text:
\section{Introduction}
This paper is concerned with
nonnegative solutions of nonlinear parabolic systems.
Nonlinear parabolic systems have been studied intensively for more than 20 years.
See \cite{AHV}--\cite{CF}, \cite{DL1}--\cite{FM}, \cite{INS}, \cite{LS}--\cite{QS}, \cite{R, S, Z} and references therein
(see \cite{QSBook} for a survey).
Their analysis is generally more complicated than
that of scalar nonlinear parabolic equations.
In this paper, by using scalar nonlinear parabolic equations,
we construct supersolutions of
\begin{equation}
\label{eq:1.1}
\left\{
\begin{array}{ll}
\partial_t u=\Delta u+v^p,\qquad & x\in\Omega,\,\,\,t>0,\vspace{5pt}\\
\partial_t v=\Delta v+u^q, & x\in\Omega,\,\,\,t>0,\vspace{5pt}\\
u=v=0, & x\in\partial\Omega,\,\,\,t>0,\vspace{5pt}\\
(u(x,0), v(x,0))=(u_0(x),v_0(x)), & x\in\Omega,
\end{array}
\right.
\end{equation}
where $p\ge 0$, $q\ge 0$, $\Omega$ is a (possibly unbounded) smooth domain
in ${\bf R}^N$ $(N\ge 1)$
and both $u_0$ and $v_0$ are nonnegative and locally integrable functions in $\Omega$.
These supersolutions enable us to obtain optimal sufficient conditions for the existence of
local-in-time solutions and global-in-time solutions.
Our arguments are simple and applicable to various nonlinear parabolic systems
without complicated calculations due to combinations of power and exponential nonlinearities.
\vspace{5pt}

Problem~\eqref{eq:1.1} is an example of  a simple reaction-diffusion system
that can be used as a model to describe heat propagation in a two component combustible mixture.
There are several results on the existence of solutions of \eqref{eq:1.1}.
Here we recall the following well-known results,
which were proved in \cite{AHV, EH01, EH02, QS}
(see also \cite[Section~32]{QSBook}).
\begin{itemize}
  \item[{\rm (A)}]
  Let $p,q\ge1$ and $r_1,r_2\in(1,\infty)$.
  Assume
  $$
  \max\{P(r_1,r_2),Q(r_1,r_2)\}\le 2,
  $$
  where
  $$
  P(r_1,r_2):=N\left(\frac{p}{r_2}-\frac{1}{r_1}\right),
  \qquad
  Q(r_1,r_2):=N\left(\frac{q}{r_1}-\frac{1}{r_2}\right).
 $$
  Then, for any $(u_0,v_0)\in L^{r_1}(\Omega)\times L^{r_2}(\Omega)$,
  problem~\eqref{eq:1.1} possesses a local-in-time solution.
  \item[{\rm (B)}]
  Let $0<pq\le 1$.
  Then problem~\eqref{eq:1.1} possesses a global-in-time solution
  for any $(u_0,v_0)\in L^\infty(\Omega)\times L^\infty(\Omega)$.
  \item[{\rm (C)}]
  Let $\Omega={\bf R}^N$ and $pq>1$.
  If
  \begin{equation}
  \label{eq:1.3}
  \frac{\max\{p,q\}+1}{pq-1}<\frac{N}{2},
  \end{equation}
  then problem~\eqref{eq:1.1} possesses a global-in-time positive solution provided that
  $(u_0,v_0)\not\equiv(0,0)$ and
  both $\|u_0\|_{L^{r_1^*}(\Omega)}$ and $\|v_0\|_{L^{r_2^*}(\Omega)}$ are sufficiently small,
  where
  \begin{equation}
  \label{eq:1.4}
  r_1^*:=\frac{N}{2}\frac{pq-1}{p+1},
  \qquad
  r_2^*:=\frac{N}{2}\frac{pq-1}{q+1}.
  \end{equation}
  \item[{\rm (D)}]
  Let $\Omega={\bf R}^N$ and $pq>1$.
  If $(u_0,v_0)\not\equiv(0,0)$ and
  $$
  \frac{\max\{p,q\}+1}{pq-1}\ge \frac{N}{2},
  $$
  then problem~\eqref{eq:1.1} admits no global-in-time positive solution.
\end{itemize}
The optimality of assumption~\eqref{eq:1.3} in (C)
follows from (D).

In this paper we aim to construct supersolutions
for a class of nonlinear parabolic systems including \eqref{eq:1.1}
and to use them to deduce sufficient conditions for the existence of local-in-time solutions.
In particular, our results for problem~\eqref{eq:1.1} stated in Section~3 improve on (A)--(C).
Furthermore, as an application of these sufficient conditions,
we obtain lower estimates on the blow-up rate for the solutions of \eqref{eq:1.1}.
In Subsection~3.2 and Section~4 we address some possible generalizations to other nonlinear parabolic systems.

Let us now outline the construction of supersolutions.
Given $(u,v)$ a positive (classical) solution of \eqref{eq:1.1},
we begin by setting $U:=u^\alpha$ and $V:=v^\beta$, where $\alpha\ge 1$ and $\beta\ge 1$.
Then $(U,V)$ satisfies
\begin{equation}
\label{eq:1.5}
\left\{
\begin{array}{ll}
\partial_t U=\Delta U+\alpha U^{1-\frac{1}{\alpha}}V^{\frac{p}{\beta}}
-\displaystyle{\frac{\alpha-1}{\alpha}\frac{|\nabla U|^2}{U}}, &x\in\Omega,\,\,t>0,\vspace{7pt}\\
\partial_t V=\Delta V+\beta V^{1-\frac{1}{\beta}}U^{\frac{q}{\alpha}}
-\displaystyle{\frac{\beta-1}{\beta}\frac{|\nabla V|^2}{V}}, &x\in\Omega,\,\,t>0,\vspace{5pt}\\
U=V=0, &x\in\partial\Omega,\,\,t>0,\vspace{3pt}\\
(U(x,0),V(x,0))=(u_0(x)^\alpha,v_0(x)^\beta),\quad &x\in\Omega.
\end{array}
\right.
\end{equation}
Let $(\tilde{U},\tilde{V})$ be a positive solution of
\begin{equation}
\label{eq:1.6}
\left\{
\begin{array}{ll}
\partial_t \tilde{U}
=\Delta \tilde{U}+\alpha \tilde{U}^{1-\frac{1}{\alpha}}\tilde{V}^{\frac{p}{\beta}},
\qquad & x\in\Omega,\,\,t>0,\vspace{5pt}\\
\partial_t\tilde{V}=\Delta\tilde{V}+\beta\tilde{V}^{1-\frac{1}{\beta}}\tilde{U}^{\frac{q}{\alpha}},
& x\in\Omega,\,\,t>0,\vspace{5pt}\\
\tilde{U}=\tilde{V}=0, & x\in\partial\Omega,\,\,t>0,\vspace{5pt}\\
(\tilde{U}(x,0),\tilde{V}(x,0))=(u_0(x)^\alpha,v_0(x)^\beta),\quad &x\in\Omega.
\end{array}
\right.
\end{equation}
Since $\alpha\ge 1$ and $\beta\ge 1$,
by \eqref{eq:1.5} and \eqref{eq:1.6}
we see that $(U,V)$ is a subsolution of \eqref{eq:1.6}.
It follows from the comparison principle that
$$
\tilde{U}(x,t)\ge u(x,t)^\alpha,
\qquad
\tilde{V}(x,t)\ge v(x,t)^\beta,
\qquad
x\in\Omega,\,\,\,t>0.
$$
Let $\overline{w}$ be a solution of
\begin{equation}
\label{eq:1.7}
\left\{
\begin{array}{ll}
\partial_t w=\Delta w+\alpha w^A+\beta w^B,\qquad & x\in\Omega,\,\,t>0,\vspace{5pt}\\
w=0, & x\in\partial\Omega,\,\,t>0,\vspace{5pt}\\
w(x,0)=u_0(x)^\alpha+v_0(x)^\beta,& x\in\Omega,
\end{array}
\right.
\end{equation}
where
$$
A:=1-\frac{1}{\alpha}+\frac{p}{\beta},
\qquad
B:=1-\frac{1}{\beta}+\frac{q}{\alpha}.
$$
Then $(\overline{w},\overline{w})$ is a supersolution of \eqref{eq:1.6}.
This implies that
$(\overline{w}^{\frac{1}{\alpha}},\overline{w}^{\frac{1}{\beta}})$ is a supersolution of \eqref{eq:1.1}.
Therefore, by the comparison principle we obtain
\begin{equation}
\label{eq:1.8}
0\le u(x,t)^\alpha\le \overline{w}(x,t),
\qquad
0\le v(x,t)^\beta\le \overline{w}(x,t),
\qquad
x\in\Omega,\,\,\,t>0.
\end{equation}
This supersolution with a suitable choice of $\alpha$ and $\beta$
enables us to obtain sufficient conditions
for the existence of local-in-time solutions and global-in-time solutions
of problem~\eqref{eq:1.1}.
Compared with the results in \cite{EH01, EH02, QS},
we see that our sufficient conditions are optimal.
By similar arguments
we can construct supersolutions of various nonlinear parabolic systems systematically
and give sufficient conditions for the existence of solutions.
See Subsection~3.2 and Section~4.
\vspace{3pt}

The rest of this paper is organized as follows.
In Section~2 we introduce notation and
prove some lemmas on the existence of the solutions of \eqref{eq:1.1}
and a scalar nonlinear parabolic equation.
In Section~3
we prove two theorems on the existence of the solutions of \eqref{eq:1.1}
that improve on (A)--(C).
Furthermore, we give lower estimates on the blow-up rate for the solutions.
In Section~4 we apply our techniques to parabolic systems with strongly coupled nonlinearities.
\section{Preliminaries}
We introduce some notation and define the notion of solution for \eqref{eq:1.1}.
Furthermore, we prove some preliminary lemmas.
In what follows, $C$ denotes a generic constant.
\vspace{3pt}

Let $1\le r\le\infty$.
We define the spaces $L^{r,\infty}(\Omega)$ and $L^{r,\infty}_{{\rm uloc}}(\Omega)$.
Let $f$ be a measurable function in a smooth domain $\Omega\subset{\bf R}^N$.
Setting $f=0$ outside of $\Omega$, we define
$$
\mu(\lambda):=\left|\{x\in{\bf R}^N\,:\,|f(x)|>\lambda\}\right|,
\qquad \lambda\ge 0,
$$
i.e. the distribution function of $f$.
Furthermore, we define
the non-increasing rearrangement of $f$ by
$$
f^*(s):=\inf\{\lambda>0\,:\,\mu(\lambda)\le s\}.
$$
Then the spherical rearrangement of $f$ is defined by
$$
f^\sharp(x):=f^*(c_N|x|^N),
$$
where $c_N$ is the volume of the unit ball in ${\bf R}^N$.
We define
$$
L^{r,\infty}(\Omega)
:=\left\{f\,:\, \mbox{$f$ is measurable in $\Omega$},\,\,\, \|f\|_{L^{r,\infty}(\Omega)}<\infty\right\},
$$
where
$$
\|f\|_{L^{r,\infty}(\Omega)}:=\sup_{s>0}\,s^{1/r}f^{**}(s),
\qquad
f^{**}(s):=\frac{1}{s}\int_0^s f^*(r)\,dr.
$$
Then the following holds (see e.g., \cite[Section~1.1]{G}).
\begin{itemize}
  \item Let $1<r<\infty$. Then $f\in L^{r,\infty}(\Omega)$ if and only if
  \begin{equation}
  \label{eq:2.1}
  0\le f^\sharp(x)\le C|x|^{-N/r},\qquad x\in{\bf R}^N.
  \end{equation}
  \item $L^r(\Omega)\subset L^{r,\infty}(\Omega)$ and
  $L^r(\Omega)\not=L^{r,\infty}(\Omega)$ if $1<r<\infty$
  and $L^{r,\infty}(\Omega)=L^r(\Omega)$ if $r\in\{1,\infty\}$.
  Furthermore,
  $$
  \|f\|_{L^{r,\infty}(\Omega)}\le\|f\|_{L^r(\Omega)}
  $$
  for any $f\in L^r(\Omega)$, where $1\le r\le\infty$.
  \item Let $1\le r\le\infty$ and let $\{r_j\}_{j=1}^k\subset[1,\infty]$ be such that
  $$
  \frac{1}{r}=\frac{1}{r_1}+\cdots+\frac{1}{r_k}.
  $$
  Then
  \begin{equation}
  \label{eq:2.2}
  \biggr\|\prod_{j=1}^kf_j\biggr\|_{L^{r,\infty}(\Omega)}\le C\prod_{j=1}^k\|f_j\|_{L^{r_j,\infty}(\Omega)}
  \end{equation}
  for $f_j\in L^{r_j,\infty}(\Omega)$ and $j=1,2,\dots,k$.
\end{itemize}
For any $x\in{\bf R}^N$ and $R>0$, we put $B(x,R):=\{y\in{\bf R}^N: |x-y|<R\}$.
We define
\begin{equation*}
\begin{split}
L^{r,\infty}_{{\rm uloc}}(\Omega) & :=\left\{f\in L^r_{{\rm loc}}(\Omega)\,:\,
\sup_{x\in\overline{\Omega}}\,\|f\|_{L^{r,\infty}(\Omega\,\cap\,B(x,1))}<\infty\right\}
\quad\mbox{if}\quad 1\le r<\infty,\\
L^{r,\infty}_{{\rm uloc}}(\Omega) & :=L^\infty(\Omega)\quad\mbox{if}\quad r=\infty.
\end{split}
\end{equation*}
For any $\rho>0$, we set
$$
|||f|||_{r,\rho}:=
\left\{
\begin{array}{ll}
\displaystyle{\sup_{x\in\overline{\Omega}}}\,\,\|f\|_{L^{r,\infty}(\Omega\,\cap\,B(x,\rho))}
 & \quad\mbox{if}\quad 1\le r<\infty,\vspace{5pt}\\
\|f\|_{L^\infty(\Omega)}
 & \quad\mbox{if}\quad r=\infty,
\end{array}
\right.
$$
which are equivalent norms of $L^{r,\infty}_{{\rm uloc}}(\Omega)$.

We denote by $S(t)$ the Dirichlet heat semigroup on $\Omega$.
Then, for any $\phi\in L^r(\Omega)$ $(r\ge 1)$,
$v(t):=S(t)\phi$ represents the unique bounded solution of
$$
\partial_t v=\Delta v\quad\mbox{in}\quad\Omega\times(0,\infty),
\quad
v=0\quad\mbox{on}\quad\partial\Omega\times(0,\infty),
\quad
v(x,0)=\phi(x)\quad\mbox{in}\quad\Omega.
$$
We first show the following.
\begin{lemma}
\label{Lemma:2.1}
There exists a positive constant $c_*>0$ such that
\begin{equation}
\label{eq:2.3}
\|S(t)\varphi\|_{L^\infty(\Omega)}\le c_*t^{-\frac{N}{2r}}|||\varphi|||_{r,\rho},
\qquad
0<t\le\rho^2,
\end{equation}
for any $\varphi\in L^{r,\infty}_{{\rm uloc}}(\Omega)$,
where $1\le r\le\infty$.
\end{lemma}
{\bf Proof.}
Let $\{z_j\}\subset{\bf R}^N$ be such that
\begin{equation}
\label{eq:2.4}
{\bf R}^N=\bigcup_{j=1}^\infty B(z_j,1),
\qquad
\sum_{j=1}^\infty e^{-\frac{|z_j|^2}{8}}<\infty.
\end{equation}
Set $u(x,t):=S(t)\varphi$ and $u_\lambda(x,t):=u(\lambda x,\lambda^2 t)$
for $\lambda>0$. Let
$$
\Gamma(x,y,t):=(4\pi t)^{-\frac{N}{2}}\exp\left(-\frac{|x-y|^2}{4t}\right).
$$
It follows from the comparison principle and \eqref{eq:2.4} that
$$
|u_\lambda(x,1)|\le\int_{{\bf R}^N}\Gamma(x,y,1)|u_\lambda(y,0)|\,dy
\le\sum_{j=1}^\infty\int_{B(x+z_j,1)}\Gamma(x,y,1)|u_\lambda(y,0)|\,dy
$$
for $x\in\lambda^{-1}\Omega$.
This together with \eqref{eq:2.2} and \eqref{eq:2.4} implies that
\begin{equation*}
\begin{split}
|u(\lambda x,\lambda^2)|=|u_\lambda(x,1)|
 & \le\sum_{j=1}^\infty
\|\Gamma(x,\cdot,1)\|_{L^{r',\infty}(B(x+z_j,1))}\|u_\lambda(0)\|_{L^{r,\infty}(B(x+z_j,1)}\\
 & \le C\sum_{j=1}^\infty\|\Gamma(x,\cdot,1)\|_{L^\infty(B(x+z_j,1))}
 |||u_\lambda(0)|||_{r,1}\\
 & \le C\sum_{j=1}^\infty e^{-\frac{|z_j|^2}{8}}|||u_\lambda(0)|||_{r,1}
 \le C |||u_\lambda(0)|||_{r,1},
\end{split}
\end{equation*}
where $1\le r\le\infty$ and $1\le r'\le\infty$ with $1/r+1/r'=1$.
Then we have
$$
\|u(\lambda^2)\|_{L^\infty(\Omega)}
\le C|||u_\lambda(0)|||_{r,1}
\le C\lambda^{-\frac{N}{r}}|||\varphi|||_{r,\lambda}.
$$
Therefore, setting $\lambda=t^{1/2}$,
we obtain
$$
\|u(t)\|_{L^\infty(\Omega)}
\le Ct^{-\frac{N}{2r}}|||\varphi|||_{r,t^{1/2}}
\le Ct^{-\frac{N}{2r}}|||\varphi|||_{r,\rho}
$$
for all $t\in(0,\rho^2]$ and $\rho>0$, which implies \eqref{eq:2.3}.
$\Box$
\vspace{3pt}

Next we define the solution of \eqref{eq:1.1} in $\Omega\times(0,T)$, where $0<T\le\infty$.
Let $u$ and $v$ be nonnegative measurable functions in $\Omega\times(0,T)$ such that
$$
u,v\in L^\infty(\tau,T-\tau:L^\infty(\Omega))
$$
for all $\tau\in(0,T/2)$.
We say that $(u,v)$ is a subsolution of \eqref{eq:1.1} in $\Omega\times(0,T)$ if
$(u,v)$ satisfies
\begin{equation*}
\begin{split}
 & u(x,t)\le [S(t)u_0](x)+\int_0^t [S(t-s)v(s)^p](x)\,ds<\infty,\\
 & v(x,t)\le [S(t)v_0](x)+\int_0^t [S(t-s)u(s)^q](x)\,ds<\infty,
\end{split}
\end{equation*}
for almost all $x\in\Omega$ and $t\in(0,T)$.
Similarly, we say that $(u,v)$ is a supersolution of \eqref{eq:1.1} in $\Omega\times(0,T)$ if
$(u,v)$ satisfies
\begin{equation*}
\begin{split}
 & \infty>u(x,t)\ge [S(t)u_0](x)+\int_0^t [S(t-s)v(s)^p](x)\,ds,\\
 & \infty>v(x,t)\ge [S(t)v_0](x)+\int_0^t [S(t-s)u(s)^q](x)\,ds,
\end{split}
\end{equation*}
for almost all $x\in\Omega$ and $t\in(0,T)$.
Furthermore, we say that $(u,v)$ is a solution of \eqref{eq:1.1} in $\Omega\times(0,T)$
if  $(u,v)$ is a subsolution and a supersolution of \eqref{eq:1.1} in $\Omega\times(0,T)$.
Similarly, we define solutions, supersolutions
and subsolutions of problem~\eqref{eq:1.7}.
\begin{lemma}
\label{Lemma:2.2}
Let $0<T\le\infty$. Then problem~\eqref{eq:1.1} possesses a solution in $\Omega\times(0,T)$
if and only if problem~\eqref{eq:1.1} possesses a supersolution in $\Omega\times(0,T)$.
This also holds true for problem~\eqref{eq:1.7}.
\end{lemma}
{\bf Proof.}
Let $0<T\le\infty$.
It suffices to prove that
the existence of a supersolution of \eqref{eq:1.1} implies the existence of a solution of \eqref{eq:1.1}.
Let $(\tilde{u},\tilde{v})$ be a supersolution of \eqref{eq:1.1} in $\Omega\times(0,T)$.
Set $u_1=S(t)u_0$ and $v_1(t)=S(t)v_0$.
Define $(u_n,v_n)$ $(n=2,3,\dots)$ inductively by
\begin{equation}
\label{eq:2.5}
u_n(t)=S(t)u_0+\int_0^t S(t-s)v_{n-1}(s)^p\,ds,
\quad
v(t)=S(t)v_0+\int_0^t S(t-s)u_{n-1}(s)^q\,ds.
\end{equation}
Then it follows inductively that
$$
0\le u_1\le u_2\le\cdots\le u_n\le\cdots\le\tilde{u}<\infty,
\quad
0\le v_1\le v_2\le\cdots\le v_n\le\cdots\le\tilde{v}<\infty,
$$
for almost all $x\in\Omega$ and $t\in(0,T)$.
This means that
$$
u(x,t):=\lim_{n\to\infty}u_n(x,t),
\qquad
v(x,t):=\lim_{n\to\infty}v_n(x,t),
$$
exist for almost all $x\in\Omega$ and $t\in(0,T)$.
Furthermore, by \eqref{eq:2.5} we see that $(u,v)$ is a solution of \eqref{eq:1.1} in $\Omega\times(0,T)$.
Similarly, we obtain the desired conclusion for problem~\eqref{eq:1.7}.
Thus Lemma~\ref{Lemma:2.2} follows.
$\Box$
\vspace{5pt}
\newline
Applying the comparison principle and Lemma~\ref{Lemma:2.2},
we obtain the following.
\begin{lemma}
\label{Lemma:2.3}
Assume that there exists a solution of \eqref{eq:1.7} in $\Omega\times(0,T)$
for some $\alpha\ge 1$ and $\beta\ge 1$, where $0<T\le\infty$.
Then problem~\eqref{eq:1.1} possesses a solution in $\Omega\times(0,T)$.
\end{lemma}

The end of this section
we give sufficient conditions
for the existence of the solutions of \eqref{eq:1.7}
by employing the argument in \cite{RS}.
\begin{lemma}
\label{Lemma:2.4}
Let $A\ge 0$, $B\ge 0$, $\alpha\ge 1$ and $\beta\ge 1$.
Put $w_0=u_0^\alpha+v_0^\beta$.
\begin{itemize}
  \item[{\rm (i)}] Suppose $\max\{A,B\}\le 1$.
  Then, for any nonnegative function $w_0\in L^1_{{\rm loc}}(\Omega)$ satisfying
  $S(t)w_0\in L^\infty(\Omega)$ for all $t>0$,
  problem~\eqref{eq:1.7} possesses a global-in-time solution.
  \item[{\rm (ii)}] Otherwise, let $\max\{A,B\}\ge1$. Then the following holds.
  \begin{itemize}
  \item[{\rm (a)}]
  Let $1\le\sigma\le\max\{A,B\}$
  be such that $w_0\in L^\sigma_{{\rm loc}}(\Omega)$ and
  $S(t)w_0^\sigma\in L^\infty(\Omega)$ for all $t>0$.
  Then there exist positive constants $\gamma_1$ and $T_1$ such that, if
  \begin{equation}
  \label{eq:2.6}
  \sup_{0<t\le T}
  \biggr[\left\|S(t)w_0^\sigma\right\|_{L^\infty(\Omega)}^{1-\frac{1}{\sigma}}
  \int_0^t\|S(s)w_0^\sigma\|_{L^\infty(\Omega)}^{\frac{\max\{A,B\}}{\sigma}-1}\,ds\biggr]\le\gamma_1
  \end{equation}
  for some $T>0$, then problem~\eqref{eq:1.7} possesses a solution $w$
  in $\Omega\times(0,\min\{T,T_1\}]$ such that
  $$
  0\le w(x,t)\le 2[S(t)w_0^\sigma]^{\frac{1}{\sigma}}+2t
  \quad\mbox{in}\quad\Omega\times(0,\min\{T,T_1\}].
  $$
  \item[{\rm (b)}]
  Let $1\le\sigma\le\min\{A,B\}$
  be such that $w_0\in L^\sigma_{{\rm loc}}(\Omega)$ and
  $S(t)w_0^\sigma\in L^\infty(\Omega)$ for all $t>0$.
  Then there exists a positive constant $\gamma_2$ such that, if
  $$
  \sup_{0<t\le T}
  \biggr[\left\|S(t)w_0^\sigma\right\|_{L^\infty(\Omega)}^{1-\frac{1}{\sigma}}
  \int_0^t\left\{\|S(s)w_0^\sigma\|_{L^\infty(\Omega)}^{\frac{A}{\sigma}-1}
  +\|S(s)w_0^\sigma\|_{L^\infty(\Omega)}^{\frac{B}{\sigma}-1}\right\}\,ds\biggr]\le\gamma_2
  $$
  for some $T>0$, then
  problem~\eqref{eq:1.7} possesses a solution $w$ in $\Omega\times(0,T]$ such that
  $$
  0\le w(x,t)\le 2[S(t)w_0^\sigma]^{\frac{1}{\sigma}}
  \quad\mbox{in}\quad\Omega\times(0,T].
  $$
  \end{itemize}
\end{itemize}
\end{lemma}
{\bf Proof.}
We first prove assertion~(ii)-(a).
Let $\gamma_1$ be a sufficiently small positive constant and assume \eqref{eq:2.6}.
Set
\begin{equation}
\label{eq:2.7}
\overline{w}(t):=2[S(t)w_0^\sigma]^{\frac{1}{\sigma}}+2t.
\end{equation}
Since
\begin{equation*}
\begin{split}
F[\overline{w}](t) & :=S(t)w_0+\int_0^tS(t-s)[\alpha \overline{w}(s)^A+\beta \overline{w}(s)^B]\,ds\\
 & \le\left[S(t)w_0^\sigma\right]^{\frac{1}{\sigma}}
 +\int_0^t S(t-s)\left[1+C\overline{w}(s)^{\max\{A,B\}}\right]\,ds\\
 & \le \left[S(t)w_0^\sigma\right]^{\frac{1}{\sigma}}+t
 +C\int_0^t S(t-s)\overline{w}(s)^{\max\{A,B\}}\,ds,\qquad t>0,
\end{split}
\end{equation*}
we have
\begin{equation*}
\begin{split}
 F[\overline{w}](t)
  & \le\frac{1}{2}\overline{w}(t)+C\int_0^t S(t-s)\overline{w}(s)^{\max\{A,B\}}ds\\
 & \le \frac{1}{2}\overline{w}(t)+C\int_0^t S(t-s)
 \left[[S(s)w_0^\sigma]^{\frac{\max\{A,B\}}{\sigma}}+s^{\max\{A,B\}}\right]\,ds\\
 & \le\frac{1}{2}\overline{w}(t)+C\int_0^t S(t-s)
 \left\|S(s)w_0^\sigma\right\|_{L^\infty(\Omega)}^{\frac{\max\{A,B\}}{\sigma}-1}S(s)w_0^\sigma\,ds
 +Ct^{1+\max\{A,B\}}\\
 & \le\frac{1}{2}\overline{w}(t)
+C\biggr[\int_0^t\left\|S(s)w_0^\sigma\right\|_{L^\infty(\Omega)}^{\frac{\max\{A,B\}}{\sigma}-1}\,ds\biggr]S(t)w_0^\sigma
+Ct^{1+\max\{A,B\}}
\end{split}
\end{equation*}
for all $t>0$.
Then it follows from \eqref{eq:2.6} that
$$
F[\overline{w}](t)
\le \frac{1}{2}\overline{w}(t)+C\gamma_1[S(t)w_0^\sigma]^{\frac{1}{\sigma}}+Ct^{1+\max\{A,B\}}
$$
for all $t\in(0,T]$.
Therefore, taking a sufficiently small $\gamma_1$ if necessary,
we can find a constant $\tau\in(0,1)$ such that
$$
F[\overline{w}](t)\le\overline{w},\qquad t\in(0,\min\{T,\tau\}].
$$
This implies
that $\overline{w}$ is a supersolution of \eqref{eq:1.7} in $\Omega\times(0,\min\{T,\tau\}]$.
Then assertion~(ii)-(a) follows from Lemma~\ref{Lemma:2.2}.
Similarly, setting
$$
\overline{w}(t):=2[S(t)w_0^\sigma]^{\frac{1}{\sigma}}
$$
instead of \eqref{eq:2.7},
we have assertion~(ii)-(b).

It remans to prove assertion~(i).
Since $0\le A,B\le 1$, it follows that
$$
\alpha x^A+\beta x^B\le (\alpha+\beta)(x+1)
$$
for any $x>0$.
This implies that
$$
e^{(\alpha+\beta)t}S(t)w_0+e^{(\alpha+\beta)t}-1
$$
is a supersolution of \eqref{eq:1.7} in $\Omega\times(0,\infty)$.
Then assertion~(i) follows from Lemma~\ref{Lemma:2.2}.
$\Box$
\section{Weakly coupled nonlinear parabolic systems}
In this section, using supersolutions constructed in Section~1 and
applying Lemmas~\ref{Lemma:2.3} and \ref{Lemma:2.4},
we study the existence of local-in-time solutions and global-in-time solutions of \eqref{eq:1.1}.
Furthermore, we obtain lower estimates on the blow-up rate for the solutions.
\subsection{Existence of the solutions}
We first give sufficient conditions for the existence of the solutions of \eqref{eq:1.1}
by using uniformly local $L^{r,\infty}$ spaces.
In this subsection, we write
$$
P=P(r_1,r_2):=N\left(\frac{p}{r_2}-\frac{1}{r_1}\right)
\qquad\mbox{and}\qquad
Q=Q(r_1,r_2):=N\left(\frac{q}{r_1}-\frac{1}{r_2}\right)
$$
for simplicity.
\begin{theorem}
\label{Theorem:3.1}
Let $u_0$ and $v_0$ be nonnegative measurable functions in $\Omega$ such that
$u_0\in L^{r_1,\infty}_{{\rm uloc}}(\Omega)$ and $v_0\in L^{r_2,\infty}_{{\rm uloc}}(\Omega)$,
where $r_1,r_2\in[1,\infty)$.
\begin{itemize}
  \item[{\rm (i)}] Let $\max\{P,Q\}\le 0$. Then there exists a global-in-time solution of \eqref{eq:1.1}.
  \item[{\rm (ii)}] Let $W_0:=u_0^{r_1}+v_0^{r_2}$. Assume
  $0<\max\{P,Q\}\le 2$ and $1<r\le\min\{r_1,r_2\}$.
  Then there exists $\sigma_*>1$ with the following property:
  for any $1<\sigma\le\sigma_*$, there exist positive constants $\gamma_1$ and $T_1$ such that, if
  \begin{equation}
  \label{eq:3.1}
  |||W_0|||_{1,\rho}\le\gamma_1\rho^{N\left(1-\frac{2}{\max\{P,Q\}}\right)}
  \end{equation}
  for some $\rho>0$, then problem~\eqref{eq:1.1} possesses a solution~$(u,v)$
  in $\Omega\times(0,\min\{\rho^2,T_1\}]$ satisfying
  \begin{equation}
  \label{eq:3.2}
  0\le u(x,t)^{r_1}+v(x,t)^{r_2}\le C[S(t)W_0^\frac{\sigma}{r}](x)^{\frac{r}{\sigma}}+C
  \end{equation}
  in $\Omega\times(0,\min\{\rho^2,T_1\}]$.
  Furthermore, in the case where $P>0$ and $Q>0$,
  for any $1<\sigma\le\sigma_*$,
  there exists a positive constant $\gamma_2$ such that, if
  \begin{equation}
  \label{eq:3.3}
  |||W_0|||_{1,\rho}
  \le\gamma_2\max\left\{\rho^{N\bigr(1-\frac{2}{P}\bigr)},\rho^{N\bigr(1-\frac{2}{Q}\bigr)}\right\}
  \end{equation}
  for some $\rho>0$, then problem~\eqref{eq:1.1} possesses a solution
  in $\Omega\times(0,\rho^2]$ satisfying
  \begin{equation}
  \label{eq:3.4}
  0\le u(x,t)^{r_1}+v(x,t)^{r_2}\le C[S(t)W_0^\frac{\sigma}{r}](x)^{\frac{r}{\sigma}}
  \end{equation}
 in $\Omega\times(0,\rho^2]$.
  \item[{\rm (iii)}]
  Let $\max\{P,Q\}>0$.
  There exist positive constants $\gamma_3$ and $T_2$ such that, if
  $$
  \int_0^T\|S(s)W_0\|_{L^\infty(\Omega)}^{\frac{\max\{P,Q\}}{N}}\,ds\le\gamma_3
  $$
  for some $T>0$, then problem~\eqref{eq:1.1} possesses a solution
  in $\Omega\times(0,\min\{T,T_2\}]$ satisfying
  $$
  0\le u(x,t)^{r_1}+v(x,t)^{r_2}\le C[S(t)W_0](x)+C
  $$
  in $\Omega\times(0,\min\{T,T_2\}]$.
  Furthermore, in the case where $P>0$ and $Q>0$,
  there exists a positive constant $\gamma_4$ such that, if
  $$
  \int_0^T\left\{\|S(s)W_0\|_{L^\infty(\Omega)}^{\frac{P}{N}}
  +\|S(s)W_0\|_{L^\infty(\Omega)}^{\frac{Q}{N}}\right\}\,ds\le\gamma_4,
  $$
  then problem~\eqref{eq:1.1} possesses a solution in $\Omega\times(0,T]$ satisfying
  $$
  0\le u(x,t)^{r_1}+v(x,t)^{r_2}\le C[S(t)W_0](x)
 $$
  in $\Omega\times(0,T]$.
  \end{itemize}
\end{theorem}
{\bf Proof.}
Let $1\le r\le\min\{r_1,r_2\}$. Set
$\alpha:=r_1/r\ge 1$ and $\beta:=r_2/r\ge 1$.
Then
\begin{equation}
\label{eq:3.5}
\begin{split}
 & w_0:=u_0^\alpha+v_0^\beta
 =u_0^{\frac{r_1}{r}}+v_0^{\frac{r_2}{r}}\in L^{r,\infty}_{{\rm uloc}}(\Omega),\\
 & A:=1-\frac{1}{\alpha}+\frac{p}{\beta}
=1+r\left(\frac{p}{r_2}-\frac{1}{r_1}\right)
=1+\frac{r}{N}P>0,\\
 & B:=1-\frac{1}{\beta}+\frac{q}{\alpha}
=1+r\left(\frac{q}{r_1}-\frac{1}{r_2}\right)
=1+\frac{r}{N}Q>0.
\end{split}
\end{equation}
In the case $\max\{P,Q\}\le0$,
it follows from Lemma~\ref{Lemma:2.4}~(i) that
problem~\eqref{eq:1.7} possesses a global-in-time positive solution $\overline{w}$.
Then, by Lemma~\ref{Lemma:2.3}
we see that problem~\eqref{eq:1.1} possesses a global-in-time positive solution.
Thus assertion~(i) follows.

We prove assertion~(ii). Let
$$
1<r\le\min\{r_1,r_2\},\qquad
1<\sigma\le\max\{A,B\},\qquad
\sigma\le r.
$$
Since $w_0^\sigma\in L^{\frac{r}{\sigma},\infty}_{{\rm uloc}}(\Omega)$,
by Lemma~\ref{Lemma:2.1} we have
$$
\|S(t)w_0^\sigma\|_{L^\infty(\Omega)}\le c_*t^{-\frac{\sigma N}{2r}}
|||w_0^\sigma|||_{\frac{r}{\sigma},\rho}
\le Ct^{-\frac{\sigma N}{2r}}|||W_0|||_{1,\rho}^{\frac{\sigma}{r}},
\qquad 0<t\le\rho^2.
$$
This together with $\sigma>1$, $\max\{P,Q\}\le 2$ and \eqref{eq:3.1} implies that
\begin{equation}
\label{eq:3.6}
\begin{split}
 & \left\|S(t)w_0^\sigma\right\|_{L^\infty(\Omega)}^{1-\frac{1}{\sigma}}
\int_0^t\|S(s)w_0^\sigma\|_{L^\infty(\Omega)}^{\frac{\max\{A,B\}}{\sigma}-1}\,ds\\
 & \le \left[C|||W_0|||_{1,\rho}^{\frac{\sigma}{r}}\right]^{\frac{\max\{A,B\}-1}{\sigma}}
 t^{-\frac{(\sigma-1)N}{2r}}
 \int_0^t s^{-\frac{\max\{P,Q\}}{2}+\frac{(\sigma-1)N}{2r}}\,ds\\
 & \le C|||W_0|||_{1,\rho}^{\frac{\max\{P,Q\}}{N}}
 t^{1-\frac{\max\{P,Q\}}{2}}
 \le C\gamma_1^{\frac{\max\{P,Q\}}{N}},
 \qquad 0<t\le\rho^2.
\end{split}
\end{equation}
Taking a sufficiently small $\gamma_1$ if necessary,
by Lemma~\ref{Lemma:2.4}~(ii)-(a)
we can find a positive constant $T_1$ such that
problem~\eqref{eq:1.7} possesses a solution $w$ in $\Omega\times(0,\min\{\rho^2,T_1\}]$
such that
\begin{equation}
\label{eq:3.7}
0<w(x,t)\le 2[S(t)w_0^\sigma]^{\frac{1}{\sigma}}+2t
\end{equation}
in $\Omega\times(0,\min\{\rho^2,T_1\}]$.
Then, by Lemma~\ref{Lemma:2.3}
we see that problem~\eqref{eq:1.1} possesses a solution $(u,v)$ in $\Omega\times(0,\min\{\rho^2,T_1\}]$.
Furthermore, by \eqref{eq:1.8} and \eqref{eq:3.7} we obtain
\begin{equation}
\label{eq:3.8}
\max\left\{u(x,t)^{\frac{r_1}{r}},v(x,t)^{\frac{r_2}{r}}\right\}
\le w(x,t)\le 2[S(t)w_0^{\sigma}]^{\frac{1}{\sigma}}+2t
\le C[S(t)W_0^\frac{\sigma}{r}]^{\frac{1}{\sigma}}+2t
\end{equation}
in $\Omega\times(0,\min\{\rho^2,T_1\}]$,
which implies \eqref{eq:3.2}.

In the case $P>0$ and $Q>0$,
it follows that $A>1$ and $B>1$. Let $1<\sigma\le\min\{A,B\}$.
Then, similarly to \eqref{eq:3.6}, by \eqref{eq:3.3} we have
\begin{equation*}
\begin{split}
 & \left\|S(t)w_0^\sigma\right\|_{L^\infty(\Omega)}^{1-\frac{1}{\sigma}}
\int_0^t\left\{\|S(s)w_0^\sigma\|_{L^\infty(\Omega)}^{\frac{A}{\sigma}-1}
+\|S(s)w_0^\sigma\|_{L^\infty(\Omega)}^{\frac{B}{\sigma}-1}\right\}\,ds\\
 & \le C|||W_0|||_{1,\rho}^{\frac{P}{N}}t^{1-\frac{P}{2}}
+C|||W_0|||_{1,\rho}^{\frac{Q}{N}}t^{1-\frac{Q}{2}}
\le C(\gamma_2^{\frac{P}{N}}+\gamma_2^{\frac{Q}{N}})
\end{split}
\end{equation*}
for all $0<t\le\rho^2$.
Then, taking a sufficiently small $\gamma_2$ if necessary,
by Lemma~\ref{Lemma:2.4}~(ii)-(b)
we can find a solution~$w$ of \eqref{eq:1.7}
in $\Omega\times(0,\rho^2]$
such that
\begin{equation}
\label{eq:3.9}
0<w(x,t)\le 2[S(t)w_0^\sigma]^{\frac{1}{\sigma}}
\quad\mbox{in}\quad\Omega\times(0,\rho^2].
\end{equation}
By Lemma~\ref{Lemma:2.3}
we can find a solution of \eqref{eq:1.1} in $\Omega\times(0,\rho^2]$.
Furthermore, similarly to \eqref{eq:3.8},
by \eqref{eq:1.8} and \eqref{eq:3.9} we obtain
$$
\max\left\{u(x,t)^{\frac{r_1}{r}},v(x,t)^{\frac{r_2}{r}}\right\}
\le w(x,t)\le C[S(t)W_0^\frac{\sigma}{r}]^{\frac{1}{\sigma}}
$$
in $\Omega\times(0,\rho^2]$,
which implies \eqref{eq:3.4}.
Thus assertion~(ii) follows.
Assertion~(iii) is also proved by the same argument with $r=\sigma=1$ as in the proof of assertion~(ii) and
by Lemma~\ref{Lemma:2.4}~(ii).
Thus Theorem~\ref{Theorem:3.1} follows.
$\Box$\vspace{5pt}
\newline
It follows that
$P=Q=2$ if and only if
\begin{equation*}
r_1=r_1^*:=\frac{N}{2}\frac{pq-1}{p+1}
\quad\mbox{and}\quad
r_2=r_2^*:=\frac{N}{2}\frac{pq-1}{q+1}.
\end{equation*}
Then, by Theorem~\ref{Theorem:3.1}~(ii) we have
\begin{corollary}
\label{Corollary:3.1}
Assume that
$$
r_1^*:=\frac{N}{2}\frac{pq-1}{p+1}>1
\quad\mbox{and}\quad
r_2^*:=\frac{N}{2}\frac{pq-1}{q+1}>1.
$$
Then there exists a constant $\delta>0$ such that,
for any nonnegative measurable functions
$u_0\in L^{r_1^*,\infty}(\Omega)$ and $v_0\in L^{r_2^*,\infty}(\Omega)$,
if
$$
\|u_0\|_{L^{r_1,\infty}(\Omega)}+\|v_0\|_{L^{r_2,\infty}(\Omega)}<\delta,
$$
then problem~\eqref{eq:1.1} possesses a global-in-time solution
and \eqref{eq:3.4} holds in $\Omega\times(0,\infty)$.
\end{corollary}
Furthermore, by Theorem~\ref{Theorem:3.1} and \eqref{eq:2.1}
we obtain
\begin{corollary}
\label{Corollary:3.2}
Let $\Omega$ be a domain in ${\bf R}^N$ such that $0\in\Omega$.
Let $u_0$ and $v_0$ be nonnegative measurable functions in $\Omega$ such that
\begin{equation}
\label{eq:3.11}
0\le u_0(x)\le d|x|^{-N/r_1},
\qquad
0\le v_0(x)\le d|x|^{-N/r_2},
\end{equation}
for all $x\in\Omega$, where $d>0$, $r_1>1$ and $r_2>1$.
In the case $\max\{P,Q\}\le 0$, problem~\eqref{eq:1.1} possesses a global-in-time solution.
On the other hand, in the case $\max\{P,Q\}>0$,
there exists a constant $d_*>0$ such that,
if $0<d\le d_*$, then
the following holds.
\begin{itemize}
  \item[{\rm (i)}]
  Let $0<\max\{P,Q\}\le 2$. Then
  problem~\eqref{eq:1.1} possesses a solution $(u,v)$ in $\Omega\times(0,T)$ for some $T>0$ satisfying
  $$
  0\le u(x,t)\le C(|x|+t^2)^{-\frac{N}{r_1}}+C,
  \qquad
  0\le v(x,t)\le C(|x|+t^2)^{-\frac{N}{r_2}}+C,
  $$
  in $\Omega\times(0,T)$.
  \item[{\rm (ii)}]
  Let $P=Q=2$.
  Then problem~\eqref{eq:1.1} possesses a global-in-time solution $(u,v)$ satisfying
  $$
  0\le u(x,t)\le C(|x|+t^2)^{-\frac{N}{r_1}},
  \qquad
  0\le v(x,t)\le C(|x|+t^2)^{-\frac{N}{r_2}},
  $$
  in $\Omega\times(0,\infty)$.
\end{itemize}
\end{corollary}
{\bf Proof.}
Assume \eqref{eq:3.11}.
It follows from \eqref{eq:2.1}
that $u_0\in L^{r_1,\infty}(\Omega)$ and $v_0\in L^{r_2,\infty}(\Omega)$.
Set
$$
W_0(x):=u_0^{r_1}+v_0^{r_2}.
$$
For any $1<\sigma\le r\le\min\{r_1,r_2\}$,
by \eqref{eq:3.11} we have
$$
0\le W_0(x)^{\frac{\sigma}{r}}\le Cd^\sigma |x|^{-\frac{\sigma N}{r}},
\qquad x\in\Omega,
$$
which implies that
$$
0\le [S(t)W_0^{\frac{\sigma}{r}}](x)\le Cd^\sigma(|x|+t^2)^{-\frac{\sigma N}{r}},
\qquad x\in\Omega,\,\,t>0.
$$
Then Corollary~\ref{Corollary:3.2} follows from Theorem~\ref{Theorem:3.1}.
$\Box$\vspace{7pt}

Next we give a sufficient condition
for problem~\eqref{eq:1.1} to posses a global-in-time
positive solution for some initial function.
Assume the following.
\begin{equation}
\label{eq:3.12}
\left\{
\begin{array}{l}
 \mbox{There exists a constant $p_*(\Omega)>1$ with the following properties:}\vspace{3pt}\\
 {{\rm(i)}}
 \quad\,\, \mbox{If $1<\min\{A,B\}\le p_*(\Omega)$,
  then problem~\eqref{eq:1.7} has no global-in-time}\vspace{3pt}\\
 \qquad\,\,\mbox{positive solutions;}\vspace{5pt}\\
  {{\rm(ii)}}
  \quad\mbox{If $p_*(\Omega)<\min\{A,B\}<\infty$, then
  problem~\eqref{eq:1.7} possesses a global-in-}\vspace{3pt}\\
  \qquad\,\,\mbox{time positive solution for some initial function.}
\end{array}
\right.
\end{equation}
The critical exponent $p_*(\Omega)$ has been identified  for various domains
{\rm(}see e.g., {\rm\cite{DL, L}}{\rm)}.
In particular,
\begin{itemize}
  \item[{\rm (i)}] $p_*(\Omega)=1+2/N$ if $\Omega={\bf R}^N$,
  \item[{\rm (ii)}] $p_*(\Omega)=1+2/(N+1)$ if $\Omega$ is a half space of ${\bf R}^N$,
  \item[{\rm (iii)}] $p_*(\Omega)=1+2/N$ if $\Omega$ is the exterior domain
  of a compact set in ${\bf R}^N$ and $N\ge 2$.
\end{itemize}
(See also {\rm\cite{P}} for {\rm (iii)}.)
\begin{theorem}
\label{Theorem:3.2}
Let $\Omega$ be a smooth domain in ${\bf R}^N$.
\begin{itemize}
  \item[{\rm (i)}] If $pq\le 1$, then
  problem~\eqref{eq:1.1} possesses a global-in-time solution
  for any initial function $(u_0,v_0)\in L^\infty(\Omega)\times L^\infty(\Omega)$.
  \item[{\rm (ii)}] Assume \eqref{eq:3.12}.
  Then problem~\eqref{eq:1.1} possesses a global-in-time positive solution
  for some initial function if
  \begin{equation}
  \label{eq:3.13}
  \frac{pq-1}{\max\{p,q\}+1}>p_*(\Omega)-1.
  \end{equation}
\end{itemize}
\end{theorem}
{\bf Proof.}
We prove assertion~(i). Assume $pq\le 1$.
Let $u_0$, $v_0\in L^\infty(\Omega)$.
Let $r\ge 1$ be such that $rp\ge 1$.
Then $u_0\in L^r_{{\rm uloc}}(\Omega)$ and $v_0\in L^{pr}_{{\rm uloc}}(\Omega)$.
Furthermore,
$$
P(r,pr)=0,\qquad
Q(r,pr)=\frac{N}{pr}(pq-1)\le 0.
$$
Therefore, assertion~(i) follows from Theorem~{\rm\ref{Theorem:3.1}}~{\rm (i)}.

We prove assertion~(ii).
Assume
$$
A:=1-\frac{1}{\alpha}+\frac{p}{\beta}>p_*(\Omega),
\qquad
B:=1-\frac{1}{\beta}+\frac{q}{\alpha}>p_*(\Omega),
$$
for some $\alpha\ge 1$ and $\beta\ge 1$.
This is equivalent to \eqref{eq:3.13}.
Then assertion~(ii) follows from Lemma~\ref{Lemma:2.3} and the definition of $p_*(\Omega)$.
Thus the proof is complete.
$\Box$
\begin{remark}
\label{Remark:3.1}
Assertions~{\rm (B)} and {\rm (C)} follow from Theorem~{\rm\ref{Theorem:3.2}}
and the fact $p_*({\bf R}^N)=1+2/N$.
Furthermore,
assertion~{\rm (A)} follows from Theorem~{\rm\ref{Theorem:3.1}}.
Indeed, if $u_0\in L^{r_1}(\Omega)$ and $v_0\in L^{r_2}(\Omega)$
for some $r_1,r_2\ge 1$, then
$$
\lim_{\rho\to 0}|||u_0|||_{r_1,\rho}=0
\quad\mbox{and}\quad
\lim_{\rho\to 0}|||u_0|||_{r_2,\rho}=0,
$$
which means that \eqref{eq:3.1} holds for all sufficiently small $\rho>0$.
Then assertion~{\rm (A)} follows from Theorem~{\rm\ref{Theorem:3.1}}~{\rm (ii)}.
\end{remark}
\subsection{$k$-component weakly coupled parabolic systems}
Our arguments to problem~\eqref{eq:1.1}
are applicable to the $k$-component nonlinear parabolic system
\begin{equation}
\label{eq:3.14}
\left\{
\begin{array}{ll}
\partial_t u_i=\Delta u_i+u_{i+1}^{p_i},\qquad & x\in\Omega,\,\,\,t>0,\vspace{5pt}\\
u_i=0, & x\in\partial\Omega,\,\,\,t>0,\vspace{5pt}\\
u_i(x,0)=u_{i,0}(x), & x\in\Omega,
\end{array}
\right.
\end{equation}
where $k\in\{1,2,3,\dots\}$,
$i\in\{1,\cdots,k\}$, $p_i\ge 0$ and $u_{k+1}=u_1$.
Assume that
$\{u_{i,0}\}$ are nonnegative measurable functions in $\Omega$
such that $u_{i,0}\in L^{r_i,\infty}_{\rm uloc}(\Omega)$ $(i=1,\dots,k)$,
where $r_i\in[1,\infty)$.
Let $\alpha_1,\alpha_2,\dots,\alpha_k\ge 1$ and $\alpha_{k+1}=\alpha_1$.
Set $U_i=u_i^{\alpha_i}$ for $i\in\{1,\dots,k\}$ and $U_{k+1}=U_1$.
Then, similarly to \eqref{eq:1.5}, we have
\begin{equation}
\label{eq:3.15}
\left\{
\begin{array}{ll}
\partial_t U_i=\Delta U_i+\alpha_i U_i^{1-\frac{1}{\alpha_i}}U_{i+1}^{\frac{p_i}{\alpha_{i+1}}}
-\displaystyle{\frac{\alpha_i-1}{\alpha_i}\frac{|\nabla U_i|^2}{U_i}}, &x\in\Omega,\,\,t>0,\vspace{7pt}\\
U_i=0, &x\in\partial\Omega,\,\,t>0,\vspace{3pt}\\
U_i(x,0)=u_{i,0}(x)^{\alpha_i},\quad &x\in\Omega.
\end{array}
\right.
\end{equation}
Set
$$
A_k:=\max_{1\le i\le k}\left(1-\frac{1}{\alpha_i}+\frac{p_i}{\alpha_{i+1}}\right),
\qquad
B_k:=\min_{1\le i\le k}\left(1-\frac{1}{\alpha_i}+\frac{p_i}{\alpha_{i+1}}\right).
$$
Then there exist positive constants $c_1$ and $c_2$ such that
$$
\sum_{i=1}^k\alpha_i \xi^{1-\frac{1}{\alpha_i}+\frac{p_i}{\alpha_{i+1}}}
\le c_1\xi^{A_k}+c_2\xi^{B_k},\qquad \xi\in[0,\infty).
$$
Let $\overline{w}$ be a solution of
$$
\left\{
\begin{array}{ll}
\partial_t w=\Delta w+c_1 w^{A_k}+c_2 w^{B_k},\qquad & x\in\Omega,\,\,t>0,\vspace{5pt}\\
w=0, & x\in\partial\Omega,\,\,t>0,\vspace{5pt}\\
w(x,0)=\sum_{i=1}^k u_{i,0}(x)^{\alpha_i},& x\in\Omega.
\end{array}
\right.
$$
Then it follows that
$$
(\overline{w}^{\frac{1}{\alpha_1}},\dots,\overline{w}^{\frac{1}{\alpha_k}})
$$
is a supersolution of \eqref{eq:3.14}.
Therefore we can apply the arguments in this section to \eqref{eq:3.15}
with $A$ and $B$ replaced by $A_k$ and $B_k$, respectively.
In particular, Theorem~\ref{Theorem:3.1} and Corollaries~\ref{Corollary:3.1} and \ref{Corollary:3.2}
hold with $P(r_1,r_2)$ and $Q(r_1,r_2)$ replaced by
$$
P_k(r_1,\dots,r_k):=N\max_{1\le i\le k}\left(\frac{p_i}{r_{i+1}}-\frac{1}{r_i}\right),
\quad
Q_k(r_1,\dots,r_k):=N\min_{1\le i\le k}\left(\frac{p_i}{r_{i+1}}-\frac{1}{r_i}\right),
$$
respectively.
We leave the details to the reader.
\subsection{Lower estimates on the blow-up rate}
As an application of the results in subsection~3.1,
we give lower estimates on the blow-up rate of the solutions of \eqref{eq:1.1}
by modifying the argument in \cite{IS},
which gave lower estimates of the life span of the solutions
to the hear equation with a nonlinear boundary condition.
\begin{theorem}
\label{Theorem:3.3}
Let $\Omega$ be a smooth domain in ${\bf R}^N$ and $pq>1$.
Let $(u,v)$ be the minimal solution of \eqref{eq:1.1} in $\Omega\times(0,T)$, where $0<T<\infty$,
such that
$$
\limsup_{t\to T}\,\left[\|u(t)\|_{L^\infty(\Omega)}+\|v(t)\|_{L^\infty(\Omega)}\right]=\infty.
$$
Let $r_1^*$ and $r_2^*$ be constants given in \eqref{eq:1.4}
and let $\ell_*\ge 1$ be such that $\ell_*\min\{r_1^*,r_2^*\}\ge 1$.
Set $\rho(t)=\sqrt{T-t}$.
Then, for any $r_1\in(\ell_*r_1^*,\infty]$ and $r_2\in(\ell_*r_2^*,\infty]$,
the following holds.
\begin{itemize}
\item[{\rm(i)}]
There exists a positive constant $c_1$ such that
$$
\liminf_{t\to T}\,\left\{(T-t)^{\frac{N}{2}(\frac{1}{r_1^*}-\frac{1}{r_1})}|||u(t)|||_{r_1,\rho(t)}
+(T-t)^{\frac{N}{2}(\frac{1}{r_2^*}-\frac{1}{r_2})}|||v(t)|||_{r_2,\rho(t)}\right\}\ge c_1.
$$
\item[{\rm (ii)}]
There exists a positive constant $c_2$ such that
$$
\left\{
\begin{array}{l}
\displaystyle{\limsup_{t\to T}}\,(T-t)^{\frac{p+1}{pq-1}-\frac{N}{2r_1}}|||u(t)|||_{r_1,\rho(t)}\ge c_2,\vspace{3pt}\\
\displaystyle{\limsup_{t\to T}}\,(T-t)^{\frac{q+1}{pq-1}-\frac{N}{2r_2}}|||v(t)|||_{r_2,\rho(t)}\ge c_2.
\end{array}
\right.
$$
\end{itemize}
\end{theorem}
{\bf Proof.}
We first prove assertion $({\rm i})$.
Let $\ell>\ell_*$ be such that $r_1\ge r_1':=\ell r_1^*$ and $r_2\ge r_2':=\ell r_2^*$.
Then
$$
P(r_1',r_2')=Q(r_1',r_2')=N\left(\frac{p}{\ell r_2^*}-\frac{1}{\ell r_1^*}\right)=\frac{2}{\ell}.
$$
Since the minimal solution $(u,v)$ blows up at $t=T$,
by Theorem~\ref{Theorem:3.1}~(ii)
we can find constants $\gamma_1>0$ and $\delta\in(0,T)$ such that
\begin{equation}
\label{eq:3.16}
|||u(t)^{r_1'}|||_{1,\rho(t)}+|||v(t)^{r_2'}|||_{1,\rho(t)}>\gamma_1\rho(t)^{-N(\ell-1)}
\end{equation}
for all $t\in(T-\delta,T)$.
Indeed, if not, then we deduce from Theorem~\ref{Theorem:3.1}~(ii)
that problem~\eqref{eq:1.1} possesses a solution of \eqref{eq:1.1} not blowing up at $t=T$.

On the other hand,
it follows from \eqref{eq:2.2} that
\begin{equation}
\label{eq:3.17}
\begin{split}
|||u(t)^{r_1'}|||_{1,\rho(t)}
 & =\sup_{x\in\overline{\Omega}}\|u(t)\|_{L^{r_1',\infty}(B(x,\rho(t)))}^{r_1'}\\
 & \le C(T-t)^{\frac{N}{2}\left(1-\frac{r_1'}{r_1}\right)}
 \,\sup_{x\in\Omega}\|u(t)\|_{L^{r_1,\infty}(B(x,\rho(t)))}^{r_1'}\\
 & =C(T-t)^{\frac{N}{2}\left(1-\frac{\ell r_1^*}{r_1}\right)}|||u(t)|||_{r_1,\rho(t)}^{\ell r_1^*}
\end{split}
\end{equation}
and
\begin{equation}
\label{eq:3.18}
|||v(t)^{r_2'}|||_{1,\rho(t)}
\le C(T-t)^{\frac{N}{2}\left(1-\frac{\ell r_2^*}{r_2}\right)}|||v(t)|||_{r_2,\rho(t)}^{\ell r_2^*}
\end{equation}
for all $t\in(0,T)$.
By \eqref{eq:3.16}, \eqref{eq:3.17} and \eqref{eq:3.18} we obtain
\begin{equation}
\label{eq:3.19}
(T-t)^{-\frac{N\ell r_1^*}{2r_1}}|||u(t)|||_{r_1,\rho(t)}^{\ell r_1^*}
+(T-t)^{-\frac{N\ell r_2^*}{2r_2}}|||v(t)|||_{r_2,\rho(t)}^{\ell r_2^*}
\ge\gamma_2\rho(t)^{-N\ell}
\end{equation}
for $t\in(T-\delta,T)$, where $\gamma_2$ is a positive constant.

Let $t\in(T-\delta,T)$ and assume that
$$
(T-t)^{-\frac{N\ell r_1^*}{2r_1}}|||u(t)|||_{r_1,\rho(t)}^{\ell r_1^*}
\ge(T-t)^{-\frac{N\ell r_2^*}{2r_2}}|||v(t)|||_{r_2,\rho(t)}^{\ell r_2^*}.
$$
Then, by \eqref{eq:3.19} we have
$$
(T-t)^{-\frac{N\ell r_1^*}{2r_1}}|||u(t)|||_{r_1,\rho(t)}^{\ell r_1^*}
\ge\frac{\gamma_2}{2}(T-t)^{-\frac{N\ell}{2}},
$$
which implies that
\begin{equation}
\label{eq:3.20}
|||u(t)|||_{r_1,\rho(t)}\ge\left(\frac{\gamma_2}{2}\right)^{\frac{1}{\ell r_1^*}}(T-t)^{-\frac{N}{2r_1^*}+\frac{N}{2r_1}}
=\left(\frac{\gamma_2}{2}\right)^{\frac{1}{\ell r_1^*}}(T-t)^{-\frac{p+1}{pq-1}+\frac{N}{2r_1}}.
\end{equation}
If not, we obtain
\begin{equation}
\label{eq:3.21}
|||v(t)|||_{r_2,\rho(t)}\ge
\left(\frac{\gamma_2}{2}\right)^{\frac{1}{\ell r_2^*}}(T-t)^{-\frac{q+1}{pq-1}+\frac{N}{2r_2}}.
\end{equation}
We deduce from \eqref{eq:3.20} and \eqref{eq:3.21} that
\begin{equation*}
\begin{split}
 & (T-t)^{\frac{p+1}{pq-1}-\frac{N}{2r_1}}|||u(t)|||_{r_1,\rho(t)}
+(T-t)^{\frac{q+1}{pq-1}-\frac{N}{2r_2}}|||v(t)|||_{r_2,\rho(t)}\\
 & \ge\min\biggr\{\left(\frac{\gamma_2}{2}\right)^{\frac{1}{\ell r_1^*}},\left(\frac{\gamma_2}{2}\right)^{\frac{1}{\ell r_2^*}}\biggr\}>0
\end{split}
\end{equation*}
for all $t\in(T-\delta,T)$,
which implies assertion $({\rm i})$.

We prove assertion $(\rm ii)$ by contradiction.
Let $\epsilon$ be a sufficiently small positive constant.
Let $r_1>\ell_* r_1^*$ and assume
\begin{equation}
\label{eq:3.22}
\sup_{t\in(T-\delta,T)}(T-t)^{\frac{p+1}{pq-1}-\frac{N}{2r_1}}|||u(t)|||_{r_1,\rho(t)}\le\epsilon.
\end{equation}
It follows from \eqref{eq:1.4} that
\begin{equation}
\label{eq:3.23}
\frac{p+1}{pq-1}-\frac{N}{2r_1}>0.
\end{equation}
Let $r_2':=r_2^*r_1/r_1^*$. Then $r_1>1$, $r_2'>\ell_*r_2^*\ge 1$ and
\begin{equation}
\label{eq:3.24}
\frac{N}{2}\left(\frac{q}{r_1}-\frac{1}{r_2'}\right)=\frac{Nr_1^*}{2r_1}\left(\frac{q}{r_1^*}-\frac{1}{r_2^*}\right)
=\frac{r_1^*}{r_1}\in(0,1).
\end{equation}
Put $\tau:=T-\delta$ and $\eta:=(T-t)/2$.
Then, by Lemma~\ref{Lemma:2.1} and \eqref{eq:3.22}--\eqref{eq:3.24} we obtain
\begin{equation*}
\begin{split}
 & |||v(t)|||_{r_2',\rho(t)} \\
 & \le|||S(t-\tau)v(\tau)|||_{r_2',\rho(t)}
+\int_\tau^t|||S(t-s)u(s)^q|||_{r_2,\rho(t)}\,ds\\
 & \le C|||v(\tau)|||_{r_2',(T-t)^{\frac{1}{2}}}+C\int_\tau^t(t-s)^{-\frac{N}{2}\left(\frac{q}{r_1}-\frac{1}{r_2'}\right)}|||u(s)|||_{r_1,\rho(t)}^q\,ds\\
 & \le C|||v(\tau)|||_{r_2',T^\frac{1}{2}}+C\epsilon^q\left\{\int_\tau^{t-\eta}+\int_{t-\eta}^t\right\}(t-s)^{-\frac{N}{2}\left(\frac{q}{r_1}-\frac{1}{r_2'}\right)}
 (T-s)^{-\frac{pq+q}{pq-1}+\frac{Nq}{2r_1}}\,ds\\
 & \le C|||v(\tau)|||_{r_2',T^\frac{1}{2}}
 +C\epsilon^q\biggr[
 \eta^{-\frac{N}{2}\left(\frac{q}{r_1}-\frac{1}{r_2'}\right)}(T-t+\eta)^{-\frac{pq+q}{pq-1}}\int_\tau^{t-\eta}(T-s)^\frac{Nq}{2r_1}\,ds\\
 & \qquad\qquad\qquad\qquad\qquad\qquad\qquad
+(T-t)^{-\frac{pq+q}{pq-1}+\frac{Nq}{2r_1}}\int_{t-\eta}^t(t-s)^{-\frac{N}{2}\left(\frac{q}{r_1}-\frac{1}{r_2'}\right)}\,ds\biggr]\\
 & \le C|||v(\tau)|||_{r_2',T^\frac{1}{2}}+C\epsilon^q(T-t)^{-\frac{q+1}{pq-1}+\frac{N}{2r_2'}}
\end{split}
\end{equation*}
for all $0<\tau<t<T$. This implies that
\begin{equation}
\label{eq:3.25}
\limsup_{t\to T}\,(T-t)^{\frac{q+1}{pq-1}-\frac{N}{2r_2'}}|||v(t)|||_{r_2',\rho(t)}\le\epsilon^q.
\end{equation}
Taking a sufficiently small $\epsilon>0$ if necessary,
we deduce from \eqref{eq:3.22} and \eqref{eq:3.25} that
\begin{equation*}
\begin{split}
 & \limsup_{t\to T}\,
\left[(T-t)^{\frac{p+1}{pq-1}-\frac{N}{2r_1}}|||u(t)|||_{r_1,\rho(t)}
+(T-t)^{\frac{q+1}{pq-1}-\frac{N}{2r_2'}}|||v(t)|||_{r_2',\rho(t)}\right]\\
 & \le\epsilon+C\epsilon^q<c_1,
\end{split}
\end{equation*}
where $c_1$ is the constant given in assertion~(i).
This contradicts assertion~(i), which means  that
$$
\limsup_{t\to T}\,(T-t)^{\frac{p+1}{pq-1}-\frac{N}{2r_1}}\|u(t)\|_{r_1,\rho(t)}\ge c
$$
for some $c>0$.
Similarly, we have
$$
\limsup_{t\to T}\,(T-t)^{\frac{q+1}{pq-1}-\frac{N}{2r_2}}\|v(t)\|_{r_2,\rho(t)}\ge c'
$$
for some $c'>0$.
Thus assertion~(ii) follows and the proof of Theorem~\ref{Theorem:3.3} is complete.
$\Box$\vspace{5pt}
\newline
It follows from Theorem~\ref{Theorem:3.3}~(ii) that
\begin{equation}
\label{eq:3.26}
\limsup_{t\to T}\,(T-t)^{\frac{p+1}{pq-1}}\|u(t)\|_{L^\infty(\Omega)}>0,
\quad
\limsup_{t\to T}\,(T-t)^{\frac{q+1}{pq-1}}\|v(t)\|_{L^\infty(\Omega)}>0.
\end{equation}
(See also \cite[Proposition~3.4]{Z}.)
For upper estimates of blow-up rate of the solutions of \eqref{eq:1.1},
see e.g., \cite{AHV, CM, CF, DL2, QSBook}, which show that
\eqref{eq:3.26} gives the optimal lower estimate on the blow-up rate of the solutions.
\section{Parabolic systems with strongly coupled nonlinearities}
In this section we apply the methods of Sections~2 and 3 to
obtain sufficient conditions for the existence of the solutions
for parabolic systems with strongly coupled nonlinearities.
\subsection{Strongly coupled power nonlinearities}
Consider the parabolic system with strongly coupled power nonlineaities
\begin{equation}
\label{eq:4.1}
\left\{
\begin{array}{ll}
\partial_t u=\Delta u+u^{p_1}v^{p_2},\qquad & x\in\Omega,\,\,\,t>0,\vspace{5pt}\\
\partial_t v=\Delta v+u^{q_1}v^{q_2}, & x\in\Omega,\,\,\,t>0,\vspace{5pt}\\
u=v=0, & x\in\partial\Omega,\,\,\,t>0,\vspace{5pt}\\
(u(x,0), v(x,0))=(u_0(x),v_0(x)), & x\in\Omega,
\end{array}
\right.
\end{equation}
where $p_i\ge 0$, $q_i\ge 0$ $(i=1,2)$,
$\Omega$ is a (possibly unbounded) smooth domain in ${\bf R}^N$ $(N\ge 1)$
and both $u_0$ and $v_0$ are nonnegative and locally integrable functions in $\Omega$.
Problem~\eqref{eq:4.1} was studied in \cite{C, DL1, EL, FM, LS},
however, compared with problems~\eqref{eq:1.1} and \eqref{eq:1.7},
much less is known about the conditions for the existence of the solutions of \eqref{eq:4.1}.
In this subsection we apply the arguments in Section~3 to problem~\eqref{eq:4.1}
and obtain sufficient conditions for the existence of the solutions.
Furthermore, we give lower estimates on the blow-up rate of the solutions.

Let $(u,v)$ be a positive (classical) solution of problem~\eqref{eq:4.1}.
Set $U:=u^\alpha$ and $V:=v^\beta$ for some $\alpha\ge 1$ and $\beta\ge 1$.
Then $(U,V)$ satisfies
$$
\left\{
\begin{array}{ll}
\partial_t U=\Delta U+\alpha U^{1+\frac{p_1-1}{\alpha}}V^{\frac{p_2}{\beta}}
-\displaystyle{\frac{\alpha-1}{\alpha}\frac{|\nabla U|^2}{U}}, &x\in\Omega,\,\,t>0,\vspace{7pt}\\
\partial_t V=\Delta V+\beta U^{\frac{q_1}{\alpha}}V^{1+\frac{q_2-1}{\beta}}
-\displaystyle{\frac{\beta-1}{\beta}\frac{|\nabla V|^2}{V}}, &x\in\Omega,\,\,t>0,\vspace{5pt}\\
U=V=0, &x\in\partial\Omega,\,\,t>0,\vspace{3pt}\\
(U(x,0),V(x,0))=(u_0(x)^\alpha,v_0(x)^\beta),\quad &x\in\Omega.
\end{array}
\right.
$$
Put
$$
A=1+\frac{p_1-1}{\alpha}+\frac{p_2}{\beta},
\qquad
B=1+\frac{q_1}{\alpha}+\frac{q_2-1}{\beta}.
$$
Let $\overline{w}$ be a solution of \eqref{eq:1.7}.
Similarly to \eqref{eq:1.1}, we immediately see that
$(\overline{w}^{\frac{1}{\alpha}},\overline{w}^{\frac{1}{\beta}})$
is a supersolution of \eqref{eq:4.1}.
Then we apply the same arguments as in Sections~2 and 3
to obtain the following theorems.
\begin{theorem}
\label{Theorem:4.1}
Let $\Omega$ be a smooth domain in ${\bf R}^N$.
Consider problem~\eqref{eq:4.1}.
Then the same statements as in Theorem~{\rm\ref{Theorem:3.1}}
and Corollaries~{\rm\ref{Corollary:3.1}} and {\rm\ref{Corollary:3.2}} hold with $P$ and $Q$
replaced by
$$
\tilde{P}:=N\left(\frac{p_1-1}{r_1}+\frac{p_2}{r_2}\right),\qquad
\tilde{Q}:=N\left(\frac{q_1}{r_1}+\frac{q_2-1}{r_2}\right),
$$
respectively.
\end{theorem}
{\bf Proof.}
Let $1\le r\le\min\{r_1,r_2\}$, $\alpha=r_1/r$ and $\beta=r_2/r$.
Then, similarly to \eqref{eq:3.5}, we have
$$
A=1+\frac{p_1-1}{\alpha}+\frac{p_2}{\beta}=1+\frac{r}{N}\tilde{P},
\qquad
B=1+\frac{q_1}{\alpha}+\frac{q_2-1}{\beta}=1+\frac{r}{N}\tilde{Q}.
$$
Then Theorem~\ref{Theorem:4.1} follows from the same arguments in Section~3.
$\Box$\vspace{5pt}
\newline
We remark that
$\tilde{P}=\tilde{Q}=2$ if and only if
$$
r_1=\tilde{r}_1^*:=\frac{N}{2}\frac{\delta}{1-q_2+p_2},
\qquad
r_2=\tilde{r}_2^*:=\frac{N}{2}\frac{\delta}{1-p_1+q_1},
$$
where $\delta:=q_1p_2-(p_1-1)(q_2-1)$.
On the other hand,
by Lemmas~\ref{Lemma:2.3} and \ref{Lemma:2.4}
we immediately obtain the following theorem.
\begin{theorem}
\label{Theorem:4.2}
Let $\Omega$ be a smooth domain in ${\bf R}^N$.
\begin{itemize}
\item[{\rm (i)}] Assume that
\begin{equation}
\label{eq:4.2}
\frac{p_1-1}{\alpha}+\frac{p_2}{\beta}\le 0,
\qquad
\frac{q_2-1}{\beta}+\frac{q_1}{\alpha}\le 0,
\end{equation}
for some $\alpha\ge 1$ and $\beta\ge 1$.
Then problem~\eqref{eq:4.1} possesses a global-in-time solution
for any initial function $(u_0,v_0)\in L^\infty(\Omega)\times L^\infty(\Omega)$.
\item[{\rm (ii)}]
Assume \eqref{eq:3.12}. Furthermore, assume that
\begin{equation}
\label{eq:4.3}
1+\frac{p_1-1}{\alpha}+\frac{p_2}{\beta}>p_*(\Omega),
\qquad
1+\frac{q_2-1}{\beta}+\frac{q_1}{\alpha}>p_*(\Omega),
\end{equation}
for some $\alpha\ge 1$ and $\beta\ge 1$.
Then problem~\eqref{eq:4.1} possesses a global-in-time positive solution
for some initial function $(u_0,v_0)$.
\end{itemize}
\end{theorem}
\begin{remark}
\label{Remark:4.1}
\eqref{eq:4.2} holds if and only if
$$
p_1\le 1,
\quad
q_2\le 1,
\quad
\delta\le 0.
$$
This is the same condition as in Theorem~{\rm 3.2} in {\rm\cite{C}}
and Theorem~{\rm 5}~{\rm II}-{\rm A} in {\rm\cite{EL}}.
Furthermore, \eqref{eq:4.3} holds if and only if
one of the following holds:
\begin{itemize}
  \item $p_1+p_2>p_*(\Omega)$ and  $q_1+q_2>p_*(\Omega)$;
  \item $p_1<1$, $p_1+p_2\le p_*(\Omega)<q_1+q_2$ and $\delta>(p_*(\Omega)-1)(1-p_1+q_1)$;
  \item $q_2<1$, $q_1+q_2\le p_*(\Omega)<p_1+p_2$ and $\delta>(p_*(\Omega)-1)(1-q_2+p_2)$.
\end{itemize}
In the case $\Omega={\bf R}^N$,
this is the same conditions as in Theorem~{\rm 5}~{\rm I}-{\rm A} and {\rm II}-{\rm B} in {\rm\cite{EL}}
and it is the optimal condition for the existence of global-in-time positive solutions of \eqref{eq:4.1}.
\end{remark}
On the other hand,
similarly to Theorem~\ref{Theorem:3.3}~(i),
we have:
\begin{theorem}
\label{Theorem:4.3}
Assume $\tilde{r}_1^*>0$ and $\tilde{r}_2^*>0$ .
Let $(u,v)$ be the minimal solution of \eqref{eq:4.1} in $\Omega\times(0,T)$, where $0<T<\infty$,
such that
$$
\limsup_{t\to T}\,\left[\|u(t)\|_{L^\infty(\Omega)}+\|v(t)\|_{L^\infty(\Omega)}\right]=\infty.
$$
Let $\ell_*\ge 1$ be such that $\ell_*\min\{\tilde{r}_1^*,\tilde{r}_2^*\}\ge 1$.
Then, for any $r_1\in(\ell_*r_1^*,\infty]$ and $r_2\in(\ell_*r_2^*,\infty]$,
there exists a positive constant $c_1$ such that
$$
\liminf_{t\to T}\,\left\{(T-t)^{\frac{N}{2}(\frac{1}{r_1^*}-\frac{1}{r_1})}|||u(t)|||_{r_1,\rho(t)}
+(T-t)^{\frac{N}{2}(\frac{1}{r_2^*}-\frac{1}{r_2})}|||v(t)|||_{r_2,\rho(t)}\right\}\ge c_1,
$$
where $\rho(t)=\sqrt{T-t}$. In particular,
\begin{equation}
\label{eq:4.4}
\liminf_{t\to T}\,\left\{(T-t)^{\frac{1-q_2+p_2}{\delta}}||u(t)||_{L^\infty(\Omega)}
+(T-t)^{\frac{1-p_1+q_1}{\delta}}||v(t)||_{L^\infty(\Omega)}\right\}\ge c_1.
\end{equation}
\end{theorem}
For upper estimates on the blow-up rate of the solutions of \eqref{eq:4.1},
see e.g., \cite[Theorem~1.1]{R}, which shows that
the lower estimate on the blow-up rate \eqref{eq:4.4} is optimal.
\begin{remark}
\label{Remark:4.2}
Similarly to Subsection~{\rm 3.2},
we can apply the arguments in this subsection to
the $k$-component nonlinear parabolic system
$$
\left\{
\begin{array}{ll}
{\displaystyle \partial_t u_i=\Delta u_i+\prod_{j=1}^ku_{j}^{p_{i,j}}},\qquad & x\in\Omega,\,\,\,t>0,\vspace{5pt}\\
u_i=0, & x\in\partial\Omega,\,\,\,t>0,\vspace{5pt}\\
u_i(x,0)=u_{i,0}(x), & x\in\Omega,
\end{array}
\right.
$$
where $i\in\{1,\cdots,k\}$, $p_{i,j}\ge 0$ $(j=1,\cdots,k)$
and $\{u_{i,0}\}$ are nonnegative and locally integrable functions in $\Omega$.
We leave the details to the reader.
\end{remark}
\subsection{Strongly coupled exponential nonlinearities}
Consider the parabolic system with strongly coupled exponential nonlinearities
\begin{equation}
\label{eq:4.5}
\left\{
\begin{array}{ll}
\partial_t u=\Delta u+e^{p_1 u}e^{p_2v},\qquad & x\in{\bf R}^N,\,\,\,t>0,\vspace{5pt}\\
\partial_t v=\Delta v+e^{q_1 u}e^{q_2 v}, & x\in{\bf R}^N,\,\,\,t>0,\vspace{5pt}\\
(u(x,0), v(x,0))=(u_0(x),v_0(x)),\qquad & x\in{\bf R}^N,
\end{array}
\right.
\end{equation}
where $p_i\ge 0$, $q_i\ge 0$ $(i=1,2)$
and both $u_0$ and $v_0$ are locally integrable functions in ${\bf R}^N$.
Set $\tilde{u}:=e^u$ and $\tilde{v}=e^v$. Then $(\tilde{u},\tilde{v})$ satisfies
$$
\left\{
\begin{array}{ll}
\partial_t\tilde{u}=\Delta\tilde{u}+\tilde{u}^{p_1+1}\tilde{v}^{p_2}
-\displaystyle{\frac{|\nabla \tilde{u}|^2}{\tilde{u}}},
\qquad & x\in{\bf R}^N,\,\,\,t>0,\vspace{8pt}\\
\partial_t\tilde{v}=\Delta\tilde{v}+\tilde{u}^{q_1}\tilde{v}^{q_2+1}
-\displaystyle{\frac{|\nabla \tilde{v}|^2}{\tilde{v}}}, & x\in{\bf R}^N,\,\,\,t>0,\vspace{8pt}\\
(\tilde{u}(x,0), \tilde{v}(x,0))=(e^{u_0(x)},e^{v_0(x)}),\qquad & x\in{\bf R}^N.
\end{array}
\right.
$$
Similarly to Lemma~\ref{Lemma:2.2},
we see that problem~\eqref{eq:4.5} possesses a solution $(u,v)$ in ${\bf R}^N\times(0,T)$,
where $0<T\le\infty$, if the problem
$$
\left\{
\begin{array}{ll}
\partial_t\hat{u}=\Delta\hat{u}+\hat{u}^{p_1+1}\tilde{v}^{p_2},
\qquad & x\in{\bf R}^N,\,\,\,t>0,\vspace{5pt}\\
\partial_t\hat{v}=\Delta\hat{v}+\hat{u}^{q_1}\hat{v}^{q_2+1},
 & x\in{\bf R}^N,\,\,\,t>0,\vspace{5pt}\\
(\hat{u}(x,0), \hat{v}(x,0))=(e^{u_0(x)},e^{v_0(x)}),\qquad & x\in{\bf R}^N
\end{array}
\right.
$$
possesses a solution $(\hat{u},\hat{v})$ in ${\bf R}^N\times(0,T)$.
Then we can apply the arguments in Subsection~4.1
and obtain sufficient conditions for the existence of the solutions of \eqref{eq:4.5}.
We leave the details to the reader again.
\,\\

\noindent
{\bf Acknowledgements.}
The first author was supported
by the Grant-in-Aid for for Scientific Research (A)(No.~15H02058),
from Japan Society for the Promotion of Science.
The second author was supported by the Grant-in-Aid for Young Scientists (B)
(No.~24740107)
from Japan Society for the Promotion of Science
and by the JSPS Program for Advancing Strategic International Networks
to Accelerate the Circulation of Talented Researchers
``Mathematical Science of Symmetry, Topology and Moduli,
Evolution of International Research Network based on OCAMI".
The third author was partially supported by WCMCS, Warsaw.

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A: The first bus of Choice Tours and Travels leaves at 11:30 from Pune.
Q-5). When does Choice Tours and Travels last bus depart from Pune ?
A: The last bus of Choice Tours and Travels departs at 23:45 from Pune.
Q-6). What are Choice Tours and Travels bus ticket fares from Pune to Latur ?
A: The starting bus fare of Choice Tours and Travels is 300. The maximum bus ticket fare of Choice Tours and Travels is 300.
Keeping your accessibility and satisfaction of road journey in mind, Choice Tours and Travels offers a number of buses throughout day and night. It also grants great convenience for any kind of commuters like business or leisure travellers . Pune to Latur is one of the major functioning routes of Choice Tours and Travels. The skilled drivers and staff ensures complete comfort of its passengers.
Choice Tours and Travels runs several types of buses from Pune to Latur. Generally, AC/Non AC sleeper buses are prefered during night travel. Besides, travelers may opt for AC/Non AC Semi sleeper and AC/Non AC Seater buses while commuting during day hours. All categories of Choice Tours and Travels buses ensure a good travel experience.


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In a slight change to our usual series on design careers, in this episode we talk about the compensation packages we have been offered at varying stages of our careers. How can you negotiate to get the most out of your compensation package, and what can be included? Find out how we have used our compensation packages to our benefit.


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Mrs. Doubtfire (1993)
October 6, 2014 December 30, 2014 / Sally
A few years ago Ben and I were talking and Mrs. Doubtfire somehow came up in our conversation. I was amazed to learn that Ben did not own this movie. He loves Robin Williams, and I always thought it was one of his best family friendly comedies. Ben then responded, "I hate that movie," which seemed even more perplexing. How could anyone hate Mrs. Doubtfire? Turns out this film was released around the time Ben's parents were divorcing, and he had a strong memory of crying in the theater while watching it. Suddenly his dislike for this movie became very obvious, and I felt like a jerk for not putting two and two together sooner.
It's strange, but I had kind of forgotten that Mrs. Doubtfire is about divorce. I've not seen the movie all the way through since I was a kid. All of my memories of it involve Robin Williams dancing with a vacuum cleaner and smashing a pie into his face. I suppose if I was pushed to explain the plot I would have remembered that the Williams is pretending to be a female British nanny in order to spend time with his children, but I honestly never gave it that much thought. I am fortunate that divorce has not affected my life in any significant way. To me, this is just a silly movie about a cross-dressing man, but I totally get that this would be depressing to watch if you were a kid (or adult) actually going through a similar situation.
While in many ways this is still a very silly movie, I can appreciate the serious side of the story as an adult. I can imagine that most parents would do whatever was necessary to remain in their children's lives after a divorce. The emotions displayed by Williams and Sally Field, who plays his ex-wife, feel incredibly real and add to the raw emotion of the situation. I felt that the film makes it clear that they are both to blame for the divorce, and are both failing their children in different ways. It's clear by the end that both Mom and Dad are learning to be better parents, and figuring out how to co-parent in a way that puts the kids' needs first. (If only every split were this amicable, right?) I also appreciated that the parents don't get back together in the end. It's a nice bit of realism in what is otherwise a rather unrealistic story.
Robin Williams of course steals the show, as he did in just about every project he was a part of. He has some great moments of ad-libbing, but also excels in the scripted portions of the story. It's amazing how he so perfectly embodies his self-created alter ego (even if I don't believe that he'd actually be able to fool his ex-wife or children). Williams also knows exactly how to play each scene, bringing heartbreaking seriousness to the scenes that need them.
Rating: B+
This was the second of two films that Sally and I decided to watch early after Robin Williams died suddenly on August 11 2014. While the first film, Hook, was a bittersweet film for me to watch because I had really enjoyed it growing up, Mrs Doubtfire was a film that never really grabbed me as a child. The story focuses on divorce and it was released at the same time my own parents were splitting up. This movie was far too real for a child going through divorce to deal with, I remember crying when I watched it the first time in theatres with my father. Watching it again, when my parent's divorce is not so raw, I can definitely appreciate the comedy more than I used to, but it still brings back more memories than I'd like to remember.
Williams plays Daniel Hillard, a perpetually out of work voice actor married to a workaholic architect, Miranda (Sally Field), with three kids, Lydia (Lisa Jakub), Chris (Matthew Lawrence) and Natalie (Mara Wilson). To celebrate his son's birthday, Daniel organises a large party, going against the wishes of his wife. When she returns to find the home a huge mess, it is the final straw for their marriage and she announces she wants a divorce. After a brief time of separation, Miranda decides she needs a nanny and advertises for one. Daniel sees this as an opportunity to spend some more time with his kids as he has only been allowed supervised visitation rights by the divorce court. Daniel transforms himself into an old woman, perfect for the role of housekeeper in the Hillard household. Can Daniel keep up the charade long enough to win his kids back or will it all blow up in his face eventually?
One of the things I like about this film is that it is realistic. There is never any talk of Daniel wanting to win back his wife. This story could so easily have fallen into that romantic comedy trap as Robin Williams attempted to help his wife remember what she loved about him and get back together. Daniel's motivation is only to see his children. This was a more realistic scenario as a lot of marriages fail and this movie reflects that really well. I am sure there would have been studio pressure to have the extra happy ending, but the fact the resisted this makes the film seem more real. Which is saying something when the plot involves a middle aged man man cross-dressing as an old Scottish woman.
As with most Robin Williams films, his ad-libbing is the highlight here. The man is able to turn it up to 11 whenever he needs too, and does it with great effect throughout the film. His scenes as Mrs Doubtfire are hilarious at times. I can't imagine how difficult it must have been for the young cast to keep a straight face, let alone the more experienced actors like Sally Field. The movie is very funny, and I'm sure if it didn't hit quite so close to home, I'd have enjoyed it so much more.
Rated B
comedy, M
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The company's assembly sites are keeping waste out of landfills, and its suppliers are moving toward renewable energy.
Apple doesn't talk about its supply chain very often, but the company does love to tout its environmental efforts. On Tuesday, Apple announced that its supplier Lens Technology has committed to using 100 percent renewable energy to manufacture glass for Apple, by the end of 2018.
Lens Technology is expected to use wind power to reach this goal. Currently, it produces glass for Apple at two factories in Changsha, Human province, in southern China. By using clean wind power, the firm will avoid releasing nearly 450,000 metric tons of carbon dioxide each year.
Apple also announced that all 14 of its final assembly sites in China are compliant with UL's Zero Waste to Landfill validation. That program make sure that all waste from a manufacturing plant is kept out of landfills through reuse, recycling, composting, or "converting into energy" (aka incinerating) when necessary. Apple says that since it began this program in January 2015, 140,000 metric tons of waste have been spared from landfills.
Two assembly sites run by Apple's contractor Foxconn, in Guanlan and Taiyuan, were the first in China to receive the Zero Waste to Landfill certification. Additionally, Foxconn committed last October to constructing 400 megawatts of solar by 2018, and Apple says it's "well on its way" to the first 80 megawatts of that committment. Apple already powers 100 percent of its own operations in China with renewable energy.
You can read more about Apple's efforts to protect the environment on Apple.com/environment.


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Home » baseball player » Russ Ford Net Worth 2018: What is this baseball player worth?
Russ Ford Net Worth 2018: What is this baseball player worth?
Russ Ford was a pro baseball player who played Pitcher. Ford was born on April 25, 1883, in Brandon, Manitoba. Ford died on January 24, 1960, in Rockingham, North Carolina. This page will take a closer look at Russ Ford's net worth.
Russ Ford Career, Earnings
Ford batted Right and threw Right. Ford debuted in the MLB on April 28, 1909 for the New York Highlanders. In all, Ford played for the New York Yankees, and Buffalo Buffeds. Ford's career ended with the Buffalo Blues in 1915.
Some of Ford's most prominent statistics in the MLB included a Win-loss record (pitching) stat of 99-71, a Earned run average stat of 2.59, and a Strikeouts stat of 710.
Russ Ford Net Worth 2018
Baseball salaries can range widely. In Major League Baseball, the median pay is near $3 million every year. Top MLB players can get $25 million or more every year, and lower paid players make $1 million or less. Outside the MLB, most contracts pay less than $10,000 a year.
Russ Ford net worth: baseball salary distribution
So what was baseball player Russ Ford's net worth at the time of death? Our estimate for Russ Ford's net worth at death is:
Want to see some related net worth articles? Check out these: Clarence Fieber, Val Heim, Edwin Maysonet, Edgar Olmos, Don Pavletich, Ox Eckhardt, Terry Fox, Sean Conroy, John Montague, Big Jeff Pfeffer, and Ray Blemker.


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California and China Strengthen Cleantech Ties as Trade War Smolders
CHENGDU, CHINA — At least a dozen cranes are visible from the Holiday Inn at Tianfu New Area, a state designated development zone on the outskirts of Chengdu, the capital of Sichuan Province.
The Tianfu New Area is so new, in fact, that the streets and sidewalks below the towering construction equipment are virtually empty.
As Beijing, Shanghai and Shenzhen have established themselves as epicenters for high-tech investment and innovation, the Chinese government has shifted its focus to building out a competitive technology ecosystem in this cloudy inland metropolis.
Chengdu is already home to several large international corporations, including Dell and IBM. The Chinese tech giant Huawei operates a research center in Chengdu, and recently rolled out multi-dimensional 5G network across the city.
Chengdu's Tianfu New Area was approved in 2014 to serve as a model for sustainable urban development and a hub for strategic new industries. Now, it is where key stakeholders are concentrating efforts to expand their work on the "Energy Internet," or the deep integration of energy and information technology, which was identified as a strategic focus area in China's 13th Five-Year Plan.
In June, a group of prominent companies, government leaders, researchers and startups from both sides of the Pacific gathered at the Tianfu New Area convention center for the first annual Energy Internet Innovation and Entrepreneurship Summit.
"The Energy Internet has become a very important part of our strategy for power [sector] development, the integration of power architecture to increase the percentage of renewable energy in our energy mix, as well as promote the efficient utilization of energy," Yinong Zhao, deputy director of the Department of Power at China's National Energy Administration, said in his opening remarks at the summit, communicated via translator.
He added that in today's globalized energy economy "strengthening dialogue is the only way to promote a win-win future."
The first day of the conference also marked the launch of Tsinghua University's Energy Internet Innovation and Entrepreneurship Center, a newly built research facility and co-working space equipped with a live power grid simulation testing system. The center — operated in partnership with the Oakland, California-based cleantech accelerator New Energy Nexus — also serves as launch pad for energy startups looking to do business in China.
The Energy Internet Summit in Chengdu, China.
Despite the ongoing U.S.-China trade war, California policy and business leaders have forged ahead with efforts to collaborate with China on the advancement of low-carbon energy technologies considered both financially lucrative and critical to mitigating the worst impacts of climate change.
U.S.-China relations have been on the rocks for over a year, with both countries imposing tariffs on billions of dollars worth of each other's goods.
Top U.S. officials held a phone call with China's lead trade negotiator this week, and both parties said they remain committed to reaching a "phase one" trade agreement. But tensions remain high as past talks have ended in near-collapse.
"Obviously it's a period of great tension right now between the U.S. and China," said David Hochschild, chair of the California Energy Commission (CEC), in an interview over breakfast in Chengdu. "That's why it's more important than ever that we're here representing California, because we want a strong relationship and a strong partnership."
As economic and technology development powerhouses, California and China have a unique opportunity to partner in building a clean energy future at home and abroad, Hochschild said.
"I think the ability to scale and innovate is incredibly strong in China and California," he said. "I actually think it's the most important strategic relationship on energy that there is right now."
Chinese energy companies visit the Bay Area
Earlier this month, New Energy Nexus hosted a delegation of Chinese energy industry professionals on a tour of the Bay Area's clean energy sector, building upon relationships established in Chengdu.
The Sichuan Energy Internet Research Institute at Tsinghua University.
"The whole idea is to introduce Chinese clean energy companies to California's cleantech ecosystem, as we have a very unique one, and to encourage and inspire the same type of ecosystem to grow within China," said Andrew Chang, director of the New Energy Nexus China program out of Shanghai.
Continue reading the full story on cleantech ties between California and China on Greentech Media.