cite{ss12}.
Now finite-time blow-up actually arises under the conditions of the following proposition.
\begin{proposition}\label{ats1}
Assume that $p \geq r$ and the Turing condition \eqref{turing2} holds then
\begin{equation}
\bar{u}\geq \max\{ 1, \bar{u}_0\}.
 \label{eqn:1.13}
\end{equation}
Whereas if anti-Turing condition \eqref{aturing} holds and $\bar{u}_0>1$, then finite-time blow-up occurs, i.e. $T<+\infty.$
 \label{lem:1.2}
\end{proposition}
\begin{proof}
Since $p>1$ and $p\geq r$, there is $r\leq \mu\leq p$ satisfying $s\geq 1$. Then we obtain
\[ \avint u^p\geq \left( \avint u^{\mu}\right)^{\frac{p}{\mu}}, \quad \left( \avint u^r\right)^\gamma\leq \left( \avint u^\mu \right)^{\frac{r}{\mu}\gamma} \]
via H\"{o}lder's inequality and hence
\begin{equation}
\displaystyle\frac{d\bar{u}}{dt}= -\bar{u}+\displaystyle{\frac{\avint u^p}{\left(\avint u^r\right)^{\gamma}}}\geq -\bar{u}+\left(\avint u^\mu\right)^{\frac{p-r\ga}{\mu}}\geq -\bar{u}+\bar{u}^{p-r\gamma}.
 \label{eqn:1.14}
\end{equation}
In case $p-r\gamma<1$ then the differential inequality  (\ref{eqn:1.14}) implies \eqref{eqn:1.13}. Whilst, in the complementary case $p-r\gamma>1$ again by virtue of (\ref{eqn:1.14}) we derive that $\bar{u}$ blows up in finite time provided $\bar{u}_0>1,$ and hence $u$ does so.
\end{proof}
\begin{rem}
Proposition \ref{ats1} illustrates that in that case the qualitative behaviour of the full system \eqref{gm1}-\eqref{gm4} and those of the non-local problem \eqref{bsd1}-\eqref{bsd3} is quite different. It should be pointed out that the anti-Turing condition \ref{aturing} the full system does not exhibit any instability, whilst an instability emerges when Turing condition \eqref{turing2} is imposed.
\end{rem}
Here by quenching in infinite time we mean
\bge\label{mkl1}
T=+\infty \quad\mbox{and}\quad \liminf_{t\uparrow+\infty}\Vert u(\cdot,t)\Vert_r=0\quad\mbox{for some}\quad r>1.
\ege
We note that property \eqref{mkl1} does not arise neither for the original system (\ref{gm1})-(\ref{gm4}) nor for the shadow system (\ref{sd1})-(\ref{sd2}), as it can be concluded by the classification of the homogeneous orbits given in  \cite{nst06}. Therefore, our first main result, see Theorem \ref{eiq} below,  which concerns the exclusion of infinite time quenching for the solutions of (\ref{bsd1})-(\ref{bsd3}) is in agreement with what is observed in systems (\ref{gm1})-(\ref{gm4}) and (\ref{bsd1})-(\ref{bsd3}).

Henceforth, $C$ and $c$ denote large and small positive constants independent of $t$, respectively.
\begin{theorem}\label{eiq}
There is $\delta_0>0$ such that any $0<\delta\leq \delta_0$ admits the estimate
\begin{equation}
\avint u^{-\delta}\leq C\quad\mbox{for any}\quad t>0,
 \label{eqn:1.16h}
\end{equation}
 \label{thm:1.1}
\end{theorem}
where constant $C$ is independent of $t.$
\begin{rem}\label{ts3}
In fact, inequality (\ref{eqn:1.16h}) implies $\displaystyle{ \avint u^{\delta}\geq c=C^{-1}}$ and then
\begin{equation}
\avint u^r\geq \left(\avint u^\delta \right)^{r/\delta}\geq c^{r/\delta}>0\quad\mbox{for any}\quad t>0,
 \label{eqn:1.17}
\end{equation}
follows by Jensen's inequality taking $\delta\leq r,$ where again $c$ is independent of time $t.$ Consequently, relation \eqref{eqn:1.17} guarantees that the nonlinear term of non-local equation \eqref{bsd1} stays away from zero and therefore the solution $u$ is bounded away from zero as well.
\end{rem}
\begin{rem}
Remark \ref{ts3} is interpreted in biological context as follows: the activator can never be diminished.
\end{rem}
Next we focus on the special case $r=p+1$. In this case problem (\ref{bsd1})-(\ref{bsd3}) admits a variational structure which is not the case for the original system (\ref{gm1})-(\ref{gm4}). In particular for $r=p+1$ problem (\ref{bsd1})-(\ref{bsd3}) has a Lyapunov functional of the form
\[ J(u)= \displaystyle\frac{1}{2}\left( \Vert \nabla u\Vert_2^2+\Vert u\Vert_2^2\right)-\displaystyle\frac{1}{(p+1)(1-\gamma)}\left(\displaystyle\avint u^{p+1} \, dx\right)^{1-\gamma}, \]
since along any solution trajectory there holds
\begin{equation}\label{27}
\displaystyle\frac{d}{dt} J(u(t))=-\Vert u_t\Vert_2^2 \leq  0.
\end{equation}
Note also that in that case Turing condition (\ref{turing2}) is reduced to
\be\label{mk4}
 \gamma>\frac{p-1}{p+1}.
\ee
Next under anti-Turing condition and via the  double well potential method (see \cite{sattinger68, Levine73}, we obtain the following:
\begin{theorem}
Let $r=p+1$ and  $\gamma<\min\{ 1, \frac{p-1}{p+1}\}.$ If $J(u_0)\leq 0$ then finite-time blow-up occurs, i.e.  $T<+\infty$.
 \label{thm:1.2}
\end{theorem}
\begin{rem}
Theorem \ref{thm:1.2} can be interpreted in the biological context as follows: if the activator's initial concentration is large and  its suppression by the inhibitor is rather small (since $0<\ga<1$) then naturally the activator's growth becomes unlimited.
\end{rem}
On the other hand, under the Turing condition \eqref{mk4} we derive the following:
\begin{theorem}\label{thm:1.2a}
Let $N\geq 3$ and $r=p+1$. If $\frac{p-1}{p+1}<\ga<1$ and $1<p<\frac{N+2}{N-2}$ then a global-in-time solution exists, i.e.
\begin{equation}
T=+\infty\quad\mbox{and}\quad \sup_{(0,T)}\Vert u(\cdot,t)\Vert_\infty\leq C.
 \label{eqn:1.16}
\end{equation}
 \label{prop1}
\end{theorem}
\begin{rem}
Theorems \ref{thm:1.2} and \ref{thm:1.2a} indicate that in case where $r=p+1$ then there is a discrepancy  between the behaviour of the full system \eqref{gm1}-\eqref{gm4} and those of the non-local problem \eqref{bsd1}-\eqref{bsd3}. Indeed, under the anti-Turing condition the full system does not exhibit any instability, whilst an instability occurs when Turing condition \eqref{mk4} holds.
\end{rem}
Turing condition \eqref{mk4} further implies that the solution orbit for problem (\ref{bsd1})-(\ref{bsd3}) is compact in $C(\overline{\Omega})$ and the $\omega$-limit set
\[ \omega(u_0)=\{u_\ast\in C(\overline{\Omega}) \mid \mbox{$\exists t_k\uparrow+\infty$ s.t. }\lim_{k\rightarrow\infty}\Vert u(\cdot,t_k)-u_\ast\Vert_\infty=0\} \]
of this orbit is nonempty, connected, compact, and lies in the set of stationary solutions, which are defined as the solutions of the following problem
\begin{equation}
-\Delta u_\ast +u_\ast =\displaystyle\frac{u_\ast^p}{\Big(\displaystyle\avint u_\ast^{p+1} \Big)^{\gamma}}, \ u_\ast>0 \quad\mbox{in}\quad\Omega, \quad \displaystyle\frac{\partial u_\ast}{\partial \nu}=0\quad\mbox{on}\quad \partial\Omega.
 \label{ssd2}
\end{equation}
Concerning (\ref{ssd2}), existence of stable spiky stationary solutions is known (see the survey paper by Wei \cite{wei08}) and thus formation of Turing patterns converging to these spiky solutions is guaranteed as long as \eqref{mk4} holds.

In the following, global-in-time existence of the solution is obtained via a priori estimates of some $L^{\ell}$-norms of solution $u(x,t).$ These a priori estimates hold in a parameter range which implies condition $\frac{p-1}{r}<1$ which, as mentioned earlier, guarantees the global-in-time existence of the solution to the original model (\ref{gm1})-(\ref{gm4}).
\begin{theorem}
If $\frac{p-1}{r}<\min\{1, \frac{2}{N}, \frac{1}{2}(1-\frac{1}{r})\}$ and $0<\gamma<1$, then problem (\ref{bsd1})-(\ref{bsd3}) has a global-in-time solution, i.e. (\ref{eqn:1.16}) holds.
 \label{thm:1.3}
\end{theorem}
\begin{rem}
The result of Theorem \ref{thm:1.3} is in agreement with the global-in-time existence result obtained in \cite{j06} for the full system, and so in that case \eqref{gm1}-\eqref{gm4} and \eqref{bsd1}-\eqref{bsd3} share the same dynamics.
\end{rem}
Now we consider the following $L^{\ell}-$norms , $\ell>0,$ of solution $u(x,t)$
 \begin{equation}
\zeta(t)=\avint u^r\,dx, \quad z(t)=\avint u^{-p+1+r}\,dx, \quad w(t)=\avint u^{p-1+r}\,dx.
 \label{eqn:1.21}
\end{equation}
By choosing proper initial data and using  phase plane analysis, we can actually derive estimates of $\zeta(t), z(t)$ and $w(t),$  see section \ref{sec:4}, identifying also some invariant regions in the plane. In particular, our results can be expressed as follows:
\begin{theorem}\label{obu1}
Let $0<\gamma<1$, $r\leq 1$, and $\frac{p-1}{r}>1$. Assume, furthermore that either (1) $w(0)<\zeta(0)^{1-\gamma}$, or (2) $\frac{p-1}{r}\geq 2$ and $w(0)<1.$ Then finite-time blow-up occurs for problem (\ref{bsd1})-(\ref{bsd3}), i.e. $T<+\infty.$
\end{theorem}
\begin{rem}
Since $\frac{p-1}{r}>1> \gamma$ is assumed, Theorem \ref{obu1} is associated with the finite-time blow-up under anti-Turing condition and is in agreement with the blow-up result  \cite[Theorem 2]{msy95}. This is actually an indication, under condition $\frac{p-1}{r}>1> \gamma$,the qualitative behaviour of the full system \eqref{gm1}-\eqref{gm4} and those of the non-local problem (\ref{bsd1})-(\ref{bsd3}) is quite similar.
\end{rem}
\begin{rem}
The biological interpretation of Theorem \ref{obu1} is as follows: a large initial concentration for the activator combined with small {\it net cross-inhibition index} can lead to its unlimited growth.
\end{rem}
\begin{theorem}
 Let $\gamma>1$, $r\geq 1$ and $\frac{p-1}{r}<1$. Assume, further that $w(0)<\zeta(0)^{1-\gamma}$ and $\zeta(0)^{1+\gamma}>z(0)$.  Then problem (\ref{bsd1})-(\ref{bsd3}) has a global-in-time solution, i.e. (\ref{eqn:1.16}) holds.
 \label{thm:1.6}
\end{theorem}
\begin{rem}
Theorem \ref{thm:1.6}, on the contrary, deals with the case of global-in-time existence under Turing condition and it is also in agreement with Jiang's result in \cite{j06}. Consequently under assumptions of Theorem \ref{thm:1.6} both the full system \eqref{gm1}-\eqref{gm4} as well as the  non-local problem (\ref{bsd1})-(\ref{bsd3}) ensemble the same long-time behaviour.
\end{rem}
In section \ref{prl} it has been already specified through linear stability analysis that under the Turing condition \eqref{turing2} the stable solution $\bar{u}=1$ of \eqref{ts1} destabilises as a solution of (\ref{bsd1})-(\ref{bsd3}). The next result shows that the preceding {\it DDI} phenomenon is realised in the form of diffusion-induced blow-up.
\begin{theorem}
Let $N\geq 3,\; \max\{1,\frac{N}{N-2}\}\leq r\leq p$ and $\frac{2}{N}<\frac{p-1}{r}<\gamma$, then there is a family of radially symmetric blowing up solutions corresponding to a family of spiky initial data.
 \label{thm:1.7}
\end{theorem}
\begin{rem}
Usually {\it DDI} phenomena are connected with pattern formation. The same happens in the case of the diffusion-induced blow-up provided by Theorem \ref{thm:1.7}. The form of the destabilising patterns are determined in section \ref{sec:7}, see particularly Remark \ref{nkl7}.
\end{rem}
\section{Proof of Theorem \ref{thm:1.1}}\label{sec:qi}

Set $\chi=u^{\frac{1}{\al}}$, $\al\neq 0$, then $\chi$ satisfies
\bge
&&\al \chi_t=\alpha\left(\Delta \chi+4 (\al-1) \vert\nabla \chi^{\frac{1}{2}}\vert^2\right)-\chi+f \quad \mbox{in}\quad \Om\times (0,T),\label{ob1}\\
&& \frac{\pl \chi}{\pl \nu}=0, \quad \mbox{on}\quad \partial\Om\times (0,T),\label{ob2a}\\
&&\chi(x,0)=u_0^{\frac{1}{\al}}(x),\quad\mbox{in}\quad \Omega,\label{ob2}
\ege
with
\begin{equation}
f=\frac{u^{p-1+\frac{1}{\al}}}{\left(\avint u^r\right)^{\ga}}
 \label{ob3}.
\end{equation}
Averaging (\ref{ob1}) over $\Om$, we obtain
\begin{equation}\label{ob4}
\alpha\frac{d}{dt}\avint \chi+4\alpha(1-\alpha)\avint\vert \nabla \chi^{\frac{1}{2}}\vert^2+\avint \chi=\avint f
\end{equation}
and hence
\bge\label{ts2}
\frac{d}{dt}\avint \chi+4(1-\alpha)\avint\vert\nabla \chi^{\frac{1}{2}}\vert^2+\frac{1}{\alpha}\avint \chi\leq 0
\ege
for $\alpha<0$ since also $f>0.$ Letting now $\delta=-\frac{1}{\alpha}$ we have
\[ \frac{d}{dt}\avint \chi+4(1+\delta^{-1})\avint\vert \nabla \chi^{\frac{1}{2}}\vert^2 \leq \delta \avint \chi. \]
Since Poincar\'e-Wirtinger's inequality reads
\bgee
\Vert \nabla w\Vert_2^2\geq \mu_2\Vert w\Vert_2^2,\quad\mbox{for any}\quad w\in H^1(\Om),
\egee
where $\mu_2$ is the second eigenvalue of the Laplace operator associated with Neumann boundary conditions, then applied for $w=\chi^{\frac{1}{2}}$ to \eqref{ts2} entails
\be\label{mk5}
\frac{d}{dt}\avint \chi+c\avint \chi\leq 0
\ee
for $0<\delta\ll 1$. Differential inequality  \eqref{mk5} implies that $\chi(t)\leq C<\infty$ for any $t>0$ and thus (\ref{eqn:1.16h}) follows by the fact that $\chi=u^{-\delta}.$

\section{Proof of Theorems \ref{thm:1.2} and \ref{thm:1.2a}}\label{sec:2}

Throughout the current section, we consider $r=p+1$.

\begin{proof}[Proof of Theorem \ref{thm:1.2}]
Since $J(u_0)\leq 0$, then via the dissipation relation \eqref{27} we derive  $J(u(t))\leq 0$ for any $0<t<T.$

We also have
\[ \frac{d}{dt}\Vert u\Vert_2^2=-2I(u) \]
where
\begin{eqnarray*}
I(u) & = & \Vert \nabla u\Vert_2^2+\Vert u\Vert_2^2-\frac{\avint u^{p+1}}{\left( \avint u^r\right)^\gamma} = \Vert \nabla u\Vert_2^2+\Vert u\Vert_2^2-\left( \avint u^{p+1}\right)^{1-\gamma} \\
& = & 2J(u)+\left( \frac{2}{(p+1)(1-\gamma)}-1\right)\left( \avint u^{p+1}\right)^{1-\gamma} \\
& \leq & -\left( 1-\frac{2}{(p+1)(1-\gamma)}\right)\Vert u\Vert_{p+1}^{(p+1)(1-\gamma)}.
\end{eqnarray*}
Since $0<\gamma<\min\{ 1, \frac{p-1}{p+1}\}$, there holds that $(p+1)(1-\gamma)>2,$ and thus by virtue of H\"{o}lder's inequality we can find $\alpha>0$ such that
\bge\label{ts4}
\frac{d}{dt}\Vert u\Vert_2^2\geq c\Vert u\Vert_2^{2+\alpha},
\ege
since also $p>1.$ Now \eqref{ts4} entails that $\Vert u\Vert_2^2$ blows up in finite time since $u_0(x)>0$ and thus $u$ exhibits a finite-time blow-up as well.
\end{proof}

\begin{proof}[Proof of Theorem \ref{prop1}]
In this case we have $0<\gamma<1$ and $(p+1)(1-\gamma)<2$. Dissipation relation  \eqref{27} suggests
\bge\label{ts5}
\frac{1}{2}(\Vert \nabla u\Vert_2^2+\Vert u\Vert_2^2)\leq J(u_0)+\frac{1}{(p+1)(1-\gamma)}\left( \avint u^{p+1}\right)^{1-\gamma}.
\ege
Furthermore, Sobolev's and Young's inequalities entail
\bge\label{ts6}
\left( \avint u^{p+1}\right)^{1-\gamma}=\Vert u\Vert_{p+1}^{(p+1)(1-\gamma)}\leq \frac{1}{4}\Vert u\Vert_{H^1}^2+C,
\ege
recalling $(p+1)(1-\gamma)<2.$ Combining now \eqref{ts5} with \eqref{ts6} we derive the estimate
\begin{equation}
\Vert u(\cdot,t)\Vert_{H^1}\leq C\quad\mbox{for}\quad 0<t<T.
 \label{eqn:1.26}
\end{equation}
Now $u$ satisfies
\bgee
&& u_t=\Delta u-u+a(t)u^p, \quad\mbox{in}\quad \Om\times(0,T),\\
&& \frac{\partial u}{\partial\nu}=0,\quad\mbox{on}\quad \partial\Om\times(0,T),\\
&& u(x,0)=u_0(x)>0,\quad\mbox{in}\quad\Om,
\egee
with $$0\leq a(t)=\frac{1}{\left(\avint u^{p+1}\right)^{\gamma}}\leq C<+\infty,$$ due to \eqref{eqn:1.17}. Then, letting $A$ to be $-\Delta+1$ with the homogeneous Neumann boundary condition, we use
\[ u(\cdot,t)=e^{-tA}u_0+\int_0^te^{-(t-s)A}a(s)u(\cdot,s)^p \ ds  \]
to apply a standard bootstrap argument. In fact, it follows that (\ref{eqn:1.16}) from (\ref{eqn:1.26}), $1<p<\frac{N+2}{N-2},$ and the proof of \cite[Lemma 8.1]{ss12}.
\end{proof}

\begin{rem}
In the case where $\gamma=\frac{p-1}{p+1}$ and $1<p<\frac{N+2}{N-2}$, we have always $T=+\infty$, whilst infinite-time blow-up, i.e. $\lim_{t\uparrow +\infty}\Vert u(\cdot,t)\Vert_\infty=+\infty,$ may occur. In fact, by the proof of Theorem \ref{prop1} we have
\[ \Vert u(\cdot,t)\Vert_{H^1}\leq C(1+t) \quad \mbox{for $0<t<T$}, \]
and then by virtue of Sobolev's imbedding we obtain $\Vert u(\cdot,t)\Vert_\infty\leq C_T,$  which entails $T=+\infty$ by the parabolic regularity. Furthermore, in case where $J(u_0)<0$  we derive
\[ \frac{d}{dt}\Vert u\Vert_2^2\geq -2J(u_0)>0, \]
by the proof of Theorem \ref{thm:1.2}, and then it follows that $\lim_{t\uparrow\infty}\Vert u(\cdot,t)\Vert_2=+\infty.$ The latter implies that $\lim_{t\uparrow+\infty}\Vert u(\cdot, t)\Vert_\infty=+\infty$ and thus infinite-time blow-up occurs in that case.
\end{rem}

\section{Proof of Theorem \ref{thm:1.3}}\label{sec:3}
We assume $\frac{p-1}{r}<\min\{ 1, \frac{2}{N}, \frac{1}{2}(1-\frac{1}{r})\}$ and $0<\gamma<1$. We also consider $N\geq 2$ since the complementary case $N=1$ is simpler.

Since $p>1$, the above assumption implies $\frac{p-1}{r}<\frac{2}{N}$ and $r>p$. Then there holds that
\[ 0<\frac{1}{r-p+1}<\min\left\{ 1, \frac{1}{p-1}\cdot\frac{2}{N-2}, \frac{1}{1-p+r\gamma}\right\}, \]
since also $0<\gamma<1$.

Choosing  $\frac{1}{r-p+1}<\alpha<\min\{ 1, \frac{1}{p-1}\cdot\frac{2}{N-2}, \frac{1}{1-p+r\gamma} \}$, we have
\[ \max\left\{ \frac{N-2}{N}, \frac{1}{\alpha r}\right\}<\frac{1}{-\alpha+1+\alpha p}, \]
and hence there is $\beta>0$ such that
\begin{equation}
\max\left\{ \frac{N-2}{N}, \frac{1}{\alpha r}\right\} <\frac{1}{\beta}<\frac{1}{-\alpha+1+\alpha p}<2,
 \label{eqn:15}
\end{equation}
which also satisfies
\begin{equation}
\frac{\beta}{\alpha r}<1<\frac{\beta}{-\alpha+1+\alpha p}.
 \label{eqn:16}
\end{equation}
Note that for $f$ defined by \eqref{ob3} holds
\[ \avint f=\frac{\avint u^{p-1+\frac{1}{\alpha}}}{\left( \avint u^r\right)^\gamma}= \frac{\avint \chi^{-\alpha+1+\alpha p}}{\left( \avint\chi^{\alpha r}\right)^\gamma}. \]
By virtue of (\ref{eqn:16})
\[ \avint \chi^{-\alpha+1+\alpha p}\leq \left( \avint \chi^\beta\right)^{\frac{-\alpha+1+\alpha p}{\beta}}\quad\mbox{and}\quad \left( \avint \chi^{\alpha r}\right)^\gamma \geq \left( \avint \chi^\beta\right)^{\frac{\alpha r}{\beta}\cdot\gamma} \]
and thus
\begin{equation}
\avint f\leq \left( \avint \chi^\beta\right)^{\frac{-\alpha+1+\alpha p-\alpha r\gamma}{\beta}}=\Vert \chi^{\frac{1}{2}}\Vert_{2\beta}^{2(1-\sigma)}
 \label{eqn:3.3}
\end{equation}
with $0<\sigma=\alpha\{1-p+r\gamma\}<1$, recalling $\frac{p-1}{r}<\gamma$ and $\alpha<\frac{1}{1-p+r\gamma}$.

Now since $1<2\beta<\frac{2N}{N-2}$ holds due to (\ref{eqn:15}), then Sobolev's and Young's inequalities entail
\[ \frac{d}{dt}\avint \chi+c\Vert \chi^{\frac{1}{2}}\Vert_{H^1}^2\leq C,\quad 0<t<T, \]
using also $0<\alpha<1$, and in particular,
\[ \avint \chi\leq C, \quad\mbox{for any}\quad 0<t<T. \]
Since $\frac{1}{\alpha}$ can be chosen to be close to $r-p+1$, we have
\begin{equation}
\Vert u(\cdot,t)\Vert_q\leq C_q, \quad 0<t<T,\quad\mbox{for any}\quad 1\leq q<r-p+1,
 \label{eqn:15h}
\end{equation}
taking into account that $\chi=u^{\frac{1}{\alpha}}.$

Since $\frac{p-1}{r}<\frac{1}{2}(1-\frac{1}{r})$ implies $\frac{r-p+1}{p}>1,$ then if there is $a>1$ such that
\bge\label{ikl1}
\Vert u(\cdot,t)\Vert_q\leq C_q, \quad 0<t<T,\quad\mbox{for any}\quad 1\leq q<a(r-p+1),
\ege
by virtue of the semigroup estimate \eqref{eqn:1.2} inequality \eqref{ikl1} can be extended for any $q\geq 1$ as long as $\frac{N}{2}(\frac{1}{\ell}-\frac{1}{q})<1$. Therefore, we obtain
\bge\label{psl2}
\Vert u(\cdot,t)\Vert_q\leq C_q, \quad 0<t<T,\quad\mbox{for any}\quad 1\leq q<a_1(r-p+1),
\ege
and for $a_1>0$ defined by
\bge\label{psl1}
 \frac{1}{a_1}=\frac{1}{a}-\frac{2}{N}\cdot\frac{r-p+1}{p},
\ege
as long as the right-hand side of \eqref{psl1} is positive, otherwise  $q=\infty$ into relation \eqref{psl2}. We eventually obtain (\ref{eqn:1.16}), and the proof is complete.

\section{Proof of Theorems \ref{obu1} and \ref{thm:1.6}}\label{sec:4}

Let $(\zeta, z,w)=(\zeta(t), z(t), w(t))$ defined by (\ref{eqn:1.21}) then, by virtue of H\"{o}lder's inequality we have
\begin{equation} \label{ob5}
wz\geq \zeta^2.
\end{equation}
\begin{proof}[Proof of Theorem \ref{obu1}]
We first consider $r\leq 1<\frac{p-1}{r}$ and $0<\gamma<1$. Since $r\leq 1$ then \eqref{ob4} for $\alpha=\frac{1}{r}$ yields
\begin{eqnarray}
\frac{1}{r} \frac{d \zeta}{dt}& = & \frac{4}{r}\left(\frac{1}{r}-1\right)\avint \vert\nabla u^{\frac{r}{2}}\vert^2-\zeta+\frac{z}{\zeta^{\gamma}}\quad\mbox{for}\quad 0<t<T,
 \label{ob6}
\end{eqnarray}
and taking \eqref{ob5} into account we derive
\begin{eqnarray}
\frac{1}{r} \frac{d \zeta}{dt}\geq  -\zeta+\frac{\zeta^{2-\gamma}}{w} =\frac{\zeta}{w}\left(-w+\zeta^{1-\gamma}\right)\quad\mbox{for}\quad 0<t<T.
 \label{ob6a}
\end{eqnarray}
Furthermore, since $\frac{p-1}{r}>1$ then \eqref{ob4} for $\alpha=\frac{1}{-p+1+r}$ reads
\be\label{mk2}
\alpha\frac{dw}{dt}=4\alpha(\alpha-1)\avint \vert \nabla u^{\frac{1}{2\alpha}}\vert^2-w+\zeta^{1-\gamma},\quad\mbox{for}\quad 0<t<T,
\ee
which, since $\alpha=\frac{1}{-p+1+r}<0,$  implies
\bgee
\alpha\frac{dw}{dt}\geq -w+\zeta^{1-\gamma},\quad\mbox{for}\quad 0<t<T,
\egee
or equivalently
\begin{equation}
\frac{1}{p-1-r} \frac{dw}{dt}\leq w-\zeta^{1-\gamma},\quad\mbox{for}\quad 0<t<T.
 \label{ob7}
\end{equation}
The condition $0<\gamma<1$, entails that the curve
\begin{equation}
\Gamma_1: w=\zeta^{1-\gamma}, \ \zeta>0,
 \label{eqn:5.4}
\end{equation}
is concave in the $w\zeta-$plane, with its endpoint at the origin $(0,0)$. Relations  (\ref{ob6a}) and (\ref{ob7}) imply that the region $\mathcal{R}=\{ (\zeta,w) \mid w<\zeta^{1-\gamma}\}$ is invariant for the system \eqref{ob6}, \eqref{mk2}, i.e. if $(\zeta(0),w(0))\in \mathcal{R}$ then $(\zeta(t),w(t))\in \mathcal{R}$ for any $t>0.$ Furthermore,  $\zeta=\zeta(t)$ and $w=w(t)$ are increasing and decreasing on $\mathcal{R},$ respectively.

In case  $w(0)<\zeta(0)^{1-\gamma}$, then
\[ \frac{dw}{dt}<0, \ \frac{d\zeta}{dt}>0, \quad\mbox{for}\quad 0\leq t<T, \]
and thus,
\[ \frac{1}{w}-\frac{1}{\zeta^{1-\gamma}}\geq \frac{1}{w(0)}-\frac{1}{\zeta(0)^{1-\gamma}}\equiv c_0>0, \quad\mbox{for}\quad 0\leq t<T.\]
Therefore by virtue of \eqref{ob6a}
\begin{equation}
\frac{1}{r}\frac{d\zeta}{dt}\geq -\zeta+\frac{\zeta^{2-\gamma}}{w}=\zeta^{2-\gamma}\left(\frac{1}{w}-\frac{1}{\zeta^{1-\gamma}}\right) \geq c_0\zeta^{2-\gamma}, \quad 0\leq t<T.
 \label{eqn:5.3}
\end{equation}
Since $2-\gamma>1$ then (\ref{eqn:5.3}) implies that $\zeta(t)$ blows up in finite time
\bgee
t_1\leq \hat{t}_1\equiv\frac{(\zeta(0))^{\ga-1}}{(1-\ga)c_0 r},
\egee
and using the inequality
\bgee
\zeta(t)=\avint u^r\,dx\leq \|u(\cdot,t)\|^r_{\infty}
\egee
we conclude that $u(x,t)$ blows up in finite time $T\leq t_1$ as well.

We consider now the second case when  $\frac{p-1}{r}\geq 2$ and thus $q=\frac{p-1-r}{r}\geq 1.$ Then by virtue of Jensen's inequality
\bgee
\avint u^r\cdot\left( \avint (u^{-r})^q\right)^{\frac{1}{q}}\geq \avint u^{r}\cdot\avint u^{-r}\geq 1,
 \label{eqn:5.5}
\egee
and thus $\zeta^{\frac{1}{r}}\geq w^{-\frac{1}{p-1-r}}$ which  entails
\be\label{mk1}
w\geq \zeta^{-\frac{p-1-r}{r}}=\zeta^{1-\frac{p-1}{r}}.
\ee
In addition, inequality $\frac{p-1}{r}\geq 2$ implies that the curve
\[ \Gamma_2: w=\zeta^{1-\frac{p-1}{r}}, \ \zeta>0, \]
is convex and approaches $+\infty$ and $0$ as $\zeta\downarrow 0$ and $\zeta\uparrow+\infty$, respectively. The crossing of $\Gamma_1$ and $\Gamma_2$ is the point $(\zeta,w)=(1,1)$, and therefore $w(0)<1$ combined with \eqref{mk1} imply $w(0)<\zeta(0)^{1-\gamma}$. Consequently, the second case is reduced to the first one and again the occurrence of finite-time blow-up is established.
\end{proof}
\begin{rem}
The existence of the invariant region $\mathcal{R}=\{ (\zeta,w) \mid w<\zeta^{1-\gamma}\}$ for the system \eqref{ob6}, \eqref{mk2} entails that if the consumption of the activator cannot be suppressed initially then this can lead to its unlimited growth.
\end{rem}
\begin{proof}[Proof of Theorem \ref{thm:1.6}]
We first note that under the assumption $r\geq 1$ relation \eqref{ob6} entails
\begin{equation}
\frac{1}{r}\frac{d\zeta}{dt}\leq -\zeta+\frac{z}{\zeta^\gamma},\quad\mbox{for}\quad 0\leq t<T.
 \label{eqn:5.6}
\end{equation}
Furthermore, since $\alpha=\frac{1}{p-1-r}<0$ results from $\frac{p-1}{r}<1$, then \eqref{mk2} implies
\bgee
\alpha\frac{dw}{dt}\geq -w+\zeta^{1-\gamma},\quad\mbox{for}\quad 0\leq t<T,
 \egee
or equivalently
\begin{equation}
\frac{1}{-p+1+r}\frac{dw}{dt}\leq w-\zeta^{1-\gamma},\quad\mbox{for}\quad 0\leq t<T.
 \label{eqn:5.8}
\end{equation}

We claim that the assumption $\zeta(0)^{1+\gamma}>z(0)$ yields that  $\zeta(t)^{1+\gamma}>z(t)$ for any $0\leq t<T.$ Indeed, let us assume there exists $t_0>0$ such that
\[ \zeta(t)^{1+\gamma}>z(t), \ 0\leq t<t_0, \quad\mbox{and}\quad \zeta(t_0)=z(t_0). \]
Then we obtain
\bge
\frac{d\zeta}{dt}<0, \quad\mbox{for}\quad 0\leq t<t_0\quad\mbox{and}\quad w(t_0)\geq \zeta(t_0)^{1-\gamma},
 \label{eqn:5.9}
\ege
by virtue of \eqref{eqn:5.6} and (\ref{ob5}).

On the other hand,  $w(0)<\zeta(0)^{1-\gamma},$  due to (\ref{eqn:5.8}), entails
\bgee
\frac{d w}{dt}<0, \quad\mbox{for}\quad 0\leq t<t_0.
 \label{eqn:5.9a}
\egee
Consequently, since also $\gamma>1$ then the curve $(w(t), \zeta(t))$ for $0\leq t\leq t_0$, remains in the region $w<\zeta^{1-\gamma}$ and hence $w(t_0)<\zeta(t_0)^{1-\gamma}$, which contradicts the second inequality of (\ref{eqn:5.9}).

Thus it follows that
\[ \frac{d\zeta}{dt}<0, \ \frac{dw}{dt}<0, \quad\mbox{for}\quad 0\leq t<T, \]
and in particular, we have
\[ \Vert u(\cdot,t)\Vert_{p-1+r}\leq C,\quad\mbox{for}\quad 0\leq t<T. \]
Since $r\geq 1$ implies $\frac{p-1+r}{p}\geq 1$, we obtain (\ref{eqn:1.16}) by the same  bootstrap argument used at the end of the previous section.
\end{proof}

\section{Proof of Theorem \ref{thm:1.7}}\label{sec:5}

In the current  section we restrict ourselves to the radial case $\Om=B(0,1)$ and we also consider $N\geq 3.$ Then the solution  of \eqref{bsd1}-\eqref{bsd3} is radial symmetric, that is $u(x,t)=u(\rho,t)$ for $0\leq \rho=\vert x\vert<1.$

We regard, as in \cite{hy95}, spiky initial data of the form
\begin{equation}
u_0(\rho)=\lambda\varphi_\delta(\rho),
 \label{eqn:6.1}
\end{equation}
with $0<\lambda\ll 1$ and
\begin{equation}\label{id}
\varphi_\delta(\rho)=\begin{cases}
      \rho^{-a},&  \de\leq \rho\leq 1 \\
      \de^{-a}\left(1+\frac{a}{2}\right)-\frac{a}{2}\de^{-(a+2)}\rho^2, & 0\leq \rho<\de,
      \end{cases}
\end{equation}
for $a=\frac{2}{p-1}$ and $0<\de<1.$

It can be easily checked that $u_0(\rho)$ is decreasing, i.e.  $u'_{0}(\rho)<0,$ and thus  $\max_{\rho\in[0,1]}u_0(\rho)=u_0(0).$ Furthermore, due to the maximum principle we have that $u(\rho,t)$ is radial decreasing too, i.e. $u_{\rho}(\rho,t)<0.$

Now having specified the form of the considered initial data, Theorem \ref{thm:1.7} can be rewritten as follows:
\begin{theorem}
Let $N\geq 3,\;1\leq r\leq p$, $p>\frac{N}{N-2}$ and $\frac{2}{N}<\frac{p-1}{r}<\gamma.$ Then there is $\lambda_0>0$ with the following property: any $0<\lambda\leq \lambda_0$ admits $0<\delta_0=\delta_0(\lambda)<1$ such that any solution of problem (\ref{bsd1})-(\ref{bsd3}) with initial data of the form (\ref{eqn:6.1}) and $0<\delta\leq \delta_0$ blows up in finite time, i.e. $T<+\infty.$
 \label{thm:6.1}
\end{theorem}
We perceive that  Theorem \ref{thm:6.1} for $r=1$ is nothing but Proposition 3.3 in \cite{ln09}, which was proven using  a series of auxiliary results and inspired by an approach introduced in \cite{fmc85, hy95}. Therefore, in order to prove Theorem \ref{thm:6.1}  we are following in short the arguments presented in \cite{ln09}, and we provide any modifications where are necessary.

The next lemma is elementary and so its proof is omitted.
\begin{lemma}\label{kkl1}
The function $\phi_{\delta}$ defined by \eqref{id} satisfies the following:
\begin{enumerate}
\item[(i)]There holds that
\begin{equation}
\displaystyle{ \Delta \varphi_\delta\geq -Na\varphi_\delta^p}
 \label{eqn:6.3}
\end{equation}
in the weak sense for any $0<\delta<1$.
\item[(ii)] If $m>0$ and $N>ma$, we have
\begin{equation}\label{ine}
\avint \varphi_\delta^m=\frac{N}{N-ma}+O\left(\de^{N-ma}\right), \quad \delta\downarrow 0.
\end{equation}
\end{enumerate}
\end{lemma}
Lemma \ref{kkl1} can used to obtain some further useful estimates. Indeed, if we consider
\begin{equation}
\mu>1+r\gamma
 \label{eqn:6.15}
\end{equation}
and set
\begin{equation}
\alpha_1=\sup_{0<\delta<1}\frac{1}{\bar{\varphi}_\delta^\mu}\avint\varphi_\delta^p, \quad\mbox{and}\quad \alpha_2=\inf_{0<\delta<1}\frac{1}{\bar{\varphi}_\delta^\mu}\avint\varphi_\delta^p,
 \label{eqn:6.6}
\end{equation}
then since $p>\frac{N}{N-2}$, relation(\ref{ine}) is applicable for $m=p$ and $m=1$, and thus due to \eqref{eqn:6.15} we obtain
\bge\label{at1} 0<\alpha_1, \alpha_2<\infty. \ege
Furthermore, there holds that
\begin{equation}
d\equiv\inf_{0<\delta<1}\left( \frac{1}{2\alpha_1}\right)^{\frac{r\gamma}{p}}\left( \frac{1}{2\bar{\varphi}_\delta}\right)^{\frac{r\gamma}{p}\mu}>0.
 \label{eqn:6.7}
\end{equation}
The following auxiliary result provides a key inequality satisfied by the initial data $u_0=u_0(\vert x\vert)$ defined by \eqref{eqn:6.1}. Indeed, we have
\begin{lemma}
If $p>\frac{N}{N-2}$ and $\frac{p-1}{r}<\gamma$, there exists $\lambda_0=\lambda_0(d)>0$ such that for any $0<\lambda\leq \lambda_0$ there holds
\begin{equation}
\Delta u_0+d\lambda^{-r\gamma}u_0^p\geq 2u_0^p.
 \label{eqn:6.4}
\end{equation}
 \label{lem:6.2}
\end{lemma}

\begin{proof}
Note that inequality (\ref{eqn:6.4}) is equivalent to
\[ \Delta \varphi_\delta+d\lambda^{-r\gamma+p-1}\varphi_\delta^p\geq 2\lambda^{p-1}\varphi_\delta^p \]
which is reduced to
\[ d\lambda^{-r\gamma+p-1}\geq Na+2\lambda^{p-1} \]
due to (\ref{eqn:6.3}). Then the result follows since $\frac{p-1}{r}<\gamma$.
\end{proof}
Henceforth we fix $0<\lambda\leq \lambda_0=\lambda_0(d)$ so that (\ref{eqn:6.4}) is satisfied.  Given $0<\delta<1$, let $T_\delta>0$ be the maximal existence time of the solution to (\ref{bsd1})-(\ref{bsd3}) with initial data of the form (\ref{eqn:6.1}).

In order, to get rid off the linear dissipative term $-u$ we introduce $z=e^tu$, which then satisfies
\bge
&&z_t= \Delta z +K(t)z^p, \quad\mbox{in}\quad Q\equiv \Omega\times (0,T_\delta),\label{tbsd3}\\
&&\displaystyle\frac{\partial z}{\partial \nu}=0, \quad\mbox{on}\quad \partial \Omega\times (0,T_\delta),\label{tbsd3a}\\
&& z(x,0)=u_0(\vert x\vert),\quad\mbox{in}\quad \Omega,\quad\label{tbsd3b}
 \ege
where
\begin{equation} \label{nnt}
K(t)=\frac{e^{(1+r\gamma-p)t}}{\Big(\displaystyle\avint z^r \Big)^{\gamma}}.
\end{equation}
It is clear that $u$ blows up in finite time if and only if $z$ does so.

Due to (\ref{eqn:1.17}) we have
\begin{equation}
0< K(t)=\frac{e^{(1-p)t}}{\Big(\displaystyle\avint u^r \Big)^{\gamma}}\leq C<\infty,
 \label{eqn:6.8}
\end{equation}
thus (\ref{tbsd3}) entails
\begin{equation}
\frac{d\bar{z}}{dt}=K(t)\avint z^p
 \label{eqn:6.8h}
\end{equation}
and we finally derive the following estimate
\begin{equation}
\bar{z}(t)\geq \bar{z}(0)=\avint u_0.
 \label{eqn:6.9}
\end{equation}
Another helpful estimate of $z$ is given by the following lemma
\begin{lemma}\label{lem3}
There holds that
\begin{equation}
\rho^Nz(\rho,t)\leq \bar{z}(t) \quad\mbox{in}\quad(0,1)\times (0,T_{\de}),
 \label{eqn:6.10}
\end{equation}
and
\begin{equation}
z_\rho\left(\frac{3}{4},t\right)\leq -c, \ \ 0\leq t<T_\delta,
 \label{eqn:6.11}
\end{equation}
for any $0<\delta<1$.
\end{lemma}
\begin{proof}
Set $w=\rho^{N-1}z_{\rho}$, then $w$ satisfies
\bgee
&& \mathcal{H}[w]=0, \quad\mbox{in}\quad (0,1)\times (0,T_{\de}),\label{eqn:6.12}\\
&& w(0,t)=w(1,t)=0,\quad\mbox{for}\quad t\in(0,T_{\de}), \label{eqn:6.12a}\\
&& w(\rho,0)<0,\quad\mbox{for}\quad 0<\rho<1, \label{eqn:6.12b}
\egee
where
\[ \mathcal{H}[w]\equiv w_t-w_{\rho\rho}+\frac{N-1}{\rho}w_\rho-p K(t)z^{p-1}w. \]
The maximum principle now implies $w\leq 0$, and hence $z_{\rho}\leq 0$ in $(0,1)\times (0,T_{\de})$. Then inequality (\ref{eqn:6.10}) follows since
\begin{eqnarray*}
\rho^Nz(\rho,t) & = & z(\rho,t)\int_0^\rho Ns^{N-1}ds\leq \int_0^\rho Nz(s,t)s^{N-1}ds \\
& \leq & \int_0^1Nz(s,t)s^{N-1}ds=\avint z=\bar{z}(t).
\end{eqnarray*}

Once $w\leq 0$ is proven, we have
\bgee
& & w_t-w_{\rho\rho}+\frac{N-1}{\rho}w_\rho=pK(t)z^{p-1}w\leq 0 \quad\mbox{in}\quad (0,1)\times (0,T_{\de}),\\
& & w\left(\frac{1}{2},t\right)\leq0,\quad w\left(1,t\right)\leq 0,\quad\mbox{for}\quad t\in (0,T_{\de}) \\
& & w(\rho,0)=\rho^{N-1}u'_{0}(\rho)\leq -c, \quad\mbox{for}\quad\frac{1}{2}<\rho<1,
\egee
which entails $w\leq -c$ in $(\frac{1}{2},1)\times (0, T_\delta)$, and finally (\ref{eqn:6.11}) holds.
\end{proof}

\begin{lemma}\label{kkl2}
Given $\varepsilon>0$ and $1<q<p$ then $\psi$ defined as
\be\label{ik1}
 \psi:=\rho^{N-1}z_\rho+\varepsilon\cdot\frac{\rho^Nz^q}{\bar{z}^{\gamma+1}},
\ee
satisfies
\begin{eqnarray}
& & \mathcal{H}[\psi]\leq -\frac{2q\varepsilon}{\bar{z}^{\gamma+1}}z^{q-1}\psi+\frac{\varepsilon \rho^Nz^q}{\bar{z}^{2(\gamma+1)}}\Big\{2q\varepsilon z^{q-1}-(\gamma+1)\bar{z}^{\gamma-r\gamma}\avint z^p  \nonumber\\
& & \quad -(p-q)z^{p-1}\bar{z}^{\gamma+1-r\gamma}\Big\} \quad \mbox{in}\quad (0,1)\times (0,T_{\de}).
 \label{eqn:6.13h}
\end{eqnarray}
 \label{lem:6.4}
\end{lemma}
The proof of Lemma \ref{kkl2} follows the same steps as the proof of inequality (28) in \cite{ln09}, which holds for $r=1,$ and thus it is omitted.

Observe that when $p>\frac{N}{N-2}$ there is $1<q<p$ such that $N>\frac{2p}{q-1}$ and thus the following quantities
\begin{equation}
A_1\equiv\sup_{0<\de<1}\frac{1}{\bar{u}_0^{\mu}} \avint u_0^p=\lambda^{\mu-p}\alpha_1, \quad A_2\equiv\inf_{0<\de<1}\frac{1}{\bar{u}_0^{\mu}} \avint u_0^p=\lambda^{\mu-p}\alpha_2
 \label{eqn:6.14}
\end{equation}
are finite due to \eqref{at1}. The following result, which is a modification of Lemma 3.3 in \cite{ln09} for $r=1,$ provides a key estimate of the $L^p-$norm of $z$ in terms of $A_1$ and $A_2$ and, since it is a core result for the proof of Theorem  \ref{thm:6.1}, we will sketch its proof shortly.

\begin{proposition}\label{lem4}
There exist $0<\delta_0<1$ and $0<t_0\leq 1$ independent of any $0<\delta\leq \delta_0,$ such that the following estimate is satisfied
\begin{equation}\label{lue}
\frac{1}{2} A_2\ol{z}^{\mu}\leq  \avint z^p\,dx\leq 2 A_1 \ol{z}^{\mu},
\end{equation}
for any $0<t<\min\{t_0,T_{\de}\}.$
\end{proposition}
The proof of the above proposition requires some auxiliary results shown below.

Take $0<t_0(\delta)<T_\delta$ to be the maximal time for which inequality (\ref{lue}) holds true in $0<t<t_0(\delta),$ then we have
\begin{equation}
\frac{1}{2}A_2\bar{z}^\mu\leq \avint z^p\leq 2A_1\bar{z}^\mu, \quad\mbox{for}\quad 0<t<t_0(\delta).
 \label{eqn:6.18}
\end{equation}
We consider the case $t_0(\delta)\leq 1,$ since otherwise there is nothing to prove.

Now the first auxiliary result states

\begin{lemma}\label{nkl2}
There exists $0<t_1<1$ such that
\begin{equation}
\bar{z}(t)\leq 2\bar{u}_0, \quad 0<t<\min\{ t_1, t_0(\delta)\},
 \label{eqn:6.18h}
\end{equation}
for any $0<\delta<1$.
 \label{lem:6.6}
\end{lemma}

\begin{proof}
Since $r\geq 1$ and $t_0(\delta)\leq 1$, it follows that
\[ \frac{d\bar{z}}{dt}\leq 2A_1e^{1+r\gamma-p}\bar{z}^{\mu-r\gamma}, \quad 0<t<t_0(\delta), \]
taking also into account relations (\ref{nnt}) and (\ref{eqn:6.8h}).

Setting $C_1=2A_1e^{1+r\gamma-p}$, we obtain
\[ \ol{z}(t)\leq \left[\bar{u}_0^{1+r\ga-\mu}-C_1(\mu-r\ga-1)t\right]^{-\frac{1}{\mu-r\ga-1}} \]
by (\ref{eqn:6.15}). Therefore, (\ref{eqn:6.18h}) holds for any $0<t<\min\{ t_1, t_0(\delta)\}$ where $t_1$ is estimated as
\bgee
t_1\leq\min\left\{\frac{1-2^{1+r\ga-\mu}}{C_1(\mu-r\ga-1)}\ol{u}_0^{1+r\ga-\mu},1\right\},
\egee
and it is independent of any $0<\delta<1.$
\end{proof}
Another fruitful estimate is provided by the next auxiliary result
\begin{lemma}\label{nkl3}
There exist $0<\delta_0<1$ and $0<\rho_0<\frac{3}{4}$ such that for any $0<\delta\leq \delta_0$  the following estimate holds
\begin{equation}
\frac{1}{\vert\Omega\vert}\int_{B(\rho_0,0)}z^p\leq \frac{A_2}{8}\bar{z}^\mu, \quad\mbox{for}\quad 0<t<\min\{ t_1, t_0(\delta)\}.
 \label{eqn:6.19}
\end{equation}
\end{lemma}
\begin{proof}
First observe that
\begin{equation}
\bar{u}_0\leq \bar{z}(t)\leq 2\bar{u}_0, \quad\mbox{for}\quad 0<t<\min\{t_1, t_0(\delta)\},
 \label{eqn:6.20}
\end{equation}
follows from (\ref{eqn:6.9}) and (\ref{eqn:6.18h}). Then, $\avint z^p$ is controlled by (\ref{lue}) for $0<\min\{ t_1, t_0(\delta)\}$. Since $p>q$ then Young's inequality guarantees that the second term of the right-hand side in (\ref{eqn:6.13h}) is negative for $0<t<\min\{ t_1, t_0(\delta)\}$, uniformly in $0<\delta<1$, provided that $0<\varepsilon\leq \varepsilon_0$ for some $0<\varepsilon_0\ll 1.$ Thus
\begin{equation}
\mathcal{H}[\psi]\leq
-\frac{2q\vep z^{q-1}}{\bar{z}^{\gamma+1}}
\psi \quad\mbox{in}\quad (0,1)\times (0,\min\{ t_1, t_0(\delta)\}).
 \label{eqn:6.21}
\end{equation}
Due to (\ref{eqn:6.10}) and (\ref{eqn:6.20}), we also have
\begin{eqnarray*}
\psi & = & \rho^{N-1}z_\rho+\varepsilon\cdot\frac{\rho^Nz^q}{\bar{z}^{\gamma+1}} \leq \rho^{N-1}z_\rho+\varepsilon\cdot\rho^{N(1-q)}\bar{z}^{q-\gamma-1} \\
& \leq & \rho^{N-1}z_\rho+C\cdot\varepsilon\rho^{N(1-q)}\quad\mbox{in}\quad (0,1)\times (0,\min\{ t_1, t_0(\delta)\})
\end{eqnarray*}
which, for $0<\varepsilon\leq \varepsilon_0,$ entails
\begin{equation}
\psi\left(\frac{3}{4},t\right)<0, \quad 0<t<\min\{ t_1, t_0(\delta)\}
 \label{eqn:6.22}
\end{equation}
by (\ref{eqn:6.11}) and provided that $0<\varepsilon_0\ll 1.$

Additionally \eqref{ik1} for $t=0$ gives
\begin{equation}
\psi(\rho,0) = \rho^{N-1}\left(\lambda \varphi_{\delta}'(\rho)+\varepsilon\lambda^{q-\gamma-1}\rho\cdot\frac{\varphi_\delta^q}{\bar{\varphi}_\delta^{\gamma+1}}\right).
 \label{eqn:6.23}
\end{equation}
Now if $0\leq \rho<\delta$ and $\varepsilon$ are chosen small enough and independent of $0<\de<\de_0,$ then the right-hand side of (\ref{eqn:6.23}) is estimated as follows:
\bgee
\rho^{N}\lambda\left( -a\delta^{-a-2}+\varepsilon\lambda^{q-\gamma-2}\cdot \frac{\varphi_\delta^q}{\bar{\varphi}_\delta^{\gamma+1}} \right)\lesssim \rho^{N}\lambda\left( -a\delta^{-a-2}+\varepsilon\lambda^{q-\gamma-2}\cdot \delta^{-aq} \right)\lesssim 0
\egee
since also
\[ \displaystyle\frac{\varphi_\delta^q}{\bar{\varphi}_\delta^{\gamma+1}}\lesssim \delta^{-aq}, \quad \delta\downarrow 0,\quad\mbox{uniformly in}\quad 0\leq \rho<\delta,\]
holds by (\ref{id}) and (\ref{ine}) for $m=1,$ taking also into account that $a+2=ap>ak.$

On the other hand, if $\delta\leq \rho\leq 1$ then we obtain
\begin{equation}
\psi(\rho,0)=\rho^N\lambda\left(-a\rho^{-a-1}+\varepsilon\lambda^{q-\gamma-1}\frac{\rho^{-aq+1}}{\bar{\varphi}_\rho^{\gamma+1}}\right),
 \label{eqn:6.24}
\end{equation}
by using again (\ref{ine}) for $m=1.$ Since $a+2=ap>aq$ implies $-a-1<-aq+1$, we  derive
\[ \psi(\rho,0)<0, \quad \de\leq \rho\leq \frac{3}{4}, \]
for any $0<\delta\leq \delta_0$ and $0<\varepsilon\leq \varepsilon_0$, provided $\varepsilon_0$ is chosen sufficiently small.

Consequently we deduce
\begin{equation}
\psi(\rho,0)<0, \quad 0\leq \rho\leq \frac{3}{4},
 \label{eqn:6.25}
\end{equation}
for any $0<\delta\leq \delta_0$ and $0<\varepsilon\leq \varepsilon_0$, provided $0<\varepsilon_0\ll 1.$

Combining (\ref{eqn:6.21}), (\ref{eqn:6.22}) and (\ref{eqn:6.25}) we end up with
\[ \psi=\rho^{N-1}z_\rho+\varepsilon\cdot\frac{\rho^Nz^q}{\bar{z}^{\gamma+1}}\leq 0 \quad \mbox{in $(0,\frac{3}{4})\times (0,\min\{ t_1, t_0(\delta)\})$},  \]
which implies
\begin{equation}
z(\rho,t)\leq \left( \frac{\varepsilon}{2}(q-1)\right)^{-\frac{1}{q-1}}\cdot \rho^{-\frac{2}{q-1}}\cdot\bar{z}^{\frac{\gamma-1}{q-1}}(t) \quad \mbox{in $(0,\frac{3}{4})\times (0,\min\{ t_1, t_0(\delta)\})$}.
 \label{eqn:6.26}
\end{equation}
Since $-\frac{2}{q-1}\cdot p+N-1>-1$ due to $N>\frac{2p}{q-1},$ we finally obtain (\ref{eqn:6.19}) for some $0<\rho_0<\frac{3}{4}$.
\end{proof}
\begin{rem}\label{nkl1}
It is worth noting that relation \eqref{eqn:6.26} implies that if $z(\rho,t)$ blows up then this can only happen in the origin $\rho=0;$ that is, only a single-point blow-up is possible. In particular if we define
\[ \mathcal{S}=\{ x_0\in \overline{\Omega} \mid \exists x_k\rightarrow x_0, \ \exists t_k\uparrow T_\de, \ \lim_{k\rightarrow\infty}z(x_k, t_k)=+\infty \}, \]
to be the blow-up set of $z$  then ${\mathcal S}=\{0\}$ in the case $z$ blows up in finite time.
\end{rem}
Next we prove the key estimate \eqref{lue} using essentially  Lemmata \ref{nkl2} and \ref{nkl3}.

\begin{proof}[Proof of Proposition \ref{lem4}]
By virtue of (\ref{eqn:6.15}) and since $\frac{p-1}{r}<\delta$, there holds that $\ell=\frac{\mu}{p}>1.$ We can easily see that $\theta=\displaystyle{\frac{z}{\ol{z}^{\ell}}}$ satisfies
\bgee
&&\theta_t=\Delta \theta+ e^{(r\ga+1-p)t}\left[\frac{z^p}{\ol{z}^{\ell}\left(\avint z^r\right)^{\ga}}-\frac{\ell z\avint z^p}{\ol{z}^{\ell+1}\left(\avint z^r\right)^{\ga}} \right], \quad\mbox{in}\quad Q_0\\
&&\frac{\partial\theta}{\partial\nu}=0\quad\mbox{on}\quad\partial\Omega\times(0, \min\{ t_0, T_\delta\}),\\
&& \theta(x,0)=\frac{z(x,0)}{\bar{z}_0^{\ell}}\quad\mbox{in}\quad \Omega.
\egee
Now due to (\ref{eqn:1.17}), (\ref{eqn:6.9}), (\ref{eqn:6.10}), (\ref{eqn:6.18}), and (\ref{eqn:6.18h}), there holds that
\[ \left\Vert \theta,\ \frac{z^p}{\ol{z}^{\ell}\left(\avint z^r\right)^{\ga}}, \ \frac{\ell z\avint z^p}{\ol{z}^{\ell+1}\left(\avint z^r\right)^{\ga}}\right\Vert_{L^\infty((\Omega\setminus B(0,\rho_0))\times \min\{ t_1, t_0(\delta)\})} \leq C \]
uniformly in $0<\delta\leq \delta_0$.

Therefore, by the standard parabolic regularity, see DeGiorgi-Nash-Moser estimates in \cite[pages 144-145]{Lie96}, there is $0<t_2\leq t_1$ independent of $0<\de\leq \de_0$ such that
\[ \sup_{0< t<\min\{t_2, t_0(\delta)\}}\left\Vert \theta^p(\cdot,t)-\theta^p(\cdot,0)\right\Vert_{L^1(\Omega\setminus B(0,\rho_0))}\leq \frac{A_2}{8}\vert\Omega\vert, \]
which implies
\begin{equation}
\left\vert \frac{1}{\vert\Omega\vert}\int_{\Omega\setminus B(0,\rho_0))}\frac{z^p}{\bar{z}^\mu}-\frac{1}{\vert\Omega\vert}\int_{\Omega\setminus B(0,\rho_0)}\frac{z_0^p}{\bar{z}_0^\mu} \right\vert \leq \frac{A_2}{8}, \quad 0<t<\min\{ t_2, t_0(\delta)\},
 \label{eqn:6.28}
\end{equation}
for any $0<\delta\leq \delta_0$. Inequalities (\ref{eqn:6.19}) and (\ref{eqn:6.28}) entail
\[ \left\vert \avint \frac{z^p}{\ol{z}^\mu}-\avint \frac{z_0^p}{\ol{z}_0^\mu}\right\vert\leq \frac{3 A_2}{8}, \quad\mbox{for}\qquad 0<t<\min\{ t_2, t_0(\delta)\}\quad\mbox{and}\quad 0<\delta\leq \delta_0, \]
and hence
\begin{equation}\label{tbsd13}
\frac{5A_2}{8}\leq \avint \frac{z^p}{\ol{z}^{\mu}} \leq \frac{11 A_1}{8}, \quad 0<t<\min\{ t_2, t_0(\de)\}, \ 0<\delta\leq \delta_0,
\end{equation}
taking into account that
\[ A_2\leq \avint\frac{z_0^p}{\bar{z}_0^\mu}\leq A_1. \]
Therefore, if we take $t_0(\delta)\leq t_2$ then it follows that
\[ \frac{1}{2}A_2\bar{z}^\mu<\frac{5}{8}A_2\bar{z}^\mu\leq \avint z^p\leq \frac{11}{8}A_1\bar{z}^\mu<2A_1\bar{z}^\mu, \quad 0<t<t_0(\delta), \]
and by a continuity argument we deduce that
\[ \frac{1}{2}A_2\bar{z}^\mu\leq \avint z^p\leq 2A_1\bar{z}^\mu,  \quad 0<t<t_0(\delta)+\eta,\]
for some $\eta>0,$ which contradicts the definition of $t_0(\delta)$.

Consequently, we obtain $t_2<t_0(\delta)$ for any $0<\delta\leq \delta_0$, and the proof is complete with $t_0=t_2$.
\end{proof}
Now we have all the ingredients to proceed to the proof of the main result of this section.
\begin{proof}[Proof of Theorem \ref{thm:6.1}]
Since $t_0\leq t_1$ in (\ref{eqn:6.18h}), we have
\begin{eqnarray}
K(t) & &\geq \frac{1}{\left(\avint z^r\right)^\gamma}\geq
\frac{1}{\left( \avint z^p\right)^{\frac{r\gamma}{p}}}\geq
\left( \frac{1}{2A_1\bar{z}^\mu}\right)^{\frac{r\gamma}{p}} = \left( \frac{1}{2A_1}\right)^{\frac{r\gamma}{p}}\cdot\left(
\frac{1}{\bar{z}}\right)^{\frac{r\gamma}{p}\mu} \nonumber\\
& & = \left( \frac{1}{2\alpha_1}\right)^{\frac{r\gamma}{p}}\cdot\left( \frac{1}{2\bar{\varphi}_\delta}\right)^{\frac{r\gamma}{p}\mu}\lambda^{-r\gamma}\geq d\lambda^{-r\gamma}\equiv D, \quad 0<t<\min\{ t_0, T_\delta\},
 \label{eqn:6.33}
\end{eqnarray}
by virtue of (\ref{eqn:6.7}) and (\ref{eqn:6.14}). Since $0<\lambda\leq \lambda_0(d)$, then inequality (\ref{eqn:6.4}) applies to derive
\begin{equation}
\Delta u_0+Du_0^p\geq 2u_0^p
 \label{eqn:6.34}
\end{equation}
for any $0<\delta\leq \delta_0$.

By using (\ref{eqn:6.33}) and (\ref{eqn:6.34}) the comparison principle yields that the solution $z$ of \eqref{tbsd3}-\eqref{tbsd3b} satisfies
\begin{equation}
z\geq \tilde{z} \quad\mbox{in}\quad Q_0\equiv\Omega\times (0, \min\{ t_0, T_\delta\}),
 \label{eqn:6.35}
\end{equation}
where $\tilde z=\tilde z(x,t)$ solves the following
\bge
&&\tilde{z}_t=\Delta \tilde{z}+D\tilde{z}^p, \quad\mbox{in}\quad Q_0,\label{eqn:6.36}\\
&&\frac{\partial \tilde{z}}{\partial\nu}=0,\quad\mbox{on}\quad\partial\Omega\times(0, \min\{ t_0, T_\delta\}),\\
&&\tilde{z}(|x|,t)=u_0(\vert x\vert)\quad\mbox{in}\quad \Om.
 \label{eqn:6.36}
\ege
Let us now introduce
\[ h(x,t):=\tilde{z}_t(x,t)-\tilde{z}^p(x,t), \]
then due to (\ref{eqn:6.34}) and (\ref{eqn:6.36})  $h$ satisfies
\begin{eqnarray*}
h_t & = & \Delta h+p(p-1) \tilde{z}^{p-2} |\nabla \tilde{z}|^2+ D p \tilde{z}^{p-1}\,h \\
& \geq & \Delta h+ D p \tilde{z}^{p-1}\,h \quad\mbox{in}\quad Q_0,
\end{eqnarray*}
and
\bgee
h(x,0)= \Delta \tilde{z}(x,0)+D\tilde{z}^p(x,0)-\tilde{z}^p(x,0)=\Delta u_0+(D-1)u_0^p\geq u_0^p>0,\quad\mbox{in}\quad \Om
\egee
with boundary condition
\bgee
\frac{\partial h}{\partial \nu}=0\quad\mbox{on}\quad\partial\Omega\times(0, \min\{ t_0, T_\delta\}).
\egee
Then the maximum principle entails that $h>0$ in $Q_0$, that is,
\begin{equation}
\tilde{z}_t>\tilde{z}^p\quad\mbox{in}\quad Q_0.
 \label{eqn:6.37}
\end{equation}
Inequality (\ref{eqn:6.37}) implies
\[ \tilde{z}(0,t)\geq\left(\frac{1}{z_0^{p-1}(0)}-(p-1)t\right)^{-\frac{1}{p-1}}=\left\{\left(\frac{\de^{a}}{\la(1+\frac{a}{2})}\right)^{p-1}-(p-1)t\right\}^{-\frac{1}{p-1}}
\]
for $0<t<\min\{ t_0, T_\delta\}$, and therefore,
\begin{equation}
\min\{ t_0, T_\delta\}<\frac{1}{p-1}\cdot
\left( \frac{\delta^a}{\lambda(1+\frac{a}{2})}\right)^{p-1}.
 \label{eqn:6.38}
\end{equation}
For $0<\delta\ll 1$, the right-hand side on (\ref{eqn:6.38}) is less than $t_0$, and then $T_\delta<+\infty$ follows. Furthermore, by \eqref{eqn:6.38} $T_\de \to 0$ as $\de\to 0$ and the proof is complete.
\end{proof}
\begin{rem}
The blowing up solution $u$ obtained in Theorem \ref{thm:6.1} exhibits a single-point blow-up at the origin $\rho=0.$ Recalling that $z=e^tu$ we obtain the occurrence of single-point blow-up for $u$ in view of Remark \ref{nkl1}.

An alternative way to prove single-point blow-up is by virtue of the following estimate
\begin{equation}
\avint z^p\,dx=\frac{1}{|B_1(0)|}\int_0^1 \rho^{N-1} z^p\,d\rho \leq C,\quad\mbox{for}\quad 0<t\leq T_{\delta},
 \label{eqn:7.1}
\end{equation}
which holds due to \eqref{lue} and \eqref{eqn:6.18h}, taking $0<\delta\ll 1$ small enough such that $T_\delta\leq t_0.$
Then since $z=z(\rho, t)$ is radially decreasing, then (\ref{eqn:7.1}) implies that ${\mathcal S}=\{0\}.$
\end{rem}
\section{Blow-up rate and blow-up pattern}\label{sec:7}
One of our purposes in the current section is to determine the blow-up rate of the diffusion-driven blowing up solution provided by Theorem \ref{thm:6.1}. We also intend to identify its blow-up pattern (profile) and thus reveal the formed patterns anticipated in this {\it DDI} event.
\begin{theorem}\label{tbu}
Let $N\ge 3,\;\max\{r, \frac{N}{N-2}\}<p<\frac{N+2}{N-2}$ and $\frac{2}{N}<\frac{p-1}{r}<\gamma.$  Then the blow-up rate of the diffusion-induced blowing-up solution of Theorem \ref{thm:6.1} is determined as follows
\be
\Vert u(\cdot, t)\Vert_\infty \ \approx \ (T_{\max}-t)^{-\frac{1}{p-1}}, \quad t\uparrow T_\de,\label{ik2}
\ee
 where $T_{\max}$ stands for the blow-up time.
\end{theorem}
\begin{proof}
We first note that
\bge\label{nkl4}
0<K(t)=\frac{e^{(1+r\gamma-p)t}}{\Big(\displaystyle\avint z^r \Big)^{\gamma}}\leq C<\infty,
\ege
by virtue of \eqref{eqn:7.1} and in view of H\"{o}lder's inequality since $p>r.$

Consider now $\Phi$ satisfying
\bgee
&&\Phi_t=\Delta \Phi+C\Phi^{p},\quad\mbox{in}\quad \Om\times (0,T_{max}),\\
&&\frac{\partial \Phi}{\partial \nu}=0,\quad\mbox{on}\quad \partial\Om\times(0,T_{max}), \\
&&\Phi(x,0)=z_0(x), \quad\mbox{in}\quad \Om,
\egee
then via comparison $z\leq \Phi$ in $\Om\times (0,T_{max}).$

Yet it is known, see \cite[Theorem 44.6]{qs07}, that
\bgee
|\Phi(x,t)|\leq C_{\eta}|x|^{-\frac{2}{p-1}-\eta}\quad\mbox{for}\quad \eta>0,
\egee
when $x\in \Om,\;0<t<T_{max},$ and thus
\bge\label{tbsd18}
|z(x,t)|\leq C_{\eta}|x|^{-\frac{2}{p-1}-\eta}\quad\mbox{for}\quad (x,t)\in \Om\times (0, T_{max}),
\ege
which by virtue of \eqref{nkl4}, \eqref{tbsd18} and using also standard parabolic estimates entails that
\bge\label{nkl5}
z\in \mathcal{BUC^{\sigma}}\left(\left\{\rho_0<|x|<1-\rho_0\right\}\times \left(\frac{T_{max}}{2}, T_{max}\right)\right)
\ege
for some $\sigma\in(0,1)$ and each $0<\rho_0<1,$ where $\mathcal{BUC^{\sigma}}(M)$ denotes in general the Banach space of all bounded and uniform $\sigma-$H\"{o}lder continuous functions $h:M\subset\R^N\to \R;$ see also \cite{qs07}.

Consequently \eqref{nkl5} implies that  $\lim_{t\to T_{max}}z(x,t)$ exists and it is finite for all $x\in B_1(0)\setminus\{0\}.$

Recalling that $\displaystyle{\frac{2p}{p-1}}<N$ (or equivalently  $p>\displaystyle{\frac{N}{N-2}},\; N>2$ ) then by using \eqref{nkl4},\eqref{tbsd18} and in view of the dominated convergence theorem we derive
\bge\label{tbsd18a}
\lim_{t\to T_{max}} K(t)=\omega\in(0,+\infty).
\ege
Applying now Theorem 44.3(ii) in \cite{qs07}, taking also into account \eqref{tbsd18a}, we can find a constant $C_{u}>0$ such that
\bge\label{ube}
\left|\left|z(\cdot,t)\right|\right|_{\infty}\leq C_{u}\left(T_{max}-t\right)^{-\frac{1}{(p-1)}}\quad\mbox{in}\quad (0, T_{max}).
\ege
On the other hand, setting $N(t):=\left|\left|z(\cdot,t)\right|\right|_{\infty}=z(0,t)$ then $N(t)$ is differentiable for almost every $t\in(0,T_{\de}),$ in view of  \cite{fmc85}, and it also satisfies
\bgee
\frac{dN}{dt}\leq K(t) N^p(t).
\egee
Now since $K(t)\in C([0,T_{\max}))$ is bounded in any time interval $[0,t],\; t<T_{\max},$ then upon integration we obtain
\bge\label{lbe}
\left|\left|z(\cdot,t)\right|\right|_{\infty}\geq C_l\left(T_{\de}-t\right)^{-\frac{1}{(p-1)}}\quad\mbox{in}\quad (0, T_{\max}),
\ege
for some positive constant $C_l.$

Since $z(x,t)=e^t u(x,t) $ then by virtue of \eqref{ube} and \eqref{lbe} we obtain
\bgee
\widetilde{C}_l\left(T_{max}-t\right)^{-\frac{1}{(p-1)}}\leq\left|\left|u(\cdot,t)\right|\right|_{\infty}\leq \widetilde{C}_u\left(T_{max}-t\right)^{-\frac{1}{(p-1)}}\quad\mbox{for}\quad t\in(0, T_{max}),
\egee
where now $\widetilde{C}_l, \widetilde{C}_u$ depend on $T_{max},$
which actually leads to \eqref{ik2}.
\end{proof}
\begin{rem}\label{nkl6}
Condition \eqref{ik2} implies that the diffusion-induced blow-up of Theorem \ref{thm:6.1} is of type I, i.e. the blow-up mechanism is controlled by the ODE part of \eqref{bsd1}.

In contrast, for the finite-time blow-up furnished by Proposition \ref{lem:1.2} and Theorems \ref{thm:1.2} and \ref{obu1} we cannot derive a blow-up as in \eqref{ik2} since the blow-up of some $L^{\ell}-$norm, $\ell\geq 1,$ in each of these cases entails that \bgee
K(t)=\frac{e^{(1-p)t}}{\Big(\displaystyle\avint u^r \Big)^{\gamma}}\to 0\quad\mbox{as}\quad t\to T_{max},
\egee
and thus the approach of Theorem \ref{tbu} fails. This might be an indication that in the preceding cases finite-time blow-up is rather of type II.
\end{rem}
\begin{rem}\label{nkl7}
First observe that \eqref{tbsd18} provides a rough form of the blow-up pattern for $z$ and thus for $u$ as well. Nonetheless, due to \eqref{nkl4} then the non-local problem \eqref{tbsd3}-\eqref{tbsd3b} can be treated as the corresponding local one for which the following more accurate asymptotic blow-up profile, \cite{mz98}, is available
\bgee
\lim_{t\to T_{max}}z(|x|,t)\sim C\left[\frac{|\log |x||}{|x|^2}\right]\quad\mbox{for}\quad |x|\ll 1.
\egee
Therefore using again that $z=e^tu$ we derive a similar asymptotic blow-up profile for the driven-induced blowing up solution $u.$ This actually reveals the form of the developed patterns which are induced as a result of the {\it DDI} and will be numerically verified  in a forthcoming paper.
\end{rem}
\section{Conclusions}\label{conc}
The main purpose of the current manuscript is to unveil under which circumstances the dynamics of the interaction of the two morphogens (activator and inhibitor), described by the Gierer-Meinhardt system \eqref{gm1}-\eqref{gm4}, can be controlled by governing only the dynamics of the activator itself given by the non-local problem \eqref{bsd1}-\eqref{bsd3}. We derive some global-in-time existence as well as blow-up results both in finite and in infinite time for \eqref{bsd1}-\eqref{bsd3}. Global-in-time existence results guarantee the controlled growth of the activator as described by \eqref{bsd1}-\eqref{bsd3} whereas finite-time and infinite-time blow-up results are relevant with the activator's unlimited growth. We discovered, that there are cases, see Proposition \ref{ats1} and Theorems \ref{thm:1.2} and \ref{thm:1.2a}, where there is a serious discrepancy between the dynamics of the full system \eqref{gm1}-\eqref{gm4} and those of the non-local problem \eqref{bsd1}-\eqref{bsd3}. On the other hand, under other circumstances, see  Theorems \ref{thm:1.3}-\ref{thm:1.7}, then both  \eqref{gm1}-\eqref{gm4} and (\ref{bsd1})-(\ref{bsd3}) ensemble the same long-time dynamics.  In particular, in Theorems \ref{obu1} and \ref{thm:1.6} we show that the occurrence of some invariant regions for an associated dynamical system is vital in order to control the dynamics of the non-local problem \eqref{bsd1}-\eqref{bsd3} and thus the activator's growth. In addition,  we prove that under a Turing condition a {\it DDI} occurs, which is exhibited in the form of a driven-diffusion blow-up. The resulting destabilization enables the formation of some patterns, as anticipated in a Turing instability case. The form of the observed patterns is completely described via the study of the blow-up profile, see Remark \ref{nkl7}. Consequently, in that case the pattern formation for activator's concentration can be efficiently predicted and prescribed by the dynamics of the non-local problem \eqref{bsd1}-\eqref{bsd3}.
\section*{Acknowledgments}
This work was supported by JSPS Grant-in-Aid Scientific Research (A) 26247013 and Core-to-Core project. Part of the current work was inspired and initiated when the first author was visiting the Department of System Innovation of  Osaka University. He would like to express his gratitude for the warm hospitality.

The authors would also like to thank the anonymous reviewers for the their stimulating comments, which  substantially improved the form of the manuscript.

redpajama_set_name: RedPajamaArXiv


text:
The Health Law Blog
Board Certified Health Law Attorney
Expert Witness Availability
Available in the Following States
Certified Business Enterprise and SDVOB
ABIM Representation
Advance Directives and Living Wills
AHCA Background Screening
AHCA Disqualification Waivers
AHCA Employment Disqualifications
Allied World Assurance Company Cases
American Professional Agency Cases
Antitrust and Trade Regulation
Appellate Practice Including Appeals of ESOs, Appeals of Final Orders, Appeals of Revocations, Appeals of Veterans Cases
Assisted Living Facility Representation, Adult Day Care Centers, Continuing Care Retirement Communities, Group Homes
Attorney's Fee Expert
Baker Act Cases, Involuntary Confinement
Clinical Investigation Fraud Defense
Clinical Privileges, Fair Hearings, and Peer Review
Complex Litigation, Complex Business Litigation, Complex Commercial Litigation, Class Action Litigation, Multidistrict Litigation
Confidentiality of Medical Records
Corporations, LLC's, Partnerships and other Business Entities
Covenants-Not-To-Compete
Darwin National Insurance Company Cases
Dentists Advantage Insurance Company Cases
Disruptive Physician Representation
Emergency Hearing Representation
Employment, Discrimination, and EEOC Complaints
Healthcare Providers Service Organization (HPSO)
Home Health Agency Cases
Hospital Law
Independent Legal Counsel and Investigations
Informal Dispute Resolution (IDR)
Insurance Defense Attorney for Health Professionals
Intervention Project for Nurses (IPN)
Liberty Insurance Underwriters, Mercer Consumer Services, Mercer Health and Benefits Admin Attorney, Mercer Insurance Attorney Cases
Medicaid and Medicare Audits
Medicaid and Medicare Qui Tam or Whistleblower Cases and False Claims Act
Medicaid and Medicare Revocation
Medicaid Appeal Hearings
Medicaid Audit Defense
Medicaid Cost Reimbursement Report Denials
Medicaid Investigations
Medicaid Litigation
Medicare Appeals
Nondisciplinary Citations and Notices of Noncompliance from Florida Department of Health (DOH)
Nurses Service Organization (NSO) Insurance Company Cases
OneBeacon Insurance Company Attorney and One Beacon Insurance Company Cases
Pain Management and DEA Defense
Peer Review Proceedings
Pharmacy Law and DEA Hearings
Physician Employment Contracts
Physician Student Loan Defense
Podiatric Insurance Company of America Attorney and Podiatric Insurance Company of America Cases
Professionals Resource Network (PRN)
Psychologists, Licensed Mental Health Counselors (LMHCs), Clinical Social Workers (CSWs), and Licensed Marriage and Family Therapists (LMFT) Representation
Recovery Audit Contractor (RAC) Attorney
The Doctors Company (TDC) Insurance Cases
Trust Risk Management Services Cases
Volunteer Immunity and Good Samaritan Act
Health Law Questions and Answers
Health Law Articles and Documents
Nursing Law Manual
Medicare Part D: Measures Needed to Strengthen Program Integrity Testimony of Ann Maxwell Assistant Inspector General
On July 14, 2015, Ann Maxwell, Assistant Inspector General for Evaluation and Inspections of the Office of Inspector General (OIG), U.S. Department of Health and Human Services (HHS), gave testimony to Congress on the Medicare Part D Program. She presented before the United States House of Representatives Committee on Energy and Commerce and the Subcommittee on Oversight and Investigations. Her testimony focused on fraud, waste and abuse trends in the Medicare Part D Program and recommendations to address the underlying vulnerabilities. With more than 39 million Americans depending on the program for their prescription drugs costing over $120 billion a year, the OIG is taking aggressive measures to combat fraud.
The Medicare Part D Program is Vulnerable to Fraud.
In June of this year, the OIG deployed more than 300 special agents and law enforcement personnel to execute arrest and search warrants for Medicare fraud across the country. To date, it was the largest national health care fraud takedown. It resulted in more than 240 people being charged with defrauding the Medicare and Medicaid programs. Much of the fraud involved prescription drugs and those charged included doctors and pharmacy owners. Twenty-eight individuals from South Florida were charged with Part D fraud totaling more than $38 million in Medicare overpayments.
To read Marshall's testimony in full, click here.
Changes CMS Can Make to Protect the Integrity of the Medicare Part D Program.
Since Part D went into effect, the OIG has raised concerns about oversight and made a number of recommendations to the Centers for Medicare & Medicaid Services (CMS). The OIG recommendations center around two main themes:
(1) Leveraging Part D data to identify vulnerabilities.
(2) Employing additional tools to enhance the oversight of the Part D program.
CMS Should Take Action on the OIG's Recommendations.
CMS is responsible for improving the program's effectiveness and protecting its beneficiaries. To do so, CMS should take action on OIG's recommendations and employ all the tools at its disposal. Expect this to involve more audits, more action by ZPICs and MFCUs, and more health care subpoenas being issued. To learn more about Medicare Part D fraud, read one of our past blogs here.
Contact Health Law Attorneys Experienced in Handling Medicare and Medicaid Audits, Investigations and other Legal Proceedings.
Medicare and Medicaid fraud is a serious crime and is vigorously investigated by the state MFCU, the Agency for Health Care Administration (AHCA), the Zone Program Integrity Contractors (ZPICs), the FBI, the Office of Inspector General (OIG) of the U.S. Department of Health and Human Services (DHHS). Often other state and federal agencies, including the U.S. Postal Service (USPS), and other law enforcement agencies participate. Don't wait until it's too late. If you are concerned of any possible violations and would like a confidential consultation, contact a qualified health attorney familiar with medical billing and audits today. Often Medicare fraud criminal charges arise out of routine Medicaid audits, probe audits, or patient complaints.
The Health Law Firm's attorneys routinely represent physicians, dentists, orthodontists, medical groups, clinics, pharmacies, assisted living facilities (ALFs), home health care agencies, nursing homes, group homes and other healthcare providers in Medicaid and Medicare investigations, audits and recovery actions.
To contact The Health Law Firm please call (407) 331-6620 or visit our website at www.TheHealthLawFirm.com.
What do you think of these large crackdowns on Medicare Part D fraud? Do you think they work as a deterrent for others committing health care fraud? Please leave any thoughtful comments below.
KeyWords: Medicaid, Medicaid fraud, Medicare Part D program, Medicaid Program, Medicaid audit, Medicaid claims, fraudulent claims, overbill Medicaid, Office of Inspector General, OIG, Centers for Medicare & Medicaid Services, CMS, defense attorney, defense lawyer, Medicaid fraud defense attorney, Medicaid fraud defense lawyer, health law firm, The Health Law Firm reviews
"The Health Law Firm" is a registered fictitious business name of George F. Indest III, P.A. - The Health Law Firm, a Florida professional service corporation, since 1999.
Like this blog? Add your public comments:
Items in bold indicate required information.
Got a Letter From the United States Medical Licensing Examination (USMLE) Secretariat Accusing You of Irregular Behavior on the November 2019 Step 2 CS Exam? You Need Legal Help!
ADA Files Class Action Lawsuit Against Delta Dental For Antitrust Violations
How Can I Be Sure I Do Not Have Any Employees or Contractors Who Are on Medicare's Exclusion List?
Hospital Countersues FCA Whistleblower for Failing to Report Information to Company First
Looking Back at 2019's Largest Healthcare Fraud Case Ever Prosecuted: Surprise! No Surprise, Florida Wins!
OIG Exclusion or Medicare, Medicaid Termination Can Ruin Your Professional Career
Lasting Effects of Voluntary Relinquishment, Investigation and Discipline on Your Medical License
Many Physicians Lose "Board Eligible" Credentials with ABIM Because of Application Time Requirements
Virginia Doctor Sentenced to 40 Years in Prison For Illegally Prescribing More Than Half a Million Opioid Pills
Psychiatric Hospital in Florida Accused of Cashing in on it's Baker Act Patients
The Health Law Blog (917)
Available in Following States
Health Law Q & A
Articles & Documents to Read & Download
Certified Business Enterprise & SDVOB
Seven Things To Know When You Receive A Notice Of Investigation From The Department Of Health
About Our Firm|Areas of Practice|Resources| Contact Us|Our Blogs
|Site Map|Home
The Health Law Firm is certified as a Service-Disabled Veteran Owned (SDVO) Business and a Certified Business Enterprise (CBE)
by the Florida Office of Supplier Diversity (OSD).
Main Office • 1101 Douglas Ave., Suite 1000, Altamonte Springs FL, 32714
By Appointment • 37 N. Orange Ave., Suite 500, Orlando, FL 32801
By Appointment • 201 E. Government St., Pensacola, FL 32502 •
Telephone: (407) 331-6620 &
• Telefax: (407) 331-3030
By Appointment • 155 E. Boardwalk Drive, Suite 424, Fort Collins, CO 80525 •
Medicare/Medicaid Audits, Health Care Law, Contracts, Hospital Privileges Hearings, Investigations, DEA Defense, Board of Medicine Defense, Fraud Defense, Administrative Hearings, PRN, IPN, Professional Licensing, Medicare/Medicaid Fraud Defense, Nursing Law, Compliance Plans, Hospital Law, Board of Dentistry, Board of Nursing, Board of Pharmacy, Board of Psychology, White Collar Crime, Pain Management and Pain Medicine Physician Defense, Pain Management Clinic Defense, Zone Program Integrity Contractor (ZPIC) Audit Defense, Recovery Audit Contractor (RAC) Audit Defense, Medicaid Fraud Control Unit (MFCU) Defense, Search Warrant and Subpoena Defense, Board of Psychology Defense, NBME Representation, U.S.M.L.E. Challenges, ABIM Representation, Medical Exam Cheating Defense...and more
Available in the following Florida cities and counties: Daytona Beach, Fort Lauderdale, Gainesville, Jacksonville, Key West, Melbourne, Miami, Ocala, Orlando, Pensacola, Panama City, Sarasota, St. Petersburg, Tallahassee, Tampa, West Palm Beach, Alachua, Baker, Bay, Bradford, Brevard, Broward, Calhoun, Charlotte, Citrus, Clay, Collier, Columbia, Dade, De Soto, Dixie, Duval, Escambia, Flagler, Franklin, Gadsden, Gilchrist, Glades, Gulf, Hamilton, Hardee, Hendry, Hernando, Highlands, Hillsborough, Holmes, Indian River, Jackson, Jefferson, Lafayette, Lake, Lee, Leon, Levy, Liberty, Madison, Manatee, Marion, Martin, Monroe, Nassau, Okaloosa, Okeechobee, Orange, Osceloa, Palm Beach, Pasco, Pinellas, Polk, Putnam, St. Johns, St. Lucie, Santa Rosa, Sarasota, Seminole, Sumter, Suwannee, Taylor, Union, Volusia, Wakulla, Walton, and Washington
By making this website information available for those who access it does not constitute doing business in or having a presence in any state or jurisdiction, nor does it constitute an advertisement sent to or a solicitation made in any state or jurisdiction. This firm is located in and maintains a presence in only those states where the firm maintains an actual physical office. Its attorneys are only admitted to practice in those states specifically listed on their resumes.
Available in the following states*: Alabama, Alaska, Arizona, Arkansas, Colorado, Connecticut, Delaware, Florida, Georgia, Hawaii, Idaho, Illinois, Indiana, Iowa, Kansas, Kentucky, Louisiana, Maine, Maryland, Massachusetts, Michigan, Minnesota, Mississippi, Missouri, Montana, Nebraska, Nevada, New Hampshire, New Jersey, New Mexico, New York, North Carolina, North Dakota, Ohio, Oklahoma, Oregon, Pennsylvania, Rhode Island, South Carolina, South Dakota, Tennessee, Texas, Utah, Vermont, Virginia, Washington, West Virginia, Wisconsin, and Wyoming
Disclaimer|Terms of Representation
The Health Law Firm logo shown above is a registered Service Mark of George F. Indest III, P.A. - The Health Law Firm, since 2007.
"The Health Law Firm" is a registered fictitious business name of and a registered service mark of The Health Law Firm, P.A., a Florida professional service corporation, since 1999. Copyright © 2020 The Health Law Firm. All rights reserved.
Content Management System by Sales & Marketing Technologies

redpajama_set_name: RedPajamaCommonCrawl


The 2018 Music & Arts Black Friday Ad is here with 4 pages. There are coupons on the front page; their deals will begin on 11/16 and will run to 11/25.
Music & Arts specializes in instrument rentals, music lessons, and band and orchestra instrument sales for students, teachers, and families. Offering products and resources for the beginning and student musician, Music & Arts also has a tab for Hot Deals on instruments and accessories.
Note: These are the pages from the 2017 Music & Arts Black Friday advertisement. See this year's ad.
You've reached the archived page of the Music & Arts Black Friday Ad from 2017. Click below to see the current Music & Arts Black Friday 2019 ad.
Sign up for the BFAds mailing list and be the first to know when the ads leak for Music & Arts.


Our glaze packs are perfect for classrooms and studios and come in a great value! Each pack can contain up to 20 different colors giving you the ability to mix and match and create whatever color and effect you please. Choose from a variety of sets below and let your imagination take control.


"close friend, roommate," mid-15c., from bed (n.) + fellow (n.). Also (late 15c) "concubine." Earlier in the "close companion" sense was bed-fere (early 14c.). Bedsister "husband's concubine" is recorded in Middle English (c.1300).
My cousin Jenny Fynnett is here, and desires to be my bedfellow to-night.
We were afraid of awakening her bedfellow, and kept perfect silence.
I had exchanged my brother John as a bedfellow for Walter Packard.
At length the elder brother wearied of this diversion and aroused his bedfellow.
The next moment his bedfellow was "covered" with two "guns."
He was my companion and playmate during the day, and my bedfellow at night.
And you have the lively picture of it, every night for your bedfellow.
He assigned me to "bunk" with the only boy in the room who had no bedfellow or "bunkie."
Older; it has a white head, and shall never die till she be buried: her wrongs shall be my bedfellow.
He wriggled from his bedfellow; she put an arm about him, drew him to her.


text:
Princeton, KY (42445)
Randy Martin lives midway between Princeton and Dawson Springs on U.S. 62. Trees in his yard were damaged, with some being uprooted and others broken off on the trunk, and his shop was also damaged.
DAVID B. SNOW | The Sun
This tree was snapped and fell across Levin Dunning Road along U.S. 62 between Princeton and Dawson Springs. Emergency crews worked through Friday night to clear trees from roads.
top story editor's pick
Severe tornado damage in Caldwell
BY DAVID B. SNOW dsnow@paducahsun.com
The tornado that first touched down near Jonesboro, Arkansas, Friday night and wound its way into Kentucky for 227 miles brought damage and harm to much of western Kentucky.
Likely the longest track of any tornado in U.S. history, the twister paralleled Interstate 69 from Mayfield into Hopkins County, bringing destruction to the cities of Mayfield and Benton as well as to areas of Caldwell County.
Caldwell County Emergency Medical Services Director Mark Harrison said Saturday morning that Dawson Road – U.S. 62 between Princeton and Dawson Springs – was likely the hardest-hit area in the county.
"The fire department's been out cutting trees all night," he said. "They've been cutting trees and dragging limbs (off roads) and trying to open up holes (so cars can drive through) all night long.
"We had people out here before the storm. We were staged. We knew it was coming."
The tornado went up Ky. 293 from near Interstate 24 in Lyon County to Princeton, where it crossed over U.S. 62 into Hopkins County.
"We had seven EMS crews on," Harrison said. "One was from Trigg County to help us. We spent the night helping to find people, treat minor (injuries) and transported the serious ones.
"…But that was the concentrated area: Dawson Road, all the way out."
Harrison said most of the injuries were caused by blunt trauma from debris.
He said the tornado came into Princeton around 7:30 p.m. to 7:45 p.m. Friday.
"We had some EMS employees that had damage to their homes," Harrison said. "One employee's house (in Dawson Springs) was just gone. He was on duty all night. His wife and daughter rode it out in the bathroom, and they made it out."
Death and destruction could also be seen in rural areas of western Kentucky. Randy Martin lives on U.S. 62 between Princeton and Dawson Springs and was a witness to the damage the storm brought. The tornado blew through his neighborhood, uprooting trees and littering roads with toppled trees and debris.
"We came home at 3 (Saturday) morning, and we usually just go up the steps," he said, pointing to his back porch. "At first, I thought (the storm) blew the steps, but no, it blew the porch."
The porch dug a small trench as it was dragged two feet, misaligning the steps from the porch.
"There is a bridge right off that hill," Martin said, pointing to the distance. "It was just covered in trees. They thought the bridge fell through, it was so covered in them."
Several trees in Martin's yard were uprooted or snapped off in the middle, and his house took a 3-foot-long gash in the siding from debris during Friday's storm.
"There was a shop (across the street)," he said. "It's gone."
prn1.jpg
prn10.jpg
GALLERY: Caldwell County tornado damage
Metal pieces from the shop could be found across the road from its location, with one piece hanging in the branches of a tree.
Martin said one of his neighbors found an empty box with a Mayfield address on it, indicating the twister carried it more than 60 miles.
Martin said he has heard of tragedy in his area.
"You see that debris up there that's real high?" he asked, pointing across the highway. "That's a brand-new double-wide trailer. It's still sitting on its blocks – never got underpinned. The house next to it, where they lived before – (the tornado) never touched it, but it picked the double-wide up and dumped it on its roof."
Martin added that someone who lived up the road from him died during the storm Friday night.
Follow David B. Snow on Twitter, @SunWithSnow, or on Facebook at facebook.com/sunwithsnow.
Tweets by @TheTimesLeader1
timesleader.net
323 D South Jefferson St.
Princeton, KY 42445
Email: newsroom@timesleader.net
© Copyright 2022 The Times Leader, 323 D South Jefferson St. Princeton, KY | Terms of Use | Privacy Policy


IFTM Top Resa in Paris is a globally renowned trade fair for the tourism and travel industry. The event focuses on the wealth of travel destinations in France and around the world, demonstrates emerging technological advances, awards innovations and achievements in the sector. Among the professionals attending IFTM Top Resa 2019 you will meet tour operators and travel agencies, coach operators, airlines and event agencies, just to name a few. This is the place to stay on track with the new travel trends.
With close to 35,000 visitors searching for accommodation during the IFTM Top Resa 2019 dates - 1 to 4 October - our team recommends early planning. We have a wide network of partner hotels in Paris, and are able to deliver flexible conditions and special room rates at hotels near Paris expo Porte de Versailles. Book your hotel for IFTM Top Resa 2019 Paris today. Contact ExpoBeds and we will promptly send you a list of IFTM hotels, all chosen to fit your requirements.


The ARCOS 1.5 is a highly modular data logger that can be flexibly adapted to various measurement tasks. The basic unit is available in 5 different versions, the CPUs (i3, i5) and RAM and the mSATA and cFAST data storage media can also be configured.
The large number of interfaces for CAN, LIN, ETH and FlexRay measurement and digital I/Os allows the number of inputs to be flexibly scaled. The number of analog inputs for thermocouples, voltage, DIO can be extended to several hundred channels via the combo boxes.
The logger configuration is supported the IPEmotion CAETEC datalog PlugIn which is available for download here.
Our Support team will be glad to answer your questions.


text:
PMC Reflects on Thirty Years Producing Loudspeakers
Posted on 16th February, 2021
PMC is celebrating its 30th anniversary this year, marking three decades of 'Best of British' loudspeaker design and manufacture covering domestic and professional markets.
PMC is a familiar name to both audiophiles and those who frequent professional sound studios. The Professional Monitor Company was established in 1991 to fulfil the BBC's requirements for a monitor speaker that could handle pop and rock levels but with ultra-high resolution by Peter Thomas, then a BBC engineer, and FWO Bauch engineer Adrian Loader. The first loudspeaker was created in Peter's garage with electronics coming from Adrian's garage and dubbed Big Box.
This initial creation was refined, renamed BB5 and installed into the BBC Maida Vale Studios. Soon after the BB5 and official formation of PMC, Peter and Adrian launched their first domestic loudspeaker, the LB1 and the rest, as they say, is history.
Peter Thomas, PMC founder and owner, told StereoNET:
When I started the company with my friend and colleague, Adrian Loader, I had no idea that our audio knowledge and passion for music would one day end up with our small company designing and building universally respected loudspeakers for audiophiles and for some of the biggest artists, producers and audio facilities in the world.
Fast-forward 30 years and the company has an enviable reputation in the home audio and custom installation sectors, with products that promise to bring studio-quality audio reproduction into the home.
Peter Thomas adds:
In terms of music reproduction, there shouldn't be a distinction between a loudspeaker designed for studio use and one for domestic use - the speaker is either reproducing the music correctly, or it isn't.
PMC's core technologies, shared between pro and domestic, include the Advanced Transmission Line (ATL) bass loading, the aerodynamically designed Laminair vent, dome midrange drivers and tweeter dispersion grilles.
You'll find these innovations featured on PMC's QB1 monitors used in studios such as Capitol in LA and 301 in Sydney, as well as the recently launched fact fenestria and twenty5i Series hi-fi speakers. Again, the same is true of the high-performance Ci Series which are not only finding fans building home cinemas but studios, too.
Dolby Atmos has also been an area where PMC has been leading the field. The British company's partnership with Dolby has resulted in PMC loudspeakers being installed in Dolby Atmos music mixing rooms around the world and becoming the reference for Atmos music mixing. Additionally, PMC has also engineered Atmos mixes for artists such as Miles Davis and Gordon Goodwin's Big Phat Band and built its own Atmos-enabled mix facilities in Nashville, New York and Los Angeles.
Peter Thomas reflects on the past 30 years:
Celebrating a 30th anniversary is a major milestone for any company, and we know that we couldn't have done any of this without the help of our staff and customers. Their support and dedication have enabled us to build a highly respected global brand. For that, we are extremely grateful and would like to say a massive and heartfelt thank you as we look forward to the next chapter in the PMC story.
Since the company was founded, its workforce has gone from just two to 60 worldwide. With its head office in Biggleswade, UK and satellite offices in Los Angeles, Nashville and New York, and two manufacturing plants in the UK, PMC can efficiently service its global customer base.
We at StereoNET raise a glass to all the team at PMC and look forward to the next 30 years of this great British brand.
Visit PMC Speakers for more information
Posted in:Hi-Fi Industry
Tags: interdyn pmc speakers


We all are first cousins and had always wanted to do music together. I think after Shane (Tyler's brother) passed it really opened our eyes that none of us really have a lot of time here so let's do what we love while we can.
Usually we all get in a room, Landon or Brandon hop on a kit and the others get guitars or bass and we start jamming as a three piece. Sometimes we write all the music and then focus on lyrics melody. Other times one of us might present a more complete idea and the others chime in and make it a Goodbye June song.
Beginning to think about LP2 but really focusing on touring to support Magic Valley.
Tyler – Gibson 335 –> Pedals —> Bad Cat Black Cat 30R.
Bluesy, gritty, Rock and roll, no frills, no schticks.
Limited, mostly everything by ear.
We got into music at a pretty early age because most of our family was involved in the church somehow. Be it a music minister, pastor, or whatever we were playing music in church at an early age. It was a sin to listen to rock music (Tyler's parents were a little more lenient) but we would sneak off in our rooms and listen to bands like Bush, Metallica, Nirvana.
Black Rebel Motorcycle Club, Brandon Flowers, Sturgill Simpson, Brent Cobb.
Zeppelin, Creedence, Leon Russell, Hank Williams Jr, Kings Of Leon.
We are currently on a 4 week tour with Badflower across the west coast, Midwest and the South.
UK-BASED PRODUCER COLDABANK DROPS INFECTIOUS SUMMER HOUSE BANGER 'LOVIN' YOU' OUT NOW VIA BIG BEAT RECORDS!


text:
Pristimantis anotis est une espèce d'amphibiens de la famille des Craugastoridae.

Répartition
Cette espèce est endémique de l'État d'Aragua au Venezuela. Elle se rencontre à Rancho Grande à environ  d'altitude dans la cordillère de la Costa.

Description
Les femelles mesurent jusqu'à .

Publication originale
 Walker & Test, 1955 : New Venezuelan frogs of the genus, Eleutherodactylus. Occasional Papers of the Museum of Zoology University of Michigan, ,  (texte intégral).

Liens externes

Notes et références

Anoure (nom scientifique)
Ceuthomantinae
Faune endémique de l'État d'Aragua

redpajama_set_name: RedPajamaWikipedia


At Barbie today is notable reception . In her huge house was a place to celebrate the last school dance. So the girl so carefully prepared . After all, everyone will see what she is mistress . To Barbie did not get trapped, help her put all things beautiful and in its place. Under the feet of piles of garbage , because Barbie have not held a broom in his hand.