\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 02, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/02\hfil Structural stability]
{Structural stability for Brinkman-Forchheimer equations}

\author[Y. Liu,  C. Lin\hfil EJDE-2007/02\hfilneg]
{Yan Liu, Changhao Lin}

\address{Yan Liu \newline
Department of Applied Mathematics, 
Guangdong University of Finance,  
Guangzhou,  510520,  China}
\email{liuyan99021324@tom.com}

\address{Changhao  Lin \newline
 School of Mathematical
 Sciences,  South China Normal
 University, 510631, China}
\email{linchh@scnu.edu.cn}

\thanks{Submitted December 7, 2006. Published January 2, 2007.}
\subjclass[2000]{35B40, 35K50,35K45}
\keywords{Continuous dependence; Brinkman-Forchheimer equations;
\hfill\break\indent
 Brinkman coefficient; Forcheimer coefficient; structural stability}

\begin{abstract}
 In this paper, we obtain the continuous dependence and
 convergence results for the Brinkman and
 Forchheimer coefficients of a differential equation
 that models the flow of fluid in a saturated porous
 medium.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

The concept of structural stability in which the study of
continuous dependence (or stability) is on changes in the model
itself rather than the initial data, has been the subject of much
recent study. Many references to the work of the nature are given in the
monograph of Ames and Straughan \cite{a1}, which stress that continuous
dependence on the model itself, or structural stability, is every
bit as important as stability with respect to perturbations of the
initial data. In particularly, the stability of flow in porous media
has attracted much more attention in the literature;
see \cite{g1,p1,p2,p3,s1,s2} and their references.

  In this paper, we are interested in the Brinkman-Forchheimer
equations governing the flow of fluid in a saturated porous
medium,
   \begin{equation} \label{e1.1}
   \begin{gathered}
   \frac{\partial u_i}{\partial t}=\lambda \Delta u_i-au_i-b| u
   | u_i-p_{, i}\\
   \frac{\partial u_i}{\partial x_i}=0
   \end{gathered}
\end{equation}
where $u_i$ is the average fluid velocity in the porous medium, a is
the Darcy coefficient, $\lambda$ is the Brinkman coefficient,  $b$ is
the Forchheimer coefficient, and $p$ is the pressure. $\lambda$, $a$ and $b$
are positive constants. Here also $\Delta$ is the laplace
operator, and $\|\cdot\|$ denotes the norm of $L^2$.

We assume that $\Omega$ is a bounded, simply connected domain with
boundary $\partial \Omega$ in $R^3$. Associated with \eqref{e1.1}, we imposed
the boundary condition
\begin{equation}
u_i=0\quad \text{on } \partial
\Omega\times \{t>0\}
\end{equation}
and the initial condition
\begin{equation}
u_i(x, 0)=f_i(x).
\end{equation}
 In this paper, the usual summation convection is employed
    with repeated Latin subscripts summed from 1 to 3. The comma is
    used to indicated partial differentiation and the
    differentiation  with respect to the direction $x_k$ is denoted
    as ``$,k$''.
\section{Continuous dependence for the Brinkman coefficient}

To study the continuous dependence on $\lambda$, we let $(u_i, p)$ and
$(v_i, q)$ solve the following boundary initial-value problems for
different Brinkman coefficients $\lambda_1$ and $\lambda_2$,
\begin{equation} \label{e2.1}
\begin{gathered}
   \frac{\partial u_i}{\partial t}=\lambda_1 \Delta u_i-au_i-b| u
   | u_i-p_{, i} \quad \text{in } \Omega\times\{t>0\}\\
   \frac{\partial u_i}{\partial x_i}=0 \quad \text{in } \Omega\times\{t>0\}\\
   u_i=0 \quad \text{on } \partial \Omega\times \{t>0\}\\
u_i(x, 0)=f_i(x), \quad x \in \Omega
\end{gathered}
\end{equation}
and
\begin{equation} \label{e2.2}
\begin{gathered}
   \frac{\partial v_i}{\partial t}=\lambda_2 \Delta v_i-av_i-b| v
   | v_i-q_{, i} \quad\text{in } \Omega\times\{t>0\}\\
   \frac{\partial v_i}{\partial x_i}=0 \quad\text{in }\Omega\times\{t>0\}\\
   v_i=0 \quad\text{on }\partial \Omega\times \{t>0\}\\
v_i(x, 0)=f_i(x), \quad x \in \Omega
\end{gathered}
\end{equation}
We define the difference variables $w_i$, $\pi$ and $\lambda$ by
\begin{equation} \label{e2.3}
w_i=u_i-v_i, \pi=p-q, \lambda=\lambda_1-\lambda_2
\end{equation}
and then $(w_i, \pi)$ satisfies the boundary initial-value
problem
\begin{equation} \label{e2.4}
   \begin{gathered}
\frac{\partial w_i}{\partial t}
=\lambda_1 \Delta u_i-\lambda_2 \Delta v_i-aw_i
 -b(|u| u_i-|v| v_i)-\pi_{, i} \quad\text{in } \Omega\times\{t>0\}\\
\frac{\partial w_i}{\partial x_i}=0 \quad\text{in }\Omega\times\{t>0\}\\
   w_i=0 \quad\text{on }\partial \Omega\times \{t>0\}\\
w_i(x, 0)=0, \quad x \in \Omega
   \end{gathered}
\end{equation}
Multiplying \eqref{e2.4}$_1$ by $w_i$ and integrating over $\Omega$, we get
\begin{equation} \label{e2.5}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|w\|^2\\
&=-\int_\Omega \lambda_1\nabla u_i\nabla w_idx
  +\int_\Omega \lambda_2\nabla v_i\nabla w_idx-a\|w\|^2
  -b\int_\Omega (|u|u_i -|v|v_i)w_idx \\
&=-\int_\Omega \lambda\nabla u_i\nabla w_idx
  +\int_\Omega\lambda_2\nabla w_i\nabla w_idx-a\|w\|^2
  -b\int_\Omega (|u|u_i-|v|v_i)w_idx
\end{aligned}
\end{equation}
Since
\begin{equation} \label{e2.6}
\begin{aligned}
&(|u|u_i-|v|v_i)w_i\\
&=\frac{1}{2}|u|(u_i-v_i+v_i)w_i-\frac{1}{2}|v|v_iw_i+\frac{1}{2}|u|
  u_iw_i+\frac{1}{2}|v|w_i(u_i-v_i-u_i)\\
&=\frac{1}{2}(|u|+|v|)w_iw_idx+
\frac{1}{2}(|u|-|v|)^2(|u|+|v|)
\end{aligned}
\end{equation}
Combining \eqref{e2.5} and \eqref{e2.6}, using the Cauchy-Schwarz inequality
and dropping some negative items, we obtain
\[
\frac{d}{dt}\|w\|^2\leq
\frac{\lambda^2}{2\lambda_2}\|\nabla u\|^2
\]
Integrating from 0 to t, we obtain
\begin{equation} \label{e2.8}
\|w\|^2\leq \frac{\lambda^2}{2\lambda_2}\int_0^t\|\nabla u\|^2d\eta
\end{equation}
Our next step is to bound $\int_0^t\|\nabla u\|^2d\eta $.
Multiplying \eqref{e2.1}$_1$ by $u_i$ and integrating over $\Omega$, we see
that
\[ %2.9
\frac{d}{dt}\|u\|^2+2\lambda_1\|\nabla u\|^2\leq 0
\]
Integrating  from 0 to t, we obtain
\begin{equation}
\|u\|^2+2\lambda_1\int_0^t\|\nabla u\|^2d\eta\leq \|f\|^2
\end{equation}
thus
\begin{equation}\label{e2.10}
 \int_0^t\|\nabla u\|^2d\eta\leq \frac{\|f\|^2}{2\lambda_1}
\end{equation}
Combining \eqref{e2.8} and \eqref{e2.10}, we obtain
\begin{equation}\label{e2.11}
\|w\|^2\leq\frac{\lambda^2}{4\lambda_1\lambda_2}\|f\|^2\end{equation}
Inequality \eqref{e2.11} shows the continuous dependence on
$\lambda$. However, the convergence result can't follow from
\eqref{e2.11} as $\lambda_1\to 0, \lambda_2=0$.

\section{Convergence as $\lambda_1\to 0$, $\lambda_2=0$}

Let $(u_i, p)$ be a solution of \eqref{e2.1} with
$\lambda_1\to 0$, $(v_i, p)$ be a solution of \eqref{e2.1} with
$\lambda_1\to 0$, $w_i, \pi$ are defined the same as in section 2.
\begin{equation} \label{e3.1}
 \begin{gathered}
   \frac{\partial w_i}{\partial t}
=\lambda_1 \Delta u_i-aw_i-b(|u| u_i-|v| v_i)-\pi_{, i}
  \quad\text{in } \Omega\times\{t>0\}\\
\frac{\partial w_i}{\partial x_i}=0 \quad\text{in }\Omega\times\{t>0\}\\
   w_i=0 \quad\text{on }\partial \Omega\times \{t>0\}\\
w_i(x, 0)=0, \quad x \in \Omega
   \end{gathered}
\end{equation}
Multiplying \eqref{e3.1}$_1$ by $w_i$ and integrating over $\Omega$, we find
\begin{equation} \label{e3.2}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|w\|^2\\
&= -\lambda_1\int_\Omega
\nabla u_i\nabla w_idx-a\|w\|^2-b\int_\Omega (|u| u_i-|v| v_i)w_idx\\
&= -\lambda_1\int_\Omega \nabla u_i\nabla u_idx
 +\lambda_1\int_\Omega \nabla u_i\nabla v_idx-a\|w\|^2
 -b\int_\Omega (|u| u_i-|v| v_i)w_idx\\
&\leq \frac{\lambda_1}{4}\int_\Omega \nabla  v_i\nabla v_idx-a\|w\|^2\,.
\end{aligned}
\end{equation}
The next step is to bound $\int_\Omega v_{i, j}v_{i, j}dx$.
We know
\begin{equation}\label{e3.3}
\int_\Omega v_{i, j}v_{i, j}dx=\int_\Omega
v_{i, j}(v_{i, j}-v_{j, i})dx\,.
\end{equation}
Using  \eqref{e2.2}$_1$ with $\lambda_2=0$, we get
\begin{equation} \label{e3.4}
\begin{aligned}
&\int_0^t\int_\Omega v_{i, j}(v_{i, j}-v_{j, i})ds\,d\eta \\
&= \frac{1}{a}\int_0^t\int_\Omega
(v_{i, j}-v_{j, i})(-v_{i, jt}-b(|v|v_i)_{, j}-q_{, ij})dx\,d\eta\\
&=-\frac{1}{a}\int_0^t\int_\Omega(v_{i, j}-v_{j, i})v_{i, jt}ds\,d\eta
  -\frac{b}{a}\int_0^t\int_\Omega(v_{i, j}-v_{j, i})\\
&\quad\times (\frac{v_kv_{k, j}}{|v|}v_i+|v|v_{i, j})ds\,d\eta
 +\frac{1}{a}\int_0^t\int_\Omega v_{i, ij}q_{, j}ds\,d\eta\\
&\quad -\frac{1}{a}\int_0^t\int_\Omega v_{j, ji}q_{, i}ds\,d\eta
 -\frac{1}{a}\int_0^t\int_{\partial\Omega} v_{i, j}q_{, j}n_idsd\eta
 +\frac{1}{a}\int_0^t\int_{\partial\Omega}v_{j, i}q_{, i}n_jdsd\eta\\
&=-\frac{1}{2a}\int_\Omega v_{i, j}v_{i, j}dx|_{\eta=t}
  +\frac{1}{2a}\int_\Omega f_{i, j}f_{i, j}dx\\
&\quad -\frac{b}{a}\int_0^t\int_\Omega\frac{v_kv_{k, j}v_iv_{i, j}}{|v|}
  ds\,d\eta-\frac{b}{a}\int_0^t\int_\Omega v_{i, j}|v|v_{i, j}ds\,d\eta
\end{aligned}
\end{equation}
thus
\begin{equation} \label{e3.5}
\int_0^t\int_\Omega v_{i, j}v_{i, j}ds\,d\eta
\leq \frac{1}{2a}\int_\Omega f_{i, j}f_{i, j}dx
\end{equation}
Combining \eqref{e3.2}, \eqref{e3.3} and \eqref{e3.5}, we get
\[
\|w\|^2\leq\frac{\lambda_1}{8a}\int_\Omega f_{i, j}f_{i, j}ds\,d\eta
\]
This inequality demonstrates the convergence  $u\to v$
when $\lambda_1\to 0$, $\lambda_2=0$.

\section{Continuous dependence for the Forchheimer coefficient b}

To study continuous dependence on $b$, we let $(u_i, p)$
 and $(v_i, q)$ solve the following boundary initial-value problem for
 different coefficients $b_1$ and $b_2$.
\begin{equation} \label{e4.1}
   \begin{gathered}
   \frac{\partial u_i}{\partial t}=\lambda \Delta u_i-au_i-b_1| u
   | u_i-p_{, i} \quad\text{in } \Omega\times\{t>0\}\\
   \frac{\partial u_i}{\partial x_i}=0 \quad\text{in }\Omega\times\{t>0\}\\
   u_i=0 \quad\text{on }\partial
\Omega\times \{t>0\}\\
u_i(x, 0)=f_i(x), \quad x \in \Omega
   \end{gathered}
\end{equation}
and
\begin{equation} \label{e4.2}
   \begin{gathered}
   \frac{\partial v_i}{\partial t}=\lambda \Delta v_i-av_i-b_2| v
   | v_i-q_{, i} \quad\text{in } \Omega\times\{t>0\}\\
   \frac{\partial v_i}{\partial x_i}=0 \quad\text{in }\Omega\times\{t>0\}\\
   v_i=0 \quad\text{on }\partial
\Omega\times \{t>0\}\\
v_i(x, 0)=f_i(x), \quad x \in \Omega
   \end{gathered}
\end{equation}
We define the difference variables
\begin{equation}
w_i=u_i-v_i, \quad \pi=p-q, \quad b=b_1-b_2\,.
\end{equation}
Then $(w_i, \pi)$ satisfy
the boundary initial-value problem
\begin{equation} \label{e4.4}
   \begin{gathered}
   \frac{\partial w_i}{\partial t}=\lambda \Delta w_i-aw_i
-(b_1| u | u_i-b_2| v | v_i)-\pi_{, i }\quad\text{in } \Omega\times\{t>0\}\\
   \frac{\partial w_i}{\partial x_i}=0 \quad\text{in }\Omega\times\{t>0\}\\
   w_i=0 \quad\text{on }\partial \Omega\times \{t>0\}\\
w_i(x, 0)=0, \quad x \in \Omega
   \end{gathered}
\end{equation}
Multiplying \eqref{e4.4}$_1$ by $w_i$ and integrating over $\Omega$, we get
\begin{equation} \label{e4.5}
\frac{1}{2}\frac{d}{dt}\|w\|^2=-\lambda\int_\Omega|\nabla
w|^2dx-a\|w\|^2- \int_\Omega (b_1|u|u_i-b_2|v|v_i)w_idx
\end{equation}
For we have
\begin{equation} \label{e4.6}
b_1|u|u_i-b_2|v|v_i=\frac{b}{2}(|u|u_i+|v|v_i)+\tilde{b}(|u|u_i-|v|v_i)
\end{equation}
where
$\tilde{b}=\frac{b_1+b_2}{2}$.
Combining \eqref{e4.5}, \eqref{e4.6} and \eqref{e2.6}, we obtain
\begin{equation} \label{e4.7}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|w\|^2\\
&= -\lambda\int_\Omega|\nabla
w|^2dx-a\|w\|^2- \frac{b}{2}\int_\Omega (|u|u_i+|v|v_i)w_idx-
\tilde{b}(|u|u_i-|v|v_i)w_idx\\
&\leq -a\|w\|^2-\frac{b}{2}\int_\Omega
(|u|u_i+|v|v_i)w_idx-\frac{\tilde{b}}{2}\int_\Omega
(|u|+|v|)w_iw_idx
\end{aligned}
\end{equation}
We then use the Cauchy-Schwarz and arithmetic geometric mean
inequalities as follows
\begin{equation} \label{e4.8}
\frac{b}{2}|\int_\Omega (|u|u_i+|v|v_i)w_idx|\leq
\frac{b^2}{8\tilde{b}}\int_\Omega(|u|^3+|v|^3)dx
+\frac{1}{2}\tilde{b}\int_\Omega(|u|+|v|)w_iw_idx
\end{equation}
We now employ \eqref{e4.8} in \eqref{e4.7}, after an integration, that
\begin{equation} \label{e4.9}
\frac{1}{2}\|w\|^2+a\int_0^t\|w\|^2d\eta\leq\frac{b^2}{8\tilde{b}}
\int_0^t\int_\Omega(|u|^3+|v|^3)ds\,d\eta
\end{equation}
From \eqref{e4.1}$_1$, one deduce that
\begin{equation} \label{e4.10}
\frac{1}{2}\|u\|^2+a\int_0^t\|u\|^2d\eta
 +b_1\int_0^t\int_\Omega|u|^3ds\,d\eta+\lambda
\int_0^t\int_\Omega u_{i, j}u_{i, j}ds\,d\eta=\frac{1}{2}\|f\|^2
\end{equation}
and so
\begin{equation} \label{e4.11}
\int_0^t\int_\Omega|u|^3ds\,d\eta\leq
\frac{1}{2b_1}\|f\|^2
\end{equation}
Similarly, from $\eqref{e4.2}_1$, we can also get
\begin{equation} \label{e4.12}
\int_0^t\int_\Omega|v|^3ds\,d\eta\leq \frac{1}{2b_1}\|f\|^2
\end{equation}
Inserting \eqref{e4.11}, \eqref{e4.12} in \eqref{e4.10}, we find
\begin{equation} \label{e4.13}
\|w\|^2+2a\int_0^t\|w\|^2d\eta \leq\frac{b^2}{4b_1b_2}\|f\|^2
\end{equation}
This inequality establish continuous dependence on $b$, we
note, however, that convergence as $b_1\to 0$, $b_2=0$ is
not established from \eqref{e4.13}.

\section{Convergence as the Forchheimer coefficient $b_1\to0 $ and $b_2=0$}

Now, let $(u_i, p)$ be the solution of \eqref{e4.1}, and $(v_i, q)$ be the
solution of \eqref{e4.2} with $b_2=0$.
The object of this section is to demonstrate convergence of the
solution $u_i$ to the solution $v_i$ as $b_1\to 0$.
We also define the variables $w_i$ and $\pi$ by
\begin{equation} \label{e5.1}
w_i=u_i-v_i, \quad \pi=p-q
\end{equation}
and then $(w_i, \pi)$ satisfy the boundary
initial-value problems
\begin{equation} \label{e5.2}
   \begin{gathered}
   \frac{\partial w_i}{\partial t}=\lambda \Delta w_i-aw_i
 -b_1|u| u_i-\pi_{, i }\quad\text{in } \Omega\times\{t>0\}\\
   \frac{\partial w_i}{\partial x_i}=0 \quad\text{in }\Omega\times\{t>0\}\\
   w_i=0 \quad\text{on }\partial \Omega\times \{t>0\}\\
w_i(x, 0)=0, \quad x \in \Omega
   \end{gathered}
\end{equation}
Multiplying  by $w_i$ and integrating over $\Omega$, we obtain
\begin{equation} \label{e5.3}
 \frac{1}{2}\frac{d}{dt}\|w\|^2=-\lambda \|\nabla
w\|^2-a\|w\|^2-b_1\int_\Omega |u|u_iw_idx
\end{equation}
Using the H\"older inequality, we get
\begin{equation} \label{e5.4}
\frac{d}{dt}\|w\|^2\leq
2b_1\big(\int_\Omega |u|^3dx\Big)^{2/3}\Big(\int_\Omega
|w|^3dx\Big)^{1/3}-2\lambda \|\nabla w\|^2-2a\|w\|^2
\end{equation}
For a function $F$ such that $F=0$ on $\partial \Omega$
(see for example \cite{f1}), we
have the Sobolev inequality
\[
\int_\Omega |F|^4dx\leq c_1 \Big(\int_\Omega|F|^2dx\Big)^{1/2}
\Big(\int_\Omega  F_{i, j}F_{i, j}dx\Big)^{3/2}
\]
Then, we use the Cauchy-Schwarz inequality, to get
\begin{equation} \label{e5.5}
\begin{aligned}
\int_\Omega |w|^3dx
&\leq \Big(\int_\Omega |w|^2dx\Big)^{1/2}
\Big(\int_\Omega |w|^4dx\Big)^{1/2}\\
&\leq c_1\Big(\int_\Omega w_iw_idx\Big)^{3/4}
\Big(\int_\Omega w_{i, j}w_{i, j}dx\Big)^{3/4}
\end{aligned}
\end{equation}
Similarly,
\begin{equation} \label{e5.6}
\int_\Omega |u|^3dx\leq  c_1\Big(\int_\Omega
u_iu_idx\Big)^{3/4}\Big(\int_\Omega u_{i, j}u_{i, j}dx\Big)^{3/4}
\end{equation}
In view of \eqref{e5.4} and \eqref{e5.5}, \eqref{e5.3} can be rewritten
as
\begin{align*}
\|w\|^2
&\leq 2b_1c_1\int_0^t
 \Big[\Big(\int_\Omega w_iw_idx\Big)^{1/4}
 \Big(\int_\Omega w_{i, j}w_{i, j}dx\Big)^{1/4}
 \Big(\int_\Omega u_iu_idx\Big)^{1/2}\\
&\quad\times
 \Big(\int_\Omega u_{i, j}u_{i, j}dx\Big)^{1/2} \Big]d\eta
 -2\lambda \int_0^t\|\nabla w\|^2d\eta
 -2a\int_0^t w^2ds\,d\eta \\
&\leq 2\lambda\int_0^t\|\nabla w\|^2d\eta-2a\int_0^t\|
 w\|^2d\eta-2b_1c_1\int_0^t
 \Big[\Big(\varepsilon_1\int_\Omega w_iw_idx\Big)^{1/4}\\
 &\quad\times \Big(\varepsilon_2\int_\Omega  w_{i, j}w_{i, j}dx\Big)^{1/4}
  \Big(\max \int_\Omega u_iu_idx\cdot
 \int_\Omega \big(\varepsilon_1\varepsilon_2\big)^{-1/2}
 u_{i, j}u_{i, j}dx\Big)^{1/2}\Big]d\eta\\
&\leq\frac{1}{4}\varepsilon_1\int_0^t\int_\Omega w_iw_ids\,d\eta
 +\frac{1}{4}\varepsilon_2\int_0^t\int_\Omega
 w_{i, j}w_{i, j}ds\,d\eta \\
&\quad +\frac{4}{2}(b_1c_1)^2\cdot(\varepsilon_1\varepsilon_2)^{-1/2}
 \int_0^t\int_\Omega u_{i, j}u_{i, j}ds\,d\eta\\
&\quad -2a\max\int_\Omega u_iu_idx\int_0^t
 \int_\Omega w^2ds\,d\eta-2\lambda\int_0^t|\nabla
 w|^2d\eta-2a\int_0^t\| w\|^2d\eta
\end{align*}
If we choose $\varepsilon_1=8a$, $\varepsilon_2=8\lambda$,
the above expression can be rewritten as
\begin{equation} \label{e5.7}
\|w\|^2\leq \frac{b_1^2c_1^2}{4(a\lambda)^{1/2}}
\max\int_\Omega u_iu_idx\cdot \int_0^t\int_\Omega u_{i, j}u_{i, j}ds\,d\eta
\end{equation}
 From \eqref{e4.10}, we have
\[
\max\int_\Omega u_iu_idx\leq\|f\|^2, \quad
\int_0^t\int_\Omega u_{i, j}u_{i, j}ds\,d\eta
\leq\frac{1}{2\lambda}\|f\|^2;
\]
therefore, from \eqref{e5.7}, we have
\[
\|w\|^2\leq \frac{(b_1c_1)^{2}}{8a^{1/2}\lambda^{3/2}}\|f\|^4,
\]
which shows the desired result.\subsection*{Acknowledgments} The
authors would like to express their gratitude to  Professor
Hongliang Tu for his advice  when in writing parts of  this article.


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