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\fancyhead[CE]{M. Habibi Vosta Kolaei and S. Azami} 
\fancyhead[CO]{The first eigenvalue of $\left(p,q\right)$-Laplacian and rescaled Yamabe flow}



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{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
\vspace*{9mm}

\begin{center}

{\Large \bf 
Evolution of the first eigenvalue of the weighted $(p,q)$-Laplacian system under rescaled Yamabe flow\\}



\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}

{\bf Mohammad Javad Habibi Vosta Kolaei$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}\vspace*{-2mm}\\
\vspace{2mm} {\small  Imam Khomeini international university} \vspace{2mm}}

{\bf  Shahroud Azami \vspace*{-2mm}\\
\vspace{2mm} {\small    Imam Khomeini international university} \vspace{2mm}}

\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.}Consider the triple $ \left(M, g, d\mu\right)$ as a smooth metric measure space and $ M $ is an $n$-dimensional compact Riemannian manifold without boundary, also $d\mu = e^{-f(x)}dV$ is a weighted measure. We are going to investigate the evolution problem for the first eigenvalue of the weighted $\left(p, q\right)$-Laplacian system along the rescaled Yamabe flow and we hope to find some monotonic quantities.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 53C44, 53C21, 58C40

\noindent{\bf Keywords and Phrases:} Yamabe flow, $(p,q)$-Laplacian system, Eigenvalue
\end{quotation}}

\section{Introduction}
Consider $\left(M, g\right)$ as a compact $n$-dimensional manifold without boundary, the \linebreak Yamabe problem which was studied first by Yamabe in \cite{yam}, is to find a metric $g$ conformal to $g_{0}$ such that it's scalar curvature $R_{g}$ is constant. Generally two metrics $g$ and $g_{0}$ were called conformal if $ g= e^{-2u}g_{0}$ where $u$ is positive and smooth function in $M$. In special case if we write $g= u_{\frac{4}{n-2}}g_{0}$, then the scalar curvature $R_{g}$ of $g$ can be written as
\begin{align}\label{yek}
R_{g} = u^{-\frac{n+2}{n-2}} \Delta_{g_{0}}u + R_{g_{0}}u.
\end{align}
Therefore, the Yamabe problem is to solve (\ref{yek}) such that $R_{g}$ is constant. The above equation solved by Trudinger \cite{tru}, Aubin \cite{Au}, and Schoen \cite{sch}. Yamabe flow was introduced by Hamilton in \cite{ha} for the first time. The Yamabe flow defined as the evolution of the metric $g=g(t)$
\begin{align}\label{do}
 \frac{\partial}{\partial t} g_{ij} = -R g_{ij}, \quad g(0)=g_{0}
\end{align}
and normalized Yamabe flow also define as well as
\begin{align}\label{se}
 \frac{\partial}{\partial t} g_{ij} = -\left(R-r\right)g_{ij}, \quad g(0)=g_{0}
\end{align}
where $ R $ is the scalar curvature. Also $ r=\frac{\int_{M} R dV}{\int_{M} dV}$ is the average of the scalar curvature of the Riemannian metric $g$. \\
First of all, Schwetlick and Struwe in \cite{sch1} proved the convergence of the Yamabe flow for the case when $ 3 \leq n \leq 5 $ with the assumption that the initial metric has large energy. Finally Brandle in \cite{Brandle} has shown that the Yamabe flow convergence to a metric with constant scalar curvature.\\
In this paper, we are going to try to find some evolution equations and  some \linebreak monotonic quantities  of rescaled Yamabe flow, coupled with harmonic flow which is defined as
\begin{equation}\label{char}
\left\{
\begin{array}{lr}
 \frac{\partial}{\partial t} g_{ij} = -\left( R - s(t) \right)g_{ij} \quad g|_{t=t_{0}}=g_{0},\\
 \frac{\partial f}{\partial t}= \Delta f, \quad f(0, x)=f_{0}(x),
 \end{array} \right.
 \end{equation}
where $s(t)$ is constant only depended on time variable $t$, easily we can see the system (\ref{char}) in different cases gives us systems (\ref{do}) or (\ref{se}).\\
It has been known before that there is a one-to-one relationship between Yamabe flow (\ref{do}), and rescaled Yamabe flow (\ref{char}), in which if we consider $g(t)$ as a solution for the flow (\ref{do}), then we can find the function $\psi(t)$ as
\begin{align*}
\psi(t)= \left( 1-\int_{0}^{t} s(\nu)d\nu \right)^{-1},
\end{align*}
and also $\bar{t}=\int_{0}^{t}\psi(\nu)d\nu$. In this case $ \bar{g}(\bar{t})=\psi(t)g(t)$ will be the solution for the rescaled Yamabe flow (\ref{char}).\\
Let $ u:M \longrightarrow \mathbb{R} \, , u \in W^{1,p}_{0}\left(M\right)$ where $  W^{1,p}_{0}\left(M\right) $ is Sobolev space, for $ p \in [ 1, \infty ) $ we have seen before the introduction of  $p$-Laplacian of $u$ as below
$$ \Delta_{p} u = div( |\nabla u|^{p-2} \nabla u ) = | \nabla u |^{ p-2} \Delta u + (p-2)|\nabla u|^{p-4} ( Hess \,u)( \nabla u, \nabla u ), $$
where
$$ (Hess\, u )(X,Y) = \nabla (\nabla u)(X,Y) = X.(Y.u)-( \nabla_{X} Y).u, \qquad X,Y \in \chi(M).$$
Also weighted $p$-Laplacian can be introduced  as
$$ \Delta_{p,f} u = e^{f} div \left( e^{-f} | \nabla u |^{p-2} \nabla u \right) = \Delta_{p} u - | \nabla u |^{p-2} \nabla f . \nabla u, $$
where $ p \in [1, \infty ) $ and $ u $ is any smooth function on $ M $. \\
Now consider $ \left( M^{n}, g\right) $ as a closed Riemannian manifold we are going to define weighted $(p,q)$-Laplacian system as
\begin{equation}\label{pang}
\left\{
\begin{array}{lr}
\Delta_{p,f} u = - \lambda |u|^\alpha |v|^\beta v   & \text{in M }, \\
\Delta_{q,f} v = - \lambda |u|^\alpha |v|^\beta u  & \text{in M }, \\
 u=v=0  &  \text{on}\  \partial \text{M},
\end{array} \right.
\end{equation}
where $ p>1$ , $ q>1$ and $ \alpha , \beta $ are real numbers such that
$$ \alpha >0 ,\, \beta>0 , \qquad \frac{\alpha +1}{p} + \frac{\beta +1}{q} = 1. $$
 In this case $\lambda$ is an eigenvalue of such system.\\
  A first positive eigenvalue of a system (\ref{pang}) is defined as
 $$ inf \lbrace A(u,v) : (u,v) \in W^{1,p}_{0}(M) \times W^{1,q}_{0}(M), B(u,v)=1 \rbrace, $$
 where the pair of $ (u,v)$ is the eigenfunctions corresponding to eigenvalue $\lambda$ and
 $$A(u,v)= \frac{\alpha+1}{p} \int_{M} |\nabla u |^{p} d\mu +\frac{\beta+1}{q} \int_{M} |\nabla v|^{q} d\mu ,$$
 $$ B(u,v) = \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu. $$
The lemma below gives us the continuousness of $A(u,v,t)$ in $C^{1}$-topology which has been mentioned before.
\begin{lemma}
If $g_{1}$ and $g_{2}$ are two metrics on Riemannian manifold $M^{n} $ which satisfy $ ( 1+ \varepsilon )^{-1} < g_{2} < (1+ \varepsilon ) g_{1}$ then for any $ p \geq q > 1$, we have
$$ \lambda (g_{2}) - \lambda (g_{1}) \leq \left( \left( 1+ \varepsilon \right)^{\frac{p+n}{2}} - \left( 1+ \varepsilon\right) ^{-\frac{n}{2}} \right) \lambda(g_{1}), $$
which means, $ \lambda (t) $ is a continues function respect to $t$-variable.
\end{lemma}
\begin{proof}
In local coordinate we have $d\mu = \sqrt{det g} dx^{1} \wedge ... \wedge dx^{n}$, therefore
\begin{align*}
(1+\varepsilon )^{-\frac{n}{2}} d\mu_{g_{1}} < d\mu_{g_{2}} < ( 1+\varepsilon)^{\frac{n}{2}} d\mu_{g_{1}}.
\end{align*}
Assume that
\begin{align*}
G(g,u,v) = \frac{\alpha +1}{p} \int_{M} |\nabla u|^{p}_{g} d\mu_{g} + \frac{\beta +1}{q} \int_{M} |\nabla v |^{q}_{g} d\mu_{g},
\end{align*}
then it implies
\begin{align*}
&\int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{1}} G(g_{2},u,v) - \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{2}} G(g_{1},u,v)\\
&= \frac{\alpha +1}{p} \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{1}} \left( \int_{M} |\nabla u|^{p}_{g_{2}} d\mu_{g_{2}} - \int_{M} |\nabla u|^{p}_{g_{1}} d\mu_{g_{1}} \right)\\
&+ \frac{\alpha +1}{p} \left( \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{1}} - \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{2}} \right) \int_{M} |\nabla u |^{p}_{g_{1}} d\mu_{g_{1}}\\
&+ \frac{\beta +1}{q} \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{1}} \left( \int_{M} |\nabla v|^{q}_{g_{2}} d\mu_{g_{2}} - \int_{M} |\nabla v|^{q}_{g_{1}} d\mu_{g_{1}} \right)\\
&+ \frac{\beta +1}{q}\left( \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{1}} - \int_{M} |u|^{\alpha}|v|^{\beta}uv d\mu_{g_{2}} \right) \int_{M} |\nabla v|^{q}_{g_{1}} d\mu_{g_{1}}, \\
\end{align*}
then by applying the lemma's assumption we get
\begin{align*}
&\int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{1}} G(g_{2},u,v) - \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{2}} G(g_{1},u,v)\\
&\leq \frac{\alpha +1}{p} \left( (1+\varepsilon)^{\frac{p+n}{2}} - ( 1+ \varepsilon)^{-\frac{n}{2}} \right) \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{1}} \int_{M} |\nabla u|^{p}_{g_{1}} d\mu_{g_{1}}\\
&+ \frac{\beta +1}{q} \left( (1+\varepsilon)^{\frac{q+n}{2}} - ( 1+ \varepsilon)^{-\frac{n}{2}} \right) \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{1}} \int_{M} |\nabla v|^{q}_{g_{1}} d\mu_{g_{1}}\\
&\leq \left( (1+\varepsilon)^{\frac{p+n}{2}} - ( 1+\varepsilon)^{-\frac{n}{2}} \right) G(g_{1},u,v) \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu_{g_{1}}.
\end{align*}
Since the eigenfunctions corresponding to $ \lambda(t) $ are normalized, then we have
\begin{align*}
\lambda(g_{2}) - \lambda(g_{1}) \leq \left( ( 1+\varepsilon)^{\frac{p+n}{2}} - ( 1+\varepsilon)^{-\frac{n}{2}} \right) \lambda(g_{1}).
\end{align*}
\end{proof}
In case which is not assumed that $ \lambda(t)$ is $C^{1}$-differentiable under (\ref{char}) in the interval $ [0,T)$, the first non-zero eigenvalue of weighted $(p,q)$-Laplacian system is not known to be $C^{1}$-differentiable anymore. For this problem we are going to apply techniques of Cao \cite{cao} and Wu \cite{wang} to study the evolution and monotonicity of $ \lambda(t)$, where $u$ and $v$ are supposed to be smooth.\\
Consider $ \left( M^{n}, g(t) \right) $ as a solution of the Yamabe flow on the smooth manifold $( M^{n}, g_{0} )$ in the interval $[0,T)$ then $A(u,v)$ defines the evolution of an eigenvalue of the system (\ref{pang}), under the variation of $g(t)$ where for the eigenfunctions associated to $\lambda(t)$ we have
\begin{align}\label{haft}
 \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu &= 1, \, \int_{M} |u|^{\alpha} |v|^{\beta} u d\mu = 0, \\
 \int_{M} |u|^{\alpha} |v|^{\beta} v d\mu &= 0. \nonumber
\end{align}
First of all, let $ t_{0} \in [0,T)$, $ (u_{0}, v_{0}) = ( u(t_{0}), v(t_{0}))$ be the eigenfunctions for the eigenvalue $ \lambda(t_{0})$ of weighted $(p,q)$-Laplacian system (\ref{pang}). We consider the smooth functions as below
$$ h(t) = u_{0} \Big[\frac{det[g_{ij}(t)]}{det [g_{ij}(t_{0})]}\Big]^{\frac{1}{2(\alpha + \beta +1)}}, \, l(t)= v_{0} \Big[\frac{det[g_{ij}(t)]}{det [g_{ij}(t_{0})]}\Big]^{\frac{1}{2(\alpha + \beta +1)}},$$
along the Yamabe flow. Now let
$$ u(t) = \frac{h(t)}{\left( \int_{M} |h(t)|^{\alpha} | l(t) |^{\beta} h(t) l(t) d\mu \right)^{\frac{1}{p}}}, \quad v(t) = \frac{l(t)}{\left( \int_{M} |h(t)|^{\alpha} |l(t)|^{\beta} h(t) l(t) d\mu \right)^{\frac{1}{q}}},$$
where $u(t)$ and $v(t)$ are smooth functions under the Yamabe flow and also satisfy in (\ref{haft}), and at time $t_{0}$, $( u(t_{0}), v(t_{0}))$ is the eigenfunctions for $\lambda(t_{0})$ of weighted \linebreak $(p,q)$-Laplacian system (\ref{pang}), $ \lambda(t_{0}) = A( u(t_{0}), v(t_{0}))$ and if $ \left( M^{n}, g(t), f \right)$ be a solution of the (\ref{char}) on the smooth manifold $ \left( M^{n}, g_{0}, f_{0} \right)$ in the interval $ [0,T) $ then we can write the smooth eigenvalue function $\lambda(u,v,t)$ along the flow (\ref{char}) as below
\begin{equation}\label{shesh}
\lambda(u,v,t)= \frac{\alpha+1}{p} \int_{M} |\nabla u |^{p} d\mu +\frac{\beta+1}{q} \int_{M} |\nabla v|^{q} d\mu,
\end{equation}
where
\begin{align*}
\lambda(u,v,t)|_{t=t_{0}} = \lambda(t_{0}).
\end{align*}
In recent years, studying the evolution equations under geometric flows became a hot topic in understanding the geometry of manifolds. Perelman in \cite{per}, studying the functional:
\begin{align*}
F\left( g(t),f(t)\right)=\int_{M}\left( R+|\nabla f|^{2} \right)e^{-f}dV,
\end{align*}
and showed that this functional is non-decreasing along the Ricci flow coupled to a backward heat-type equation.
There are some other works in variational formulas. Second author in \cite{S.Azami} has studied the eigenvalues problem of $p$-Laplace operator acting on the space of functions under the Yamabe flow, also P. Ho in \cite{Pak} studied the first non-zero eigenvalue of Laplacian of $g_{0}$ with negative scalar curvature in terms of conformal Yamabe metrics, also he has worked on some other geometric operators in case of compact Riemannian manifolds. Some other works have done on evolution of eigenvalues of geometric operators along geometric flows \cite{abol, song, raz}.

 In this article we investigate the evolution problem for the weighted $(p,q)$-Laplacian system under the rescaled Yamabe flow (\ref{char})  and our main results will be classify as below;
\begin{theorem}\label{yet}
Consider $ \left( M, g(t), f(t), d\mu = e^{-f} dV \right) $,  $ t \in [ 0, T ) $ as a solution of the flow  (\ref{char}) on the smooth compact Riemannian manifold $\left( M^{n}, g_{0}, f_{0} \right) $ without boundary, and  $ s(t) > 0 $ is scalar function and also $ \Delta f - \gamma R \leq 0 $, $R \geq 0$ in $ M\times [ 0, T ) $. Suppose that $\lambda (t) $ denotes the evolution the first non-zero eigenvalue of the weighted $(p,q)$-Laplacian system then for $p\leq q$ the quantity
\begin{equation*}
\lambda(t)\left( \int_{0}^{t} -\rho(\nu) d\nu + \frac{1}{R_{min}(0)} \right)^{\frac{p-2\gamma}{2}} e^{\frac{q}{2}\int_{0}^{t} s(\nu) d\nu},
\end{equation*}
is increasing along the flow  (\ref{char})  where $ \rho(t) = e^{\int_{0}^{t} -s(\nu) d\nu}, \quad \gamma < \frac{p-2}{2}$ is constant and also $\tau$ is constant which is equal to $ \frac{1}{R_{min}(0)}$.
\end{theorem}

\begin{theorem}\label{dot}
Consider $\left( M, g_{0} \right)$ as a compact Riemannian manifold of dimension $ n \geq 3 $ without boundary in a case that $\max\limits_{M} R_{g_{0}} < 0 $, and $ g_{t}$, $t\in [0,T)$ is a Yamabe metric  which has same volume as $g_{0}$. If we denote the first eigenvalue of weighted $(p,q)$-Laplacian system under the  flow (\ref{char}) with $s(t)=r$, respect to $g_{0}$ and $g_{t}$ by $ \lambda_{1}(g_{0})$ and $ \lambda_{1}(g_{t})$, respectively, then we have
\begin{align*}
e^{-c} \lambda_{1}(g_{t}) \geq \lambda_{1}(g_{0}) \geq e^{c} \lambda_{1}(g_{t}),
\end{align*}
where
\begin{align*}
c= \left( 2n+2p-2r \right) \Big[ 1- \frac{\min\limits_{M} R_{g_{0}}}{\max\limits_{M} R_{g_{0}}}\Big] -r.
\end{align*}
\end{theorem}

\begin{theorem}\label{set}
Let $\left( M^{2}, g(t), f(t), d\mu \right), \quad t \in (0, T)$, be a solution of (\ref{char}) on the smooth  compact surface $\left( M^{2}, g_{0}, f_{0}\right)$ without boundary also we assume that $ p\geq q$. If $\lambda(t)$ denotes the evolution of the first eigenvalue of the weighted $(p, q)$-Laplacian system under the  flow (\ref{char}) with $s(t)=r$, then
\begin{itemize}
\item If $R<0$ and $ \Delta f - \gamma R \leq 0$ where $ \gamma < \frac{q-n}{2}$ then
\begin{align*}
ln\left(\lambda(t)\right) - \left(\frac{p-q}{2}-\gamma \right)rt + \frac{c}{r}\left(\frac{p}{2}-\gamma \right)e^{rt},
\end{align*}
is inceasing.
\item If $R<0$ and $ \Delta f - \gamma R \geq 0$ where $ \gamma < \frac{q-n}{2}$ then
\begin{align*}
ln\left( \lambda(t)\right) +\gamma rt -\left(\frac{q}{2}-\gamma \right)\frac{c}{r}e^{rt},
\end{align*}
is decreasing.
\item If $R>0$ and  $ \Delta f - \gamma R \leq 0$ where $ \gamma < \frac{q-n}{2}$   then
\begin{align*}
ln\left(\lambda(t) \right) + \frac{r}{2}pt + \frac{c}{r}\left(\frac{q}{2}-\gamma \right)e^{rt},
\end{align*}
is increasing.
\item If $R>0$ and $ \Delta f - \gamma R \geq 0$ where $ \gamma < \frac{q-n}{2}$  then
\begin{align*}
ln\left( \lambda(t) \right) - \left(\frac{p-q}{2} - \gamma \right)rt - \left(\frac{p}{2} - \gamma \right)\frac{c}{r}e^{rt},
\end{align*}
is decreasing,\\
\end{itemize}
where $c$ is constant and $ r=\frac{\int_{M}R dV}{\int_{M} dV}$ is the average of the scalar curvature.
\end{theorem}


In other sections we may add some more assumptions for more details.
\section{Variation of $\lambda (t)$}
In this section we are going to give some useful formulas of variation of $\lambda(t)$ along rescaled Yamabe flow. We start with the below proposition.

\begin{proposition}\label{hej}
Let $ \left( M^{n}, g(t), f(t)\right)$ be a solution of the (\ref{char}) on the smooth closed manifold $\left( M^{n},g_{0}, f_{0}\right)$. If $\lambda(t)$ denotes the evolution of the first non-zero eigenvalue under the  flow (\ref{char}), then we have
\begin{align*}
\frac{d}{dt} \lambda(u,v,t)|_{t=t_{0}} &=\frac{n}{2}\lambda(t_{0}) \int_{M} R |u|^{\alpha} |v|^{\beta} uv d\mu +\frac{ \left( \alpha +1 \right)}{2} \int_{M} (R-s(t)) |\nabla u |^{p} d\mu\\
&+\frac{\left( \beta +1 \right)}{2} \int_{M} (R-s(t)) |\nabla v |^{q} d\mu -\frac{\alpha +1}{p} \int_{M} |\nabla u |^{p} \big[ \Delta f + \frac{1}{2}nR \big] d\mu\\
&-\frac{\beta +1}{q} \int_{M} |\nabla v|^{q} \big[ \Delta f + \frac{1}{2}nR \big] d\mu.
\end{align*}
\end{proposition}
\begin{proof}
From what we explained before, $\lambda(u,v,t)$ is differentiable then by derivation from the formula (\ref{shesh}) respect to time variable $ t $, it satisfies
\begin{align}\label{sizdah}
\frac{d}{dt} \lambda(u,v,t)|_{t=t_{0}} &= \frac{\alpha +1}{2} \Big[ \int_{M} \Big\lbrace -g^{ij} g^{jk} \frac{\partial}{\partial t} (g_{lk}) \nabla_{i} u \nabla_{j} u + 2< \nabla u_{t}, \nabla u > \Big\rbrace |\nabla u |^{p-2} d\mu \Big]\\ \nonumber
&+\frac{\beta +1}{2} \Big[ \int_{M} \Big\lbrace -g^{ij} g^{jk} \frac{\partial}{\partial t} (g_{lk}) \nabla_{i} v \nabla_{j} v + 2< \nabla v_{t}, \nabla v > \Big\rbrace |\nabla v |^{q-2} d\mu \Big]\\ \nonumber
&+ \frac{\alpha +1}{p} \int_{M} |\nabla u|^{p} [ -f_{t} d\mu + \frac{1}{2} tr_{g} (\frac{\partial g}{\partial t}) d\mu ]\\ \nonumber
&+ \frac{\alpha +1}{q} \int_{M} |\nabla v|^{q} [ -f_{t} d\mu + \frac{1}{2} tr_{g} (\frac{\partial g}{\partial t}) d\mu ],
\end{align}
where $u_{t} = \frac{\partial u}{\partial t}$ and $ f_{t} = \Delta f $.
We can also calculate the term  $ \int_{M} < \nabla u_{t}, \nabla u > |\nabla u|^{p-2} d\mu $ as below
\begin{align*}
\int_{M} < \nabla u_{t}, \nabla u > |\nabla u|^{p-2} d\mu &= - \int_{M} u_{t} div \left( e^{-f} |\nabla u|^{p-2} \nabla u \right) dV,\\
&= -\int_{M} u_{t} e^{f} div \left( e^{-f} |\nabla u|^{p-2} \nabla u \right) d\mu = -\int_{M} u_{t} \Delta_{p,f} u d\mu , \\
&= -\int_{M} u_{t} \left( -\lambda |u|^{\alpha } |v|^{\beta} v d\mu \right) = \lambda \int_{M} |u|^{\alpha} |v|^{\beta} u_{t} v d\mu .
\end{align*}
Similarly we can also calculate
$$ \int_{M} < \nabla v_{t}, \nabla v > |\nabla v |^{q-2} d\mu = \lambda \int_{M} |u|^{\alpha} |v|^{\beta} uv_{t} d\mu . $$
It has been known that $ \int_{M} |u|^{\alpha} |v|^{\beta} uv d\mu =1 $, by derivation from both sides of this equation respect to time variable $ t $, we can see that
$$ \int_{M} [ (\alpha +1 ) |u|^{\alpha} |v|^{\beta} u_{t} v + (\beta +1 ) |u|^{\alpha} |v|^{\beta} u v_{t} ] d\mu + \int_{M} |u|^{\alpha} |v|^{\beta} u v \frac{1}{2} tr_{g} (- (R-s)g_{ij} ) d\mu =0, $$
which finally implies that
\begin{align}\label{chardah}
\left( \alpha +1 \right) \int_{M} < \nabla u_{t}, \nabla u > |\nabla u|^{p-2} d\mu &+ \left( \beta +1 \right) \int_{M} < \nabla v_{t}, \nabla v > |\nabla v|^{q-2} d\mu \\
&=\frac{n}{2} \lambda \int_{M} |u|^{\alpha} |v|^{\beta} uv (R-s) d\mu. \nonumber
\end{align}
Now by plugging the  flow (\ref{char}), into the formula (\ref{sizdah}), we have
\begin{align*}
\frac{d}{dt} \lambda (u,v,t) &= (\alpha +1) \int_{M} < \nabla u_{t}, \nabla u > |\nabla u |^{p-2} d\mu + (\beta +1) \int_{M} < \nabla v_{t}, \nabla v > |\nabla v |^{q-2} d\mu\\
&+ \frac{\left( \alpha +1 \right)}{2} \int_{M} (R-s(t)) |\nabla u |^{p} d\mu +\frac{\left( \beta +1 \right)}{2} \int_{M} (R-s(t)) |\nabla v |^{q} d\mu \\
&+\frac{\alpha +1}{p} \int_{M} |\nabla u |^{p} \big[ -\Delta f - \frac{1}{2}nR \big] d\mu +\frac{\beta +1}{q} \int_{M} |\nabla v|^{q} \big[ -\Delta f - \frac{1}{2}nR \big] d\mu\\
 &+ \frac{1}{2}ns(t)\lambda(t).
\end{align*}
which by replacing the equality (\ref{chardah}), into above equation implies what we looking for.
\end{proof}
\begin{remark}
In special case if we consider $ s(t) = r $ where $ r=\frac{\int_{M} R dV}{\int_{M} dV}$, it gives us the evolution formula under the normalized Yamabe flow (\ref{se}), as below
\begin{align*}
\frac{d}{dt} \lambda(u,v,t)|_{t=t_{0}} &=\frac{n}{2}\lambda(t_{0}) \int_{M}  R |u|^{\alpha} |v|^{\beta} uv d\mu + \frac{\left( \alpha +1 \right)}{2} \int_{M} (R-r) |\nabla u |^{p} d\mu\\
&+\frac{\left( \beta +1 \right)}{2} \int_{M} (R-r) |\nabla v |^{q} d\mu +\frac{\alpha +1}{p} \int_{M} |\nabla u |^{p} \big[ -\Delta f - \frac{1}{2}nR \big] d\mu\\
&+\frac{\beta +1}{q} \int_{M} |\nabla v|^{q} \big[ -\Delta f - \frac{1}{2}nR \big] d\mu.
\end{align*}
\end{remark}
Now we are going to give the proof of theorem \ref{yet} as \\
\begin{proof}
Under consideration $\Delta f - \gamma R \leq 0 $ where $\gamma < \frac{p-n}{2}$, we have
\begin{align}\label{hasht}
\frac{d}{dt} \lambda(u, v, t)|_{t=t_{0}} &\geq \frac{n}{2} \lambda(t_{0}) \int_{M}  R |u|^{\alpha} |v|^{\beta} uv d\mu\\
&+\frac{\left( \alpha +1 \right)}{p} \left( \frac{p}{2} - \left( \gamma + \frac{n}{2}\right) \right) \int_{M}  R |\nabla u |^{p} d\mu \nonumber\\
&+\frac{\left( \beta +1 \right)}{q} \left( \frac{q}{2} - \left( \gamma + \frac{n}{2} \right) \right) \int_{M}  R |\nabla v|^{q} d\mu \nonumber\\
&- \frac{q}{2}s(t_{0}) \lambda(t_{0}).\nonumber
\end{align}
Also the evolution of $ R $ under the flow (\ref{char})  is written as
\begin{align}\label{bistt}
\frac{\partial}{\partial t} R = ( n-1) \Delta R + R^{2} - Rs(t).
\end{align}
Since the solution to the ODE,  $ \frac{dy}{dt} =  y^{2} - s(t)y $ is
\begin{align*}
y(t) = \frac{\rho(t)}{\int_{0}^{t} - \rho(\nu) d\nu + \tau},
\end{align*}
where $\rho(t) = e^{\int_{0}^{t} -s(\nu) d\nu}, \quad y(0) = R_{min}(0)$ and $\tau$ is a constant equal to $ \frac{1}{R_{min}(0)}$, then
 by Maximum principle to (\ref{bistt}), we get $ R(x,t) \geq y(t) $, then by (\ref{hasht}) and $ p\leq q$ we have
\begin{align*}
\frac{d}{dt} \lambda (u,v,t)|_{t=t_{0}} \geq \lambda(u, v, t_{0}) \left( \frac{p-2\gamma}{2}y(t_{0}) - \frac{qs(t_{0})}{2} \right),
\end{align*}
which implies that in any sufficiently small neighborhood of $ t_{0}$ as $I$, we get \linebreak $ \frac{d}{dt} \lambda (u,v,t) \geq \lambda(u,v,t)\left( \frac{p-2\gamma}{2}y(t) - \frac{qs(t)}{2}\right) $  and also we have
\begin{align}\label{hevdah}
\lambda(u,v,t_{0}) = \lambda(t_{0}), \quad \lambda(u,v,t_{1}) \geq \lambda(t_{1}).
\end{align}
On the other hand, by integration from both sides,  it can be easily seen
\begin{align*}
ln\frac{\lambda(t_{0})}{\lambda(t_{1})} \geq \frac{\eta(t_{1})}{\eta(t_{0})},
\end{align*}
where
\begin{align*}
\eta(t) = \left( \int_{0}^{t} -\rho(\nu) d\nu + \frac{1}{R_{min}(0)} \right)^{\frac{p-2\gamma}{2}} e^{\frac{q}{2}\int_{0}^{t} s(\nu) d\nu}.
\end{align*}
Now since $ t_{1} < t_{0} $ and $ t_{0} $ is arbitrary thus the quantity
\begin{align*}
\lambda(t)\left( \int_{0}^{t} -\rho(\nu) d\nu + \frac{1}{R_{min}(0)} \right)^{\frac{p-2\gamma}{2}} e^{\frac{q}{2}\int_{0}^{t} s(\nu) d\nu},
\end{align*}
is increasing.
\end{proof}\\
And now, we prove  the  Theorem \ref{dot}.\\
\begin{proof}
It was known that if $ g\longrightarrow g_{\infty} $ as $ t \longrightarrow \infty $ under the  Yamabe flow (\ref{se}), in a case that $g_{\infty}$ is conformal to $ g_{0}$ and has constant negative scalar curvature, then we have
\begin{align*}
\frac{d}{dt} \left( \int_{M} dV_{g} \right) = \int_{M} \frac{\partial}{\partial t} \left( dV_{g} \right) = -\frac{n}{2} \int_{M} \left( R_{g} - r_{g} \right) dV_{g} = 0,
\end{align*}
in particular
\begin{align*}
\int_{M} dV_{g_{\infty}} = \int_{M} dV_{g_{0}}.
\end{align*}
On the other hand, $ R_{g_{t}} = \zeta^{\frac{4}{n-2}} R_{g_{\infty}}$ where we can take $\zeta$ to be $ \left( \frac{R_{g_{t}}}{R_{g_{\infty}}} \right)^{\frac{n-2}{4}} $. This implies that the metric $ \zeta^{\frac{4}{n-2}} g_{t}$ has scalar curvature being equal to
\begin{align*}
R_{\zeta^{\frac{4}{n-2}}g_{t}} = \zeta^{-\frac{4}{n-2}}R_{g_{t}} = R_{g_{\infty}}.
\end{align*}
By what proved in \cite{ka}, which says that if $ g_{1}$ and $g_{2}$ are two metrics conformal to $g_{0}$ such that $R_{g_{1}} = R_{g_{2}} <0$, then $g_{1} = g_{2}$. Therefore by what mention above we have
\begin{align*}
\int_{M} dV_{g_{0}} = \int_{M} dV_{g_{\infty}} = \int_{M} dV_{\zeta^{\frac{2n}{n-2}}g_{t}} = \zeta^{\frac{2n}{n-2}} \int_{M} dV_{g_{t}} = \zeta^{\frac{2n}{n-2}}\int_{M} dV_{g_{0}},
\end{align*}
where by assumption above this implies that $\zeta=1$ or equivalently $ g_{t} = g_{\infty}$.
Note that by \cite{Pak}, we obtain
\begin{align*}
\min\limits_{M}R_{g_{0}} \leq r_{g(t)} \leq \max\limits_{M}R_{g_{0}} \qquad t\geq 0.
\end{align*}
Also by  process of the proof of theorem 1.1, in \cite{Pak}, we conclude that
\begin{align*}
&\left( \max\limits_{M} R_{g_{0}} \right) \int_{0}^{s} \left( \max\limits_{M} R_{g(\nu)} - r_{g(s)} \right) d\nu \\
&\geq \left( r_{g(t)} - r_{g_{0}} \right) - \left(\max\limits_{M} R_{g_{0}} - \min\limits_{M} R_{g_{0}} \right) \\
& \geq -2\left( \max\limits_{M} R_{g_{0}} - \min\limits_{M} R_{g_{0}} \right).
\end{align*}
As $ t\longrightarrow \infty $, by what we mention above $ g(t) \longrightarrow g_{\infty} $, we get
\begin{align*}
-2\left( 1- \frac{\min\limits_{M} R_{g_{0}}}{\max\limits_{M} R_{g_{0}}} \right) \geq \int_{0}^{\infty} \left(\max\limits_{M} R_{g(\nu)} - r_{g(\nu)} \right) d\nu.
\end{align*}
Similarly
\begin{align*}
2\left( 1- \frac{\min\limits_{M}R_{g_{0}}}{\max\limits_{M}R_{g_{0}}} \right) \leq \int_{0}^{\infty} \left(\min\limits_{M} R_{g(\nu)} - r_{g(\nu)} \right) d\nu.
\end{align*}
Now by above results and proposition \ref{hej}, we finally obtain
\begin{align*}
ln\frac{\lambda_{1}(g_{t})}{\lambda_{1}(g_{0})} = ln\frac{\lambda_{1}(g_{\infty})}{\lambda_{1}(g_{0})} \geq \left( 2n + 2p - 2s \right) \big[ 1- \frac{\min\limits_{M}R_{g_{0}}}{\max\limits_{M}R_{g_{0}}} \big] -s.
\end{align*}
The inverse inequality holds in a similar way, so we prove what we were looking for.
\end{proof}
\subsection{Variation of $\lambda(t)$ under the normal Yamabe flow on the surface}
In this section we are going to give the proof of theorem \ref{set}.\\
\begin{proof}
We only give a proof for first section of theorem \ref{set}, the other sections follow similar process. First of all, we have to mention that if  $\left( M^{2}, g(t), f(t), d\mu \right)$ denotes the solution of the  flow (\ref{char}) with  $s(t)=r$, on the smooth Riemannian compact surface, then we can find some bounds for the scalar curvature tensor $R$ as below
\begin{itemize}
\item $ r<0; \quad r-ce^{rt} \leq R \leq r+ce^{rt}$,

\item $ r=0; \quad -\frac{c}{1+ct} \leq R \leq c$,

\item $ r>0; \quad -ce^{rt} \leq R \leq r+ ce^{rt}$,
\end{itemize}
where $c$ is constant and $r$ is as similar as we found in the normalized Yamabe flow (\ref{se}).\\
Now under $\Delta f \leq\gamma  R$, $R<0$, and $ p\geq q$, we have
\begin{align*}
\frac{d}{dt}\lambda(u, v, t)|_{t=t_{0}} \geq \lambda(t_{0}) \left( r\left( \frac{p-q}{2} - \gamma \right) -ce^{rt_{0}}\left( \frac{p}{2}-\gamma \right)\right),
\end{align*}
which implies that in any sufficient small neighborhood of $t_{0}$ as $ I=[t_{1}, t_{0}]$ we have
\begin{align*}
\lambda(u, v, t_{0}) = \lambda(t_{0}), \quad \lambda(u, v, t_{1}) \geq \lambda(t_{1}),
\end{align*}
where implies that
\begin{align*}
ln\left(\lambda(t_{0})\right) - \left( \frac{p-q}{2} -\gamma \right)rt_{0} + \frac{c}{r}\left(\frac{p}{2}-\gamma \right)e^{rt_{0}}  \geq ln\left(\lambda(t_{1})\right) - \left( \frac{p-q}{2} -\gamma \right)rt_{1} + \frac{c}{r}\left(\frac{p}{2}-\gamma \right)e^{rt_{1}},
\end{align*}
which means
\begin{align*}
ln\left(\lambda(t)\right) - \left( \frac{p-q}{2} -\gamma \right)rt + \frac{c}{r}\left(\frac{p}{2}-\gamma \right)e^{rt},
\end{align*}
is increasing.
\end{proof}

\section{ Homogeneous 3-manifolds}
Locally homogeneous $3$-manifolds have been contained into $9$ classes which are divided in two groups. The first is contained $H(3)$, $H(2) \times \mathbb{R}^{1}$ and $SO(3) \times \mathbb{R}^{1}$, and the other includes $\mathbb{R}^{3}$, SU(2), SL($2$, $\mathbb{R}$), $Heisenberg$, E(1, 1) and E(2), in which the second group are called Bianchi classes.\\
In this section we are going to give evolution of the first eigenvalue of the weighted $ ( p,q)$-Laplacian system (\ref{pang}), in a case of Bianchi classes.
\begin{remark}
Consider the evolution formula of $\lambda(t)$ under the  flow (\ref{char}), then in homogeneous condition we have
\begin{align*}
\frac{d}{dt} \lambda(u, v, t)|_{t=t_{0}} &= \frac{\alpha +1}{2} \left( R-s(t_{0}) \right) \int_{M} |\nabla u|^{p} d\mu + \frac{\beta +1}{2} \left(R - s(t_{0}) \right) \int_{M} |\nabla v|^{q} d\mu \\
&- \frac{\alpha +1}{p} \int_{M} \Delta f |\nabla u|^{p} d\mu - \frac{\beta +1}{q} \int_{M} \Delta f |\nabla v|^{q} d\mu.
\end{align*}
Now under consideration $\Delta f \leq R $ and $ p\geq q$ under the Yamabe flow (\ref{do}), where $s(t) =0$, we finally get
\begin{align}\label{bist}
\frac{d}{dt}\lambda(u,v,t)|_{t=t_{0}} &\geq R\left(\frac{p}{2}-1\right)\frac{\alpha +1}{p}\int_{M}|\nabla u|^{p} d\mu \\
&+R\left(\frac{q}{2}-1\right)\frac{\beta +1}{q}\int_{M}|\nabla v|^{q} d\mu. \nonumber
\end{align}
\end{remark}
First let us consider $g_{0}$ as a given metric in the Bianchi classes, \cite{mil} has provided before a frame $\lbrace X_{i} \rbrace_{i=1}^{3}$ in which both the Ricci tensors and metric are diagonalized and this property is preserved by Ricci flow. In this case, we consider the metric $g$ as
\begin{align*}
g(t)=A(t)\left(\theta_{1}\right)^{2} + B(t)\left(\theta_{2}\right)^{2} +C(t)\left(\theta_{3}\right)^{2},
\end{align*}
where $\lbrace \theta_{i} \rbrace_{i=1}^{3}$ is the frame of $1$-forms dual to $\lbrace X_{i} \rbrace_{i=1}^{3}$.  Now we study the behavior of the first eigenvalue of weighted $(p, q)$-Laplacian in each classes separately.\\

{\bf Case 1: $\mathbb{R}^{3}$}\\
In this case all metrics are flat, so for all $t\geq 0$ we have $g(t)=g_{0}$ where $g_{0}$ is constant, therefore $\lambda(t)$ is constant.\\

{\bf Case 2: Heisenberg} \\
This class is isomorphic to the set of upper-triangular $ 3\times 3$ matrices endowed with the usual matrix multiplication. Under the metric $g_{0}$ we choose a frame $\lbrace X_{i} \rbrace_{i=1}^{3}$ in which
\begin{align*}
[X_{2}, X_{3}]=X_{1}, \quad [X_{3}, X_{1}]=0, \quad [X_{1}, X_{2}]=0,
\end{align*}
 also under the normalization $ A_{0}B_{0}C_{0}=1 $ we have
\begin{align*}
R=-\frac{1}{2}A^{2}, \quad R_{11}= \frac{1}{2}A^{3}, \quad R_{22}= -\frac{1}{2} A^{2}B, \quad  R_{33}= -\frac{1}{2}A^{2}C, \quad
 ||Ric||^{2} = \frac{3}{4}A^{4},
\end{align*}
where under the Yamabe flow (\ref{do}) we find
\begin{align*}
A^{2}=\frac{1}{A_{0}^{2}-t},
\end{align*}
by replacing $R$ into the inequality (\ref{bist}) and integrating, we can get
\begin{align*}
\lambda(t)\left(A_{0}^{-2}-t\right)^{\left(\frac{p}{2}-1\right)},
\end{align*}
is increasing.\\

{\bf Case 3: E(2)} \\
Manifold E(2) is the group of isometries of Euclidian plane. In this case we have an Einstein metric and Ricci flow converges exponentially to flat metrics. Dependent to the metric $g_{0}$ we choose the frame $\lbrace X_{i} \rbrace_{i=0}^{3}$ such that
\begin{align*}
[X_{2}, X_{3}]=X_{1}, \quad [X_{3}, X_{1}]=X_{2}, \quad [X_{1}, X_{2}]=0,
\end{align*}
 In this case under the normalization $A_{0}B_{0}C_{0}=1$ we have
\begin{align*}
R=-\frac{1}{2}\left( 1- \frac{B_{0}}{A_{0}} \right)^{2}A^{2}, \quad R_{11}= \frac{1}{2}A\left( A^{2} - B^{2} \right), \quad R_{22}=\frac{1}{2}B\left( B^{2} - A^{2}\right),\\  R_{33}=-\frac{1}{2}C\left( A-B\right)^{2}, \quad ||Ric||^{2} = \frac{1}{2}\left( A^{2} -B^{2} \right)^{2} + \frac{1}{4}\left(A-B\right)^{4},
\end{align*}
and also under the Yamabe flow (\ref{do}), we obtain
\begin{align*}
A^{2}=\frac{A_{0}^{2}}{1-\left(A_{0}-B_{0}\right)^{2}t},
\end{align*}
then by replacing $R$ into the inequality (\ref{bist}) and integrating, the quantity
\begin{align*}
\lambda(t)\left(1-\left(A_{0}-B_{0}\right)^{2}t\right)^{\left(1-\frac{B_{0}}{A_{0}}\right)^{-2}\left(\frac{p}{2}-1\right)},
\end{align*}
is increasing.\\

{\bf Case 4: E(1,1)}\\  Manifold E(1,1) is the group of isometries of the plane with flat Lorentz metric, there is no Einstein metric here and Ricci flow fails to converge, they all are asymptotically cigar degeneracies. For a given metric $g_{0}$ similarly by a frame $\lbrace X_{i} \rbrace_{i=0}^{3}$ we have
\begin{align*}
[X_{1}, X_{2}]=0, \quad [X_{2}, X_{3}]=-X_{1}, \quad [X_{3}, X_{1}]= X_{2}.
\end{align*}
 Also under the normalization $A_{0}B_{0}C_{0}=1$ we conclude
\begin{eqnarray*}
&&R=-\frac{1}{2}\left(1+\frac{B_{0}}{A_{0}}\right)^{2}A^{2}, \quad R_{11}= \frac{1}{2}A\left(A^{2}-B^{2}\right), \quad R_{22}=\frac{1}{2}B\left(B^{2}-A^{2}\right),\\
&& R_{33}=-\frac{1}{2}C\left(A+B\right)^{2}, \quad ||Ric||^{2} = \frac{3}{4}A^{4},
\end{eqnarray*}
where under the Yamabe flow (\ref{do}), we get
\begin{align*}
A^{2} = \frac{A_{0}^{2}}{1-\left(A_{0}+B_{0}\right)^{2}t}.
\end{align*}
Now by replacing $R$ into the inequality (\ref{bist}) and integrating, we conclude that
\begin{align*}
\lambda(t)\left(1-\left(A_{0}+B_{0}\right)^{2}t\right)^{\left(1+\frac{B_{0}}{A_{0}}\right)^{-2}\left(\frac{p}{2}-1\right)},
\end{align*}
is increasing.\\

{\bf Case 5: SU(2)}\\ Similarly in this class we have Einstein metrics and Ricci flow converges exponentially in to these metrics, also by the frame $\lbrace X_{i} \rbrace_{i=0}^{3}$ we have
\begin{align*}
[X_{2}, X_{3}]=X_{1}, \quad [X_{3}, X_{1}]=X_{2}, \quad [X_{1}, X_{2}]=X_{3},
\end{align*}
 In this case under the normalization $A_{0}B_{0}C_{0}=1$, we get
\begin{eqnarray*}
&&R=\eta A^{2}, \quad R_{11}= \frac{1}{2}A[A^{2} - \left(B-C\right)^{2}],\\&& R_{22}=\frac{1}{2}B[B^{2}-\left(A-C\right)^{2}], \quad
R_{33}=\frac{1}{2} C[C^{2}-\left(A-B\right)^{2}], \\&& ||Ric||^{2} = \frac{1}{4}\big\lbrace [ A^{2}- \left(B-C\right)^{2}]^{2} + [B^{2} - \left(A-C\right)^{2}]^{2} + [C^{2}-\left( A-B\right)^{2}]^{2} \big\rbrace,
\end{eqnarray*}
where:
\begin{align*}
\eta = \frac{1}{2} \Big\lbrace 1- \left( \frac{B_{0}}{A_{0}} - \frac{C_{0}}{A_{0}} \right)^{2} +  \left( \frac{B_{0}}{A_{0}} \right)^{2} - \left( 1- \frac{C_{0}}{A_{0}}\right)^{2}  +  \left(\frac{C_{0}}{A_{0}}\right)^{2} - \left( 1- \frac{B_{0}}{A_{0}}\right)^{2} \Big\rbrace,
\end{align*}
and under the Yamabe flow (\ref{do}), we have
$
A^{2} =\frac{1}{A_{0}^{-2} + \eta t},
$
by replacing $R$ into the inequality (\ref{bist}) and integrating, we find that
if $A_{0} \geq 4B_{0}= 4C_{0}$ then
\begin{align*}
\lambda(t)\left(A_{0}^{-2} + \eta t\right)^{-\frac{1}{\eta}\left(\frac{p}{2}-1\right)},
\end{align*}
is increasing.\\

{\bf Case 6: SL(2,$\mathbb{R}$)}\\
There is no Einstein metric here and the Ricci flow doesn't converge and develops a pancake degeneracy, also by the frame $\lbrace X_{i} \rbrace_{i=0}^{3}$, we get
\begin{align*}
[X_{2}, X_{3}]=-X_{1}, \quad [X_{3}, X_{1}]=X_{2}, \quad [X_{1}, X_{2}]=X_{3},
\end{align*}
 in this case we also have
\begin{eqnarray*}
&&R=\eta A^{2}, \quad R_{11}=\frac{1}{2} A[A^{2}-\left(B-C\right)^{2}], \quad R_{22}=\frac{1}{2}B[B^{2}-\left(A+C\right)^{2}]\\
&&R_{33}=\frac{1}{2}C[C^{2}-\left(A+B\right)^{2}], \\&& ||Ric||^{2}=\frac{1}{4}\big\lbrace [A^{2}-\left(B-C\right)^{2}]^{2} + [ B^{2}-\left(A+C\right)^{2}]^{2} +[C^{2}-\left(A+B\right)^{2}]^{2}\big\rbrace,
\end{eqnarray*}
in which
\begin{align*}
\eta = -\frac{1}{2}\Big\lbrace 1+\left(\frac{B_{0}}{A_{0}}\right)^{2} + \left(\frac{C_{0}}{A_{0}}\right)^{2} + 2\frac{B_{0}}{A_{0}} + 2\frac{C_{0}}{A_{0}} -2\frac{B_{0}C_{0}}{A_{0}} \Big\rbrace,
\end{align*}
and also under the Yamabe flow (\ref{do}), we find
\begin{align*}
A^{2} = \frac{1}{A_{0}^{-2}-\eta t}.
\end{align*}
Now if $B_{0}=C_{0}$ then by replacing $R$ into the inequality (\ref{bist}) and integrating, we conclude
\begin{align*}
\lambda(t)\left(A_{0}^{-2} - \eta t\right)^{\frac{1}{\eta}\left(\frac{p}{2}-1\right)},
\end{align*}
is increasing.
\begin{center}
\begin{thebibliography}{99} % Enter references in alphabetical order and according to the following format.
\bibitem{abol} A. Abolarinwa, \textit{Evolution and monotonicity of the first eigenvalue of $ p $-Laplacian under the Ricci-harmonic map flow}, J. Appl. Anal., 21 (2)(2015), 147-160.
\bibitem{Au} T. Aubin, \textit{Equations differentiells non lineaires et probleme de Yamabe concernart la courbure scalair},  J. Math. Pures Appl.,  55(9) (1976), 269-296.
\bibitem{S.Azami} S. Azami, \textit{Eigenvalues variation of the $p$-Laplacian under the Yamabe flow}, Cogent Mathematics (2016), 3:1236566.
\bibitem{Brandle} S. Brandle, \textit{Convergence of the Yamabe flow for arbitrary initial energy}, J. Differential Geom. 69 (2005), 217-278.
 \bibitem{cao} X. Cao, \textit{Eigenvalues of $(-\Delta +\frac{R}{2})$ on manifolds with nonnegative curvature operator}, Math. Ann., 337(2) (2007), 435-442.

\bibitem{ha} R. Hamilton, \textit{Three-manifolds with positive Ricci curvature}, J. Diff. Geom. 17 (1982), 255-306.

\bibitem{Pak} P. T.  Ho, \textit{First eigenvalues of geometric operators under the Yamabe flow}, Annals of global Analysis and Geometry, 54 (2018), 449-472 .
\bibitem{song} S. B. Hou, \textit{Eigenvalues under the backward Ricci flow on locally homogeneous closed $3$-manifolds}, Acta Mathematica Sinica, 34(7) (2018), 1179-1194.
 \bibitem{ka} J. L. Kazdan and F. W. Warner, \textit{Scalar curvature and conformation of Riemannian structure}, J. Differential Geometry, 10 (1975), 113-134.
 \bibitem{raz} F. Korouki and A. Razavi, \textit{Bounds for the first eigenvalue of $\left( -\Delta -R\right)$ under the Ricci flow on Bianchi classes}, Bull. Braz. Math. Soc., New Series,  2019.
  \bibitem{per} G. Perelman, \textit{The entropy formula for the Ricci flow and its geometric application}, arXiv:math/0211159v1 (2002).

  \bibitem{mil} J. Milnor, \textit{Curvatures of left invariant metrics on Lie groups}, Adv. Math. 21(3)(1976), 293-329 .
 \bibitem{Muller} R. Muller, \textit{The Ricci Flow coupled with harmonic map heat flow}, Ann. Sci. Ec. Norm, sup 45 (2012), 101-142.

 \bibitem{sch} R. Schoen, \textit{Conformal deformation of a Riemannian metric to constant scalar curvature}, J. Differential Geom. 20(1984), 479-495.
  \bibitem{sch1} H. Schwetlick and M. Struwe, \textit{Convergence of the Yamabe flow for "large" energies}, J. Riene Angew. Math. 562 (2003), 59-100.
 \bibitem{tru} N. S. Trudinger, \textit{Remarks concerning the conformal deformation of Riemannian structures on compact manifolds}, Ann. Scuola Norm. Sup. Pisa., 22(3) (1968), 265-274.
  \bibitem{wang} Y. Z. Wang, \textit{Gradient estimates on the weighted $p$-Laplace heat equation}, J. Differential equations, 264 (2018), 506-524.

\bibitem{yam} H. Yamabe, \textit{On a deformation of Riemannian structures on compact manifolds}, Osaka J. Math., 12 (1960), 21-37.

\end{thebibliography}
\end{center}



{\small

\noindent{\bf Mohammad Javad Habibi Vosta Kolaei}

\noindent Department of Mathematics

\noindent PhD Student of Mathematics

\noindent Imam Khomeini international university

\noindent Qazvin, Iran(Islamic Republic of)

\noindent MJ.Habibi@Edu.ikiu.ac.ir}\\

{\small
\noindent{\bf  Shahroud Azami  }

\noindent  Department of Mathematics

\noindent Associate Professor of Mathematics

\noindent Imam Khomeini international university


\noindent Qazvin, Iran(Islamic Republic of)

\noindent azami@Sci.ikiu.ac.ir}\\



\end{document}