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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 144, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/144\hfil A variational formulation]
{A variational formulation for traveling waves and its applications}

\author[H.-X. Meng, Y.-X. Wang \hfil EJDE-2014/144\hfilneg]
{Hai-Xia Meng, Yu-Xia Wang}  % in alphabetical order

\address{Hai-Xia Meng \newline
School of Mathematics and Physics,
Lanzhou Jiaotong University\\
Lanzhou, Gansu 730070,  China}
\email{menghx08@lzu.edu.cn}

\address{Yu-Xia Wang \newline
School of Mathematical Sciences, University of Electronic Science 
and Technology of China, Chengdu, Sichuan 611731,  China}
\email{yxwang\_10@lzu.edu.cn}


\thanks{Submitted February 28, 2014. Published June 20, 2014.}
\subjclass[2000]{35K57}
\keywords{Variational formulation;
traveling waves; wave velocity}

\begin{abstract}
 In this article, we give a variational formulation for traveling wave
 solutions that decay exponentially at one end of the cylinder for
 parabolic equations. The variational formulation allows us to obtain the
 monotone dependence of the velocity on the domain and the nonlinearity,
 since the velocity is related to the infimum. In particular, we apply this
 method to  Ginzburg-Landau-type problems and a scalar reaction-diffusion-advection
 equation in infinite cylinders. For the former, we not only obtain
 the existence, non-existence, boundedness and regularity of the
 solutions, but also obtain the monotone dependence of the velocity on
 the nonlinearity and the domain. For the later, we  obtain the monotone
 dependence of the velocity on the nonlinearity and the domain besides the
 existence, uniqueness, monotonicity and asymptotic behavior at infinity
 of the solutions. Moreover, we deduce that the influence of the advection
 on the traveling waves is different from a flow along the cylinder axis
 considered in many articles.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

This article concerns the  reaction diffusion equation
\begin{equation}\label{rde1}
u_{t}=\Delta u+f(u)
\end{equation}
in an infinite cylinder $\Sigma$ with
 either Neumann or Dirichlet  boundary condition
\begin{equation}\label{bc1}
(\nu\cdot\nabla u)|_{\partial \Sigma}=0  \quad  \textrm{or}
\quad u|_{\partial \Sigma}=0.
 \end{equation}
Here $u=u(x,t)\in \mathbb{R}$,
$x=(y,z)\in\Sigma=\Omega\times\mathbb{R}$,
$\Omega\subset \mathbb{R}^{n-1}$ ($n\geq3$) is a bounded domain with smooth
boundary; $\nu$ is the outward normal to $\partial\Sigma$.  The nonlinearity
$f:\mathbb{R}\mapsto \mathbb{R}$ is in $C^{1}$, and $f(0)=0$, then
$u=0$ is the trivial solution of \eqref{rde1} and \eqref{bc1}.

It is known that traveling wave solution is an important class of
solutions to investigate the long time behavior of solutions of
Cauchy problems; see, for example, \cite{bf, bll, bn, Ber4, cd, he,llw08,
llw09, pan, ro, zhaowu}. In this article, we use a variational
formulation to study the existence of traveling wave solutions which
are characterized by a fast exponential decay at one end of the
cylinder and properties of obtained traveling wave solutions. 
And the variational formulation is given
to \eqref{rde1}, the nonlinearity $f$ of which is quite general, 
so the variational formulation can be applied to the study of
traveling wave solutions of many other problems.

Heinze \cite{Hei1} first proposed the idea of coverting the
existence of traveling wave solutions into the existence of
constraint minimizer in two dimensional strip with Dirichlet
boundary condition. In \cite{Hei2}, Heinze studied a model for the
heater in boiling systems and extended the results of \cite{Hei1} to
the mixed nonlinear Neumann and Dirichlet boundary problems in
infinite cylinder. And \cite{Hei2} obtained the existence of traveling wave
solutions of problem
\begin{equation}\label{tws1}
\begin{gathered}
\partial_{t}u(y,z,t)=\Delta u(y,z,t)
+f(u(y,z,t),y),\quad  (y,z,t)\in \Omega\times \mathbb{R}\times\mathbb{R}^{+},\\
\partial_{\nu}u(y,z,t)=g(u(y,z,t),y),\quad 
(y,z,t)\in \Gamma_{1}\times \mathbb{R}\times\mathbb{R}^{+},\\
u(y,z,t)=0,\quad  (y,z,t)\in \Gamma_2\times \mathbb{R}\times\mathbb{R}^{+},
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^{n-1}$ with $C^{1}$ boundary.
The boundary $\partial\Omega$ consists of two parts
$\Gamma_{1}, \Gamma_2$ corresponding to different boundary conditions.
$\Gamma_{1}$ and $\Gamma_2$ may be empty.

Let $u(y,z,t)=\bar{u}(y,c(z+ct))$ with $c\neq0$ as the unknown wave velocity,
then Equation \eqref{tws1} is written as
\begin{gather*}
\partial_{z}\bar{u}=\partial_{zz}\bar{u}+\lambda_{1}(\Delta_{y} \bar{u}
+f(\bar{u},y)),\quad  (y,z)\in \Omega\times\mathbb{R},\\
\partial_{\nu}\bar{u}=g(\bar{u},y),\quad 
(y,z)\in \Gamma_{1}\times\mathbb{R},\\
\bar{u}=0,\quad (y,z)\in \Gamma_2\times\mathbb{R}.
\end{gather*} %\label{101}
Then by defining the following two functionals
\begin{gather*}
I[u]=\frac{1}{2}\int_{\Sigma}e^{-z}
|\partial_{z}u|^2\mathrm{d}z\,\mathrm{d}y,\\
J[u]=\int_{\Omega}e^{-z}\Big(\frac{1}{2}|\nabla
_{y}u|^2-F(u,y)\Big)\mathrm{d}z\,\mathrm{d}y
-\int_{\mathbb{R}\times\partial\Gamma_{1}}e^{-z}G(u,y)\mathrm{d}z\,\mathrm{d}T_{y},
\end{gather*}
where $F(u,y)=\int_{0}^{u}f(s,y)\mathrm{d}s$,
$G(u,y)=\int_{0}^{u}g(s,y)\mathrm{d}s$,
Heinze \cite{Hei2} obtained the minimization problem
\begin{equation}\label{mp1}
\inf_{\{u\in X|J[u]=b\}}I[u]
\end{equation}
in the weighted space $X=H^{1}(\mathbb{R}\times\Omega, e^{-z})$. 
Moreover, He also obtained $\lambda_{1}=\frac{1}{c^2}=\inf_{\{u\in
X|J[u]=-1\}}I[u]$ by letting $b=-1$ in \eqref{mp1}.

For nonlinear Neumann boundary condition, the existence of traveling
wave solutions was  obtained by
Kyed \cite{Mad} for the problem
\begin{gather*}
\partial_{t}u-\Delta u=0,\quad  \textrm{in } 
\Omega\times \mathbb{R}\times\mathbb{R}^{+},\\
\frac{\partial u}{\partial \nu}=f(u),\;
\quad \textrm{on } \partial\Omega\times \mathbb{R}\times\mathbb{R}^{+},
\end{gather*}
which appears in the study of transient boiling processes 
by variational methods. Here $\Omega$ is a bounded domain in 
$\mathbb{R}^{n-1}$ with $C^{1}$ boundary.

Let $u(y,z,t)=\bar{u}(y,z+ct)$, 
$(y,z,t)\in \Omega\times \mathbb{R}\times\mathbb{R}^{+}$, then traveling wave 
equation is
\begin{gather*}
\Delta \bar{u}-c\partial_{z}\bar{u}=0,\quad  \textrm{in }  \Omega\times\mathbb{R},\\
\frac{\partial \bar{u}}{\partial \nu}=f(\bar{u}),\quad \textrm{on } 
 \partial\Omega\times\mathbb{R}.
\end{gather*}
Define
\begin{gather*}
\mathcal{\varepsilon}[u]=\frac{1}{2}\int_{\Sigma}e^{-z}
|Du|^2\mathrm{d}z\mathrm{d}y,\\
J[u]=\int_{\mathbb{R}\times\Gamma}e^{-z}F(u)\mathrm{d}S(y)\mathrm{d}z,
\end{gather*}
where $F(u)=\int_{0}^{u}f(s)\mathrm{d}s$, then the variational formulation 
is $\min_{u\in\mathcal{C}} \mathcal{\varepsilon}[u]$,
 $\mathcal{C}:=\{u\in H^{1}(\mathbb{R}\times\Omega, e^{-z})|J[u]=1\}$.

In 2004, Muratov \cite{Mur1} showed that traveling wave solutions
with a fast exponential decay at one end of the cylinder are critical points 
of certain functional. And this type of traveling wave solutions
 are called variational traveling wave solutions. Furthermore, under
certain assumptions on the shape of the solutions, \cite{Mur1} showed that there
exists a reference frame in which the solution of the initial value
problem converges to the variational traveling wave at least in a
sequence of time. Recently, Lucia et al. \cite{Luc} studied
variational traveling wave solutions for Ginzburg-Landau-type
problems
\begin{equation}\label{glt}
u_{t}=\Delta u+f(u),   \quad  f(u)=-\nabla_{u}V(u)
\end{equation}
with
\begin{equation}\label{bc2}
(\nu\cdot\nabla u)|_{\partial \Sigma}=0  \quad  \textrm{or} 
\quad u|_{\partial \Sigma}=0,
 \end{equation}
where $u=u(x,t)\in \mathbb{R}^{m},\;V:\mathbb{R}^{m}\mapsto \mathbb{R}$,
$x=(y,z)\in\Sigma=\Omega\times \mathbb{R}$, 
$\Omega\subset \mathbb{R}^{n-1}$ $(n\geq3)$ is a bounded domain with boundary 
of class $C^2$. Let $u(y,z,t)=\bar{u}(y,z-ct)$, then the traveling wave
equation is
\[
\bar{u}_{zz}+\Delta_{y}\bar{u}+c\bar{u}_{z}+f(\bar{u})=0
\]
with the boundary condition \eqref{bc2}.
By defining
\[
\Phi_{c}[u]=\int_{\Sigma}e^{cz}
\Big(\frac{1}{2}\sum_{i=1}^{m}|\nabla u_{i}|^2+V(u)\Big)\mathrm{d}y\,\mathrm{d}z
\]
and
\[
\Gamma_{c}[u]=\frac{1}{2}\int_{\Sigma}e^{cz}
\sum_{i=1}^{m}|\frac{\partial
u_{i}}{\partial z}|^2\mathrm{d}x,
\]
the constraint minimization problem is
\begin{equation}\label{rc1}
\Phi_{c}[u_{c}]=\inf_{\{u\in H^{1}_{c}(\Sigma;\mathbb{R}^{m})|\Gamma_{c}[u]=1\}}
\Phi_{c}[u]\leq0.
 \end{equation}
Then under the following three assumptions
\begin{itemize}
\item[(H1)] The function $V:\mathbb{R}^{m}\mapsto \mathbb{R}$ satisfies
$V\in C^{0}(\mathbb{R}^{m})$, $V(0)=\nabla_{u}V(0)=0$,
$V(u)\geq-C|u|^2$
for some $C\geq0$;

\item[(H2)] There exists a convex compact set 
$\mathcal{K}\subset \mathbb{R}^{m}$ which contains the origin, such that 
$V\in C^{1,1}(\mathcal{K})$ and for all
$u\not\in\mathcal{K}$, $V(u)\geq V(\Pi_{\mathcal{K}}(u))$,
 where
$\Pi_{\mathcal{K}}:\mathbb{R}^{m}\mapsto \mathbb{R}^{m}$ is the projection 
on the set
$\mathcal{K}$, that is, $\Pi_{\mathcal{K}}(u)$ is the closest point to $u$
which lies in $\mathcal{K}$;

\item[(H3)] There exist $c>0$ such that $c^2+4\upsilon_{0}>0$, and
$u\in H_{c}^{1}(\Sigma,\mathbb{R}^{m}),\; u\not\equiv0$ such that
$\Phi_{c}[u]\leq0$,
 where
\[
\upsilon_{0}=\mu_{0}+\liminf_{|u|\to0}
\frac{2V(u)}{|u|^2},
\]
\end{itemize}
where $\mu_{0}$ is the smallest eigenvalue of $-\Delta_{y}$
with the boundary condition \eqref{bc2},
they obtained the existence, non-existence, and many properties of 
variational traveling waves.

Furthermore, Muratov and Novaga \cite{Mur2} discussed front propagation 
problem for a reaction-diffusion-advection equation in infinite  cylinder
\begin{gather}\label{rdap}
u_{t}+\mathbf{v}\cdot\nabla u=\Delta u+f(u,y), \quad \; \mathbf{v}=(-\nabla_{y}\varphi,0),\quad \;
 \varphi:\overline{\Omega}\mapsto \mathbb{R},\\
\label{bc3}
u|_{\partial\Sigma_{\pm}}=0, \quad \; \nu\cdot\nabla
u|_{\partial\Sigma_{0}}=0,
\end{gather}
where $u=u(x,t)\in \mathbb{R}$, 
$x=(y,z)\in\Sigma=\Omega\times \mathbb{R}$,
$\Omega\subset \mathbb{R}^{n-1}$ is a bounded domain
with $C^2$ boundary; $\mathbf{v}$ is an imposed advection flow;
$\partial\Sigma_{\pm}=\partial\Omega_{\pm}\times \mathbb{R}$,
$\partial\Sigma_{0}=\partial\Omega_{0}\times \mathbb{R}$, 
$\partial\Omega_{\pm}$ and $\partial\Omega_{0}$ are
defined as parts of $\partial\Omega$ by
$\nu\cdot\nabla_{y}\varphi>0$, $\nu\cdot\nabla_{y}\varphi<0$ and
$\nu\cdot\nabla_{y}\varphi=0$, respectively. They were concerned
 with a particular situation in which the flow $\mathbf{v}$ is  
transverse to the axis of the cylinder; i.e., $\mathbf{v}$ does not have a
component along $z$.

For traveling wave solutions of the form $u(x,t)=\bar{u}(y,z-ct)$,
substituting it into  \eqref{rdap},  the traveling wave
equation is
 \[
\bar{u}_{zz}+c\bar{u}_{z}+\nabla_{y}\varphi\cdot\nabla_{y}
\bar{u}+\Delta_{y}\bar{u}+f(\bar{u},y)=0
\]
with boundary conditions \eqref{bc3}. By defining the following two functionals
\begin{gather*}
\Phi_{c}[u]=\int_{\Sigma}e^{cz+\varphi(y)}
\Big(\frac{1}{2}|\nabla u|^2+V(u,y)\Big)\mathrm{d}y\,\mathrm{d}z,
\\
\Gamma_{c}[u]=\frac{1}{2}\int_{\Sigma}e^{cz+\varphi(y)}
|\frac{\partial u}{\partial z}|^2\mathrm{d}x,
\end{gather*}
 the constraint minimizer problem was given by 
\begin{equation}\label{haoo}
\Phi_{c}[u_{c}]=\inf_{\{u\in H^{1}_{c}(\Sigma)|\Gamma_{c}[u]=1\}}\Phi_{c}[u]\leq0.
\end{equation}
Then under the following three assumptions
\begin{itemize}
\item[(A1)] The function $f:[0,1]\times\bar{\Omega}\to \mathbb{R}$ satisfies
\[
f(0,y)=0, \quad  f(1,y)\leq0,\quad \forall y\in\Omega;
\]

\item[(A2)] For some $\alpha\in(0,1)$
\[
f\in C^{0,\alpha}([0,1]\times\bar{\Omega}), \quad  
f_{u}\in C^{0,\alpha}([0,1]\times\bar{\Omega}),\quad 
\varphi\in C^{1,\alpha}(\bar{\Omega}),
\]
where $f_{u}=\frac{\partial f}{\partial u};$

\item[(A3)] There exist $c>0$ satisfying $c^2+4\upsilon_{0}>0$, and $u\in
H_{c}^{1}(\Sigma)$ such that $\Phi_{c}[u]\leq0$ and $ u\not\equiv0$,
where
$$ \upsilon_{0}=\min _{\{\psi\in
H^{1}(\Omega),\psi|\partial\Omega_{\pm}=0\}}R(\psi),\,\,\,
R(\psi)=\frac{\int_{\Omega}e^{
 \varphi(y)}\left(|\nabla_{y}\psi|^2
 -f_{u}(0,y)\psi^2\right)\mathrm{d}y}{\int_{\Omega}e^{
 \varphi(y)}\psi^2\mathrm{d}y},
$$
\end{itemize}
they showed only three propagation scenarios are possible: 
no propagation, a ``pulled" front, or a ``pushed" front, and the 
choice of the scenario is completely characterized via a minimization 
problem \eqref{haoo}. At the same time, they obtained the uniqueness, 
monotonicity and the exponential decay behavior besides the existence
 of the solutions if the functional has non-trivial minimizers. 
Furthermore, they discussed traveling wave solutions characterized 
by a certain ``minimal speed" if the functional does not have non-trivial 
minimizers in \cite{Mur2}.

However, in both \cite{Luc} and \cite{Mur2}, they did not consider the 
relations between nonlinear function, domain and wave speed. Furthermore, 
they did not consider influence of advection on traveling waves in \cite{Mur2} 
where the advection exists. In this paper, ignited by \cite{Hei2}, we give a 
variational formula \eqref{mp1} to
investigate the existence of traveling wave solutions. Here, the traveling wave 
solution with the form of $u(y, z, t) = \bar{u}(y, c(z + ct))$ different from 
the form $u(x,t)=\bar{u}(y,
z-ct)$ considered in \cite{Luc} and \cite{Mur2}. Due to the differences, our 
variational formulation relates the wave velocity to the infimum, which enables
us to obtain some new results. In the next section, we will give
this variational formulation to \eqref{rde1}. In final, we will apply this 
variational formulation
 to a system, i.e., Ginzburg-Landau type problems (\cite{Luc}) , scalar
reaction-diffusion-advection equation in infinite cylinder (\cite{Mur2}). 
For the former, our variational formulation
not only asserts the existence, non-existence, boundedness and
regularity of the obtained solutions but also deduces the monotone
dependence of the velocity on the nonlinearity and the domain under
the same assumptions (i.e. {\rm (H1)}, {\rm (H2)} and {\rm (H3)}) in
\cite{Luc}. For the later, we  obtain the monotone
dependence of the velocity on the nonlinearity and the domain
 besides the existence, uniqueness, monotonicity and
asymptotic behavior at infinity of the obtained solutions under the
same assumptions (i.e. {\rm (A1)}, {\rm (A2)} and {\rm (A3)}) in
\cite{Mur2}. Moreover, we obtain some results about the influence
of advection on the traveling waves, which are different from the case of 
a flow along the cylinder
axis considered in many papers (e.g. \cite{bn, Ber4}). The influence of 
the advection, which transverses to cylinder axis, on traveling waves does 
not be considered in any other literatures.

\begin{remark} \label{rmk1.1}  \rm 
In this article, we only
 give the variational formulation for Equation \eqref{rde1} with boundary \eqref{bc1}.
 In fact, by the same analysis to these of \cite{Luc,Mur2}, this variational 
formulation can be applied to deduce the
 existence  of traveling wave solutions decaying sufficiently rapidly exponentially
  at one end of the cylinder under suitable conditions for \eqref{rde1} and
 \eqref{bc1}. Moreover,  we can obtain a variational representation of the wave 
velocity and the monotone dependence    of the wave velocity on the nonlinearity 
and the domain. For simplicity, we omit the detailed procedures.
\end{remark}

Finally, we give the notation used in the paper. Throughout the paper 
$C^{k}$, $C_{0}^{\infty},\;C^{k,\alpha}$
 denote the usual spaces of continuous functions with $k$ continuous derivatives, 
smooth functions with compact   support, continuously differentially functions 
with  H\"older-continuous derivatives of order $k$ for
  $\alpha\in(0, 1)$, respectively. Unless  otherwise specified in the paper, 
``$\cdot$" denotes a scalar product and $|\cdot|$
  denotes the Euclidean norm in $\mathbb{R}^{n}$. The symbol $\nabla$ is 
reserved for the gradient in $\mathbb{R}^{n}$,
  while $\nabla_{y}$ stands for the gradient in $\Omega\subset \mathbb{R}^{n-1}$. 
Similarly, the symbol $\Delta$ stands
  for the Laplacian in $\mathbb{R}^{n}$, and $\Delta_{y}$ stands for the 
Laplacian in $\Omega\subset \mathbb{R}^{n-1}$.
  The numbers $C$, etc., will denote generic positive constants.

\section{Preliminaries and main results}

\subsection{Variational formulation}

To derive the variational formulation, we firstly introduce the following 
exponentially weighted Sobolev spaces in which we will be working.

\begin{definition} \label{def2.1}  \rm 
 Let $H_{1}^{1}(\Sigma,\mathbb{R}^{m})$ denote
the completion of the restrictions of the functions in
$\big(C^{\infty}_{0}(\mathbb{R}^{n})\big)^m$ to 
$\Sigma$  with respect to the norm
\begin{gather*}
\|u\|^2_{H_{1}^{1}(\Sigma,\mathbb{R}^{m})}
=\|u\|^2_{L_{1}^2(\Sigma,\mathbb{R}^{m})}
+\|\nabla u\|^2_{L_{1}^2(\Sigma,\mathbb{R}^{m})},\\
\|u\|^2_{L_{1}^2(\Sigma,\mathbb{R}^{m})}=\int_{\Sigma}e^{-z
}\sum_{i=1}^{m}|u_{i}|^2\mathrm{d}x.
\end{gather*}
For the Dirichlet boundary condition, replace $C^{\infty}_{0}(\mathbb{R}^{n})$ 
with $C^{\infty}_{0}(\Sigma)$ above.
\end{definition}

\begin{definition} \label{def2.2}  \rm 
 Denote by $H_2^{1}(\Sigma,\mathbb{R})$
the completion of the restrictions of $C^{\infty}_{0}(\mathbb{R}^{n})$ to 
$\Sigma$ with respect to the norm
\begin{gather*}
\|u\|^2_{H_2^{1}(\Sigma,\mathbb{R})}=\|u\|^2_{L_2^2(\Sigma,\mathbb{R})}
+\|\nabla u\|^2_{L_2^2(\Sigma,\mathbb{R})},\\
\|u\|^2_{L_2^2(\Sigma,\mathbb{R})}=\int_{\Sigma}e^{-z+\varphi(y)
}|u|^2\mathrm{d}x.
\end{gather*}
For the Dirichlet boundary condition, replace $C^{\infty}_{0}(\mathbb{R}^{n})$ 
with $C^{\infty}_{0}(\Sigma)$ above.
\end{definition}

We are concerned with traveling wave solutions of the form 
$u(x,t)=u(y, z,t)=\bar{u}(y, c(z+ct))$ with the  wave velocity $c\neq0$.
Substituting it into Equation \eqref{rde1}, one can see that the
traveling wave equation becomes
\begin{equation}\label{twe1}
\bar{u}_{z}=\bar{u}_{zz}+\frac{1}{c^2}(\Delta_{y}\bar{u}+f(\bar{u}))
\end{equation}
with boundary condition \eqref{bc1}. Moreover,  we can always assume
$c>0$ by a possible change of $z$ to $-z$.

Then we define two important functionals as follows:

\begin{definition}  \label{def2.3} \rm
Define two functionals in $H_{1}^{1}(\Sigma,\mathbb{R})$ by
\begin{gather*}
\Gamma[u]=\frac{1}{2}\int_{\Sigma}e^{-z}
\left|\frac{\partial
u}{\partial z}\right|^2\mathrm{d}z\,\mathrm{d}y,\\
J[u]=\int_{\Sigma}e^{-z}
\Big(\frac{1}{2}|\nabla_{y} u|^2-F(u)\Big)\mathrm{d}z\,\mathrm{d}y,
\end{gather*}
where $F(u)=\int_{0}^{u}f(s)\mathrm{d}s$.
\end{definition}

Now based on the above preliminaries, we can give the variational
formulation \eqref{mp1}, so that the existence of traveling waves is converted
into the existence of constraint minimizers.

\begin{theorem} \label{thm2.4}
We consider the constraint minimization problem
\begin{equation}\label{mp2}
\inf_{\{u\in H^{1}_{1}(\Sigma,\mathbb{R})|\Gamma[u]=b\}}J[u],
\end{equation}
where $b$ is a positive constant. Let $\lambda$ be the Lagrange multiplier, 
then Equation \eqref{twe1} is the variational equation corresponding 
to \eqref{mp2} and
\begin{equation}\label{ve1}
\lambda\Gamma[u]+J[u]=0,
\end{equation}
where $\lambda=c^2$.
\end{theorem}

\begin{proof} By \cite{Str}, we can easily obtain that  \eqref{twe1} is 
the variational equation corresponding to \eqref{mp2} with $\lambda$ 
as the Lagrange multiplier. In the following, we only need to show  \eqref{ve1}.
 By multiplying Equation \eqref{twe1} by $e^{-z}u_{z}$ and integrating over 
$\Sigma$, we obtain
\[
\int_{\Sigma}e^{-z}u_{z}\big[ u_{z}-u_{zz}-\frac{1}{c^2}(\Delta_{y}u+f(u)
)\big]\mathrm{d}z\mathrm{d}y=0.
\]
So \eqref{ve1} follows easily by the boundary condition \eqref{bc1} and 
integrating by parts.
\end{proof}

Since $\Gamma[u]=b$, we can always consider $b=1$ without loss of
generality, which can be achieved by a suitable shift in $z$, then
the Lagrange multiplier satisfies
\begin{equation}\label{lm1}
\lambda=c^2=-\inf_{\{u\in H^{1}_{1}(\Sigma,\mathbb{R})|\Gamma[u]=1\}}J[u]>0.
\end{equation}

\subsection{Ginzburg-Landau type problems}

In this subsection, we apply our formulation \eqref{lm1} to 
the Ginzburg-Landau-type problems. Under assumptions 
{\rm (H1)}, {\rm (H2)} and {\rm (H3)}, we not only obtain the existence,
 non-existence, boundedness and regularity of the obtained solutions, 
but also obtain some new results, i.e. the monotone dependence of the 
velocity on the nonlinearity and the domain.

For traveling wave solutions of the form
$u(y,z,t)=\bar{u}(y,c(z+ct))$, substituting  it into 
\eqref{glt}, we obtain traveling wave equation
 \begin{equation}\label{twe2}
\bar{u}_{z}=\bar{u}_{zz}+\frac{1}{c^2}\left(\Delta_{y}\bar{u}
-\nabla_{\bar{u}}V\right)
\end{equation}
with boundary condition \eqref{bc2}.
Then by Definition \ref{def2.3}, we have the following definition.

\begin{definition} \label{def2.5}  \rm 
 Let $u\in H_{1}^{1}(\Sigma,\mathbb{R}^{m})$, and define functionals by
\begin{gather*}
\Gamma_{1}[u]=\frac{1}{2}\int_{\Sigma}e^{-z}\sum_{i=1}^{m}
\left|\frac{\partial
u_{i}}{\partial z}\right|^2\mathrm{d}\,y\mathrm{d}z, \\
J_{1}[u]=\int_{\Sigma}e^{-z}
\Big(\frac{1}{2}\sum_{i=1}^{m}|\nabla_{y} u_{i}|^2+V(u)\Big)\mathrm{d}y\,\mathrm{d}z.
\end{gather*}
\end{definition}
By \eqref{mp2}-\eqref{lm1}, we know that
\begin{equation}\label{mp3}
\lambda=-\inf_{u\in\mathcal{B}}
J_{1}[u]>0,
\end{equation}
where $\lambda=c^2$ is the Lagrange multiplier,
$\mathcal{B}=\{u\in
H^{1}_{1}(\Sigma,\mathbb{R}^{m})|\Gamma_{1}[u]=1\}$.

Then to show the existence, non-existence, boundedness and
regularity analogous to \cite{Luc} by our variational formulation
\eqref{mp3}, we first show that  $u\in
H^{1}_{1}(\Sigma;\mathbb{R}^{m})$ under the assumption
$c^2+4\upsilon_{0}>0$.

 Linearizing Equation \eqref{twe2} around
$u=0$ for $z\to -\infty$, then we obtain that the solutions of 
\eqref{twe2} are approximately superposition of functions
$u_{k}(y,z)=e^{-\lambda_{k}z}v_{k}(y)$, where $\lambda_{k}$
satisfies
\begin{equation}\label{ev1}
\lambda_{k}^2+\lambda_{k}-\frac{1}{c^2}\upsilon_{k}=0,
\end{equation}
 $v_{k}(y)$ and $\upsilon_{k}\in \mathbb{R}$ are the eigenfunction and 
the eigenvalue defined by
\[
 -\Delta_{y}v_{k}+H(0)v_{k} =\upsilon_{k}v_{k},\quad
 H(u)=(\nabla_{u}\otimes\nabla_{u})V(u),
\]
with the boundary condition \eqref{bc2} respectively. 
Where $H(u)$ is the Hessian of the potential $V(u)$ (here we also assume that 
$V$ is twice differentiable at the origin). From Equation \eqref{ev1}, we obtain
\[
 \lambda_{k}^{\pm}(c)=\frac{-1\pm\sqrt
 {1+\frac{4}{c^2}\upsilon_{k}}}{2},
\]
so  by the same discussion to that of \cite{Luc}, we know if
$c^2+4\upsilon_{0}>0$, $u\in H^{1}_{1}(\Sigma,\mathbb{R}^{m})$.

Secondly, we can obtain one important inequality that is an analogue 
of the Poincar\'e inequality.

\begin{proposition} \label{prop2.6}
Let $u\in H_{1}^{1}(\Sigma,\mathbb{R}^{m})$, then
\begin{itemize}
\item[(i)]
\begin{equation}\label{ie1}
 \frac{1}{4}\int_{-\infty}^{-R}\int_{\Omega}
e^{-z}
\sum_{i=1}^{m}|u_{i}|^2\mathrm{d}y\mathrm{d}z
\leq\int_{-\infty}^{-R}\int_{\Omega}
e^{-z}\sum_{i=1}^{m}\left|\frac{\partial
u_{i}}{\partial z}\right|^2\mathrm{d}y\mathrm{d}z;
\end{equation}

\item[(ii)]
\begin{equation}\label{ie2}
 \int_{\Omega}
\sum_{i=1}^{m}u^2_{i}(y,-R)\mathrm{d}y\leq e^{-R}
\int_{-\infty}^{-R}\int_{\Omega}
e^{-z}\sum_{i=1}^{m}\left|\frac{\partial
u_{i}}{\partial z}\right|^2\mathrm{d}y\mathrm{d}z
\end{equation}
for any $R\in (-\infty,+\infty)\bigcup\{-\infty,+\infty\}$.
\end{itemize}
\end{proposition}

\begin{proof} We first prove  (i). As
\begin{align*}
&\int_{-\infty}^{-R}\int_{\Omega}
e^{-z}
|u_{i}|^2\mathrm{d}y\,\mathrm{d}z\\
&=-e^{R}\int_{\Omega}u_{i}^2(y,-R)\mathrm{d}y
+2\int_{-\infty}^{-R}\int_{\Omega} e^{-z} u_{i}\frac{\partial
u_{i}}{\partial z}\mathrm{d}y\mathrm{d}z\\
&\leq 2\Big(\int_{-\infty}^{-R}\int_{\Omega}
e^{-z}
|u_{i}|^2\mathrm{d}y\mathrm{d}z\Big)^{1/2}
\Big(\int_{-\infty}^{-R}\int_{\Omega}e^{-z}|\frac{\partial
u_{i}}{\partial z}|^2\mathrm{d}y\,\mathrm{d}z\Big)^{1/2},
  \end{align*}
Then \eqref{ie1} follows.

Now, we give the proof of (ii). Note that
\[
\int_{-\infty}^{-R}\int_{\Omega}
e^{-z} \big(u_{i}-\frac{\partial
u_{i}}{\partial z}\big)^2\mathrm{d}y\,\mathrm{d}z\geq0,
\]
we can obtain
\begin{align*}
\int_{-\infty}^{-R}\int_{\Omega}e^{-z}|\frac{\partial
u_{i}}{\partial z}|^2\mathrm{d}y\mathrm{d}z
&\geq2\int_{-\infty}^{-R}\int_{\Omega}
e^{-z}
u_{i}\frac{\partial
u_{i}}{\partial z}\mathrm{d}y\mathrm{d}z-\int_{-\infty}^{-R}\int_{\Omega}
e^{-z}
|u_{i}|^2\mathrm{d}y\mathrm{d}z\\
&=e^{R}\int_{\Omega}
u^2_{i}(y,-R)\mathrm{d}y.
\end{align*}
Thus, \eqref{ie2} is obtained.
\end{proof}

Let $I_{1}[u]=c^2\Gamma_{1}[u]+J_{1}[u]$, we first note here that there exist 
$c>0$ such that $c^2+4\upsilon_{0}>0$, and $u\in
H_{1}^{1}(\Sigma,\mathbb{R}^{m}),\; u\not\equiv0$ such that
$I_{1}[u]\leq0$ by assumption {\rm (H3)}. The functionals $I_{1}[u]$ and 
$J_{1}[u]$ have the same weak lower semicontinuous since $\Gamma_{1}[u]$ 
is weak lower semicontinuous. Furthermore, $J_{1}[u]$ is coercive because 
of assumption {\rm (H1)} and Equation \eqref{ie1}. Hence, $J_{1}[u]$ 
has non-trivial constraint minimizers by \cite{Luc}. If $\bar{u}$ is a
 minimizer of \eqref{mp3}, $I_{1}[\bar{u}]=0$, then by assumptions 
{\rm (H1)} and  {\rm (H2)}, our variational formulation yields the 
non-existence, boundedness and regularity of the obtained solutions by a 
similar discussion to that of \cite{Luc}.

Finally, we give the results of monotone dependence by our
variational formulation.

\begin{theorem}\label{thm2.7} 
We assume that the following functions $V,\;\tilde{V}$ and $\bar{V}$ 
satisfy assumptions  {\rm (H1)}-{\rm (H3)}, then we have
\begin{itemize}
\item[(i)] If $\tilde{V}\geq\bar{V}$, then $\tilde{\lambda}\leq\bar{\lambda}$,
that is, $\tilde{c}\leq\bar{c}$;

\item[(ii)] If $\tilde{\Omega},\;\bar{\Omega}\subset \mathbb{R}^{n-1}\;(n\geq3)$ 
are bounded domain with boundary of class $C^2$ and 
$\tilde{\Omega}\subset\bar{\Omega}$, then 
$\tilde{\lambda}\geq\bar{\lambda}$; that is, $\tilde{c}\geq\bar{c}$;

\item[(iii)] Let boundary condition \eqref{bc2} only be Dirichlet boundary 
condition, then $\lambda$ is the most for the ball compared to all domains 
$\Omega$ with the same volume.
 \end{itemize}
\end{theorem}

\begin{proof} (i) By assumption $\tilde{V}\geq\bar{V}$, we obtain corresponding 
functionals satisfying $\tilde{J}_{1}[\bar{u}]\geq\bar{J}_{1}[\bar{u}]$, and 
by Equation \eqref{mp3}, we have $\tilde{\lambda}\leq\bar{\lambda}$; that is, 
$\tilde{c}\leq\bar{c}$.

 (ii) Let $\bar{u}$ be a non-trivial minimizer of \eqref{mp3} corresponding 
to $\tilde{\Omega}$. Then 
$1=\tilde{\Gamma}_{1}[\bar{u}]\leq\bar{\Gamma}_{1}[\bar{u}]$,
 therefore, there exists a shift $\bar{u}_{a}=\bar{u}(z+a,y),\;a\leq0$ such that
 \[
 \bar{\Gamma}_{1}[\bar{u}_{a}]
 =e^{a}\bar{\Gamma}_{1}[\bar{u}]=1.
\]
Let us extend the minimizer in $\tilde{\Omega}$ by $0$ to $\bar{\Omega}$, 
then since $a\leq0,\;\bar{J}_{1}[\bar{u}_{a}]=e^{a}\bar{J}_{1}[\bar{u}]
\geq-\tilde{\lambda}$, we have $\tilde{\lambda}\geq\bar{\lambda}$; that is, 
$\tilde{c}\geq\bar{c}$. Where $f=-\nabla_{u}V(u)$.

 (iii) By spherical rearrangement in the coordinate $y$, we can obtain this 
process decreases the functional $J_{1}[u]$ and preserves $\Gamma_{1}[u]$ 
by \cite{Kaw}, so $\lambda$ is the most for the ball in all other domains 
$\Omega$ with the same volume by \eqref{mp3}. Where $f=-\nabla_{u}V(u)$.
\end{proof}

 \begin{remark} \label{rmk2.8}  \rm 
From the above discussion, we can obtain the boundedness of the obtained 
solutions analogous to \cite[Theorem 3.3]{Luc}, that is, we have 
$|\bar{u}(y,z)|\leq Ce^{-\lambda z}$ for some $C>0$ and $\lambda<0$. 
Then $\bar{u}(z,.)\to0$ as $z\to-\infty$ in $C^{1}(\bar{\Omega})$ is obtained.
\end{remark}

 \subsection{Scalar reaction-diffusion-advection equations}\label{one0}

 In this subsection under the assumptions {\rm (A1)}, {\rm
(A2)} and {\rm (A3)}, we apply our variational formulation to the
scalar reaction-diffusion equation  \eqref{rdap} with boundary
conditions \eqref{bc3}.

For traveling wave solutions of the form
$u(y,z,t)=\bar{u}(y,c(z+ct))$, substituting  it into 
\eqref{rdap}, we obtain the following traveling wave equation
 \begin{equation}\label{twe3}
\bar{u}_{z}=\bar{u}_{zz}+\frac{1}{c^2}
\left(\Delta_{y}\bar{u}+\nabla_{y}\varphi\cdot\nabla_{y}\bar{u}
+f(\bar{u},y)\right)
\end{equation}
with boundary conditions \eqref{bc3}.

\begin{definition} \label{def2.9}  \rm  
Let $u\in H_2^{1}(\Sigma)$, define functionals 
\begin{gather*}
\Gamma_2[u]=\frac{1}{2}\int_{\Sigma}e^{-z+\varphi(y)}
|\frac{\partial u}{\partial z}|^2\mathrm{d}y\,\mathrm{d}z,\\
J_2[u]=\int_{\Sigma}e^{-z+\varphi(y)}
\Big(\frac{1}{2}|\nabla_{y} u|^2+V(u,y)\Big)\mathrm{d}y\,\mathrm{d}z,
\end{gather*}
where
\[
V(u,y)=\begin{cases}
0,& u<0,\\
-\int^{u}_{0}f(s,y)\mathrm{d}s, &0\leq u\leq1,\\
-\int^{1}_{0}f(s,y)\mathrm{d}s, &u>1.
\end{cases}
\]
\end{definition}

By \eqref{mp2}-\eqref{lm1} and a similar discussion in Section 2, we have 
the following:
\begin{gather}\label{ve2}
\lambda\Gamma_2[u]+J_2[u]=0, \\
\label{mp4}
\lambda=c^2=-\inf_{\{u\in H^{1}_2(\Sigma)|\Gamma_2[u]=1\}}
J_2[u]>0\,.
\end{gather}
Equation \eqref{twe3} is the variational equation corresponding to \eqref{mp4}.

\begin{theorem}\label{thm2.10} 
Assume that hypotheses  {\rm (A1)}-{\rm (A3)} hold, then there exists a unique 
value of $c^{\star}\geq c$, where $c$ is defined by hypothesis {\rm (A3)}, 
and a unique function $\bar{u}\in C^2(\Sigma)\bigcap C^{1}(\bar{\Sigma})$,
 $\bar{u}\not\equiv0$, such that $(c^{\star},\bar{u})$ is the solution 
of \eqref{twe3} with boundary conditions \eqref{bc3}. 
Moreover, $\bar{u}\in H^2(\Sigma)\bigcap W^{1,\infty}(\Sigma),\;\bar{u}_{z}>0$ 
in $\Sigma$, and
\begin{equation}\label{v1}
 \lim_{z\to+\infty}(.,z)=v,\quad
 \lim_{z\to-\infty}(.,z)=0
\end{equation}
in $C^{1}(\bar{\Omega})$.
\begin{equation}\label{zin}
 \bar{u}(y,z)=a_{0}\psi_{0}(y)
 e^{-\lambda_{-}(c^{\star},\upsilon_{0})}+0(e^{-\lambda z})
\end{equation}
for some $a_{0}>0$ and $\lambda< \lambda_{-}(c^{\star},\upsilon_{0})$,
uniformly in $C^{1}(\bar{\Omega}\times[-\infty,R])$, as $R\to-\infty$.
\[
 \bar{u}(y,z)=v(y)+\tilde{a}_{0}\tilde{\psi}_{0}(y)
 e^{-\lambda_{+}(c^{\star},
 \tilde{\upsilon}_{0})}+0(e^{-\lambda z})
\]
for some $\tilde{a}_{0}>0$ and
 $\lambda> \lambda_{+}(c^{\star},\tilde{\upsilon}_{0})$,
uniformly in $C^{1}(\bar{\Omega}\times[R,+\infty])$, as $R\to+\infty$. 
Where $v:\Omega\mapsto\mathbb{R}$ is a local minimizer of 
$E(v)=\int_{\Omega}e^{\varphi(y)}(\frac{1}{2}|\nabla_{y}v|^2+V(v(y),y))\mathrm{d}y$ 
with $E(v)<0$, $\tilde{\upsilon}_{0},\;\tilde{\psi}_{0}$ and 
$\lambda_{+}(c^{\star},\tilde{\upsilon}_{0})$ are obtained by linearizing
 \eqref{twe3} around $u=v$ at large $z$.
\end{theorem}

\begin{proof} 
Due to the discussion in Section 3.1 and the maximum principle, we only need to
show
 $u\in H^{1}_2(\Sigma)$ under the assumption $c^2+4\upsilon_{0}>0$ in  
{\rm (A3)} by \cite{Mur2}.

Linearizing \eqref{twe3} around $u=0$ at large $(-z)$, then
we obtain
\[
u(y,z)\sim \Sigma_{k} a_{k}\psi_{k}(y)e^{-\lambda_{k}z},\quad (k=0,1,2,\dots)
\]
with $(\lambda_{k},\;\upsilon_{k})$
satisfying
\begin{equation}\label{ev2}
\lambda_{k}^2+\lambda_{k}-\frac{1}{c^2}\upsilon_{k}=0,
\end{equation}
and $\upsilon_{k}\in \mathbb{R}$ is the eigenvalue defined by
\[
 \Delta_{y}\psi_{k}+
 \nabla_{y}\varphi\cdot\nabla_{y}\psi_{k}
 +f_{u}(0,y)\psi_{k}
 =-\upsilon_{k}\psi_{k}
\]
 with boundary conditions \eqref{bc3}. Then by \eqref{ev2}, we obtain
\[
 \lambda_{k}^{\pm}(c)=\frac{-1\pm\sqrt
 {1+\frac{4}{c^2}\upsilon_{k}}}{2}.
\]
Hence, we obtain $u\in H^{1}_2(\Sigma)$ under the assumption
$c^2+4\upsilon_{0}>0$ by the same discussion to that of
\cite{Mur2}. Then, the existence, uniqueness, monotonicity and
asymptotic behavior at infinity of the obtained traveling wave
solutions are deduced by \cite{Mur2}.
\end{proof}

Finally, we give the new results deduced by our variational formulation.

\begin{theorem}\label{thm2.11}
 We assume that all the  nonlinearities $f,\tilde{f},\bar{f}$ satisfy assumptions
  {\rm (A1)}--{\rm (A3)}. Then we have
\begin{itemize}
\item[(i)] If
 $\tilde{V}(u,y)\leq\bar{V}(u,y)$, then 
$\tilde{\lambda}\geq\bar{\lambda}$; that is, $\tilde{c}\geq\bar{c}$. Where 
\[
\tilde{V}(u,y)\;(\bar{V}(u,y))=\begin{cases}
0,& u<0,\\
-\int^{u}_{0}\tilde{f}(s,y)\mathrm{d}s\;
(-\int^{u}_{0}\bar{f}(s,y)\mathrm{d}s),& 0\leq u\leq1,\\
-\int^{1}_{0}\tilde{f}(s,y)\mathrm{d}s\;
(-\int^{1}_{0}\bar{f}(s,y)\mathrm{d}s), & u>1.
\end{cases}
\]

\item[(ii)] If $\tilde{\Omega},\bar{\Omega}\subset \mathbb{R}^{n-1}$ 
are bounded domain with boundary of class $C^2$, $f_{y}(u,y)=0$ and 
$\tilde{\Omega}\subset\bar{\Omega}$, then $\tilde{\lambda}\geq\bar{\lambda}$;
 that is, $\tilde{c}\geq\bar{c}$;

\item[(iii)] Let $\Omega_{0}=\emptyset$ and $f_{y}(u,y)=0$, then
 $\lambda$ is the most for the ball in all other domains $\Omega$ with 
the same volume.
 \end{itemize}
\end{theorem}

The proof of the above theorem is basically the same proof as the one
of Theorem \ref{thm2.7}.
In the following, we consider the influence of the advection on traveling wave.

\begin{theorem}\label{shun3}
 We assume $\varphi=\tilde{\varphi},\;\bar{\varphi}$ and nonlinearity 
$f$ satisfy assumptions  {\rm (A1)}-{\rm (A3)}, then if
\begin{equation}\label{ping1}
e^{\tilde{\varphi}(y)}
 V(e^{-\frac{\tilde{\varphi}(y)}{2}}w,y)\geq e^{\bar{\varphi}(y)}
 V(e^{-\frac{\bar{\varphi}(y)}{2}}w,y)
\end{equation}
for
 \begin{equation}\label{fa}
|\nabla_{y}\tilde{\varphi}|^2+2\Delta_{y}
 \tilde{\varphi}\geq
 |\nabla_{y}\bar{\varphi}|^2+2\Delta_{y}
 \bar{\varphi},
\end{equation}
then we have $\tilde{\lambda}\leq\bar{\lambda}$; that is
$\tilde{c}\leq\bar{c}$:
If
\begin{equation}\label{ping2}
e^{\tilde{\varphi}(y)}
 V(e^{-\frac{\tilde{\varphi}(y)}{2}}w,y)\leq e^{\bar{\varphi}(y)}
 V(e^{-\frac{\bar{\varphi}(y)}{2}}w,y)
\end{equation}
for
\begin{equation}\label{fay}
|\nabla_{y}\tilde{\varphi}|^2+2\triangle_{y}
 \tilde{\varphi}\leq
 |\nabla_{y}\bar{\varphi}|^2+2\triangle_{y}
 \bar{\varphi},
\end{equation}
 then we have $\tilde{\lambda}\geq\bar{\lambda}$, that is $\tilde{c}\geq\bar{c}$,
 where $\tilde{c}$ and $\bar{c}$ are wave speeds corresponding  to 
$\tilde{\varphi}$ and $\bar{\varphi}$ respectively.
\end{theorem}

\begin{proof} 
Let $w=e^{\varphi(y)/2}u$; i.e. $u=e^{-\varphi(y)/2}w$ and replace $u$ 
by $w$ in the functional $\Gamma_2[u]$ and $J_2[u]$, we obtain
\begin{equation}\label{shun1}
\Gamma_2[w]=\frac{1}{2}\int_{\Sigma}e^{-z}
|\frac{\partial w}{\partial z}|^2\mathrm{d}y\,\mathrm{d}z
\end{equation}
and
\begin{equation}\label{shun2}
  \begin{split}
J_2[w]&=\frac{1}{2}\int_{\Sigma}e^{-z}
\Big(|\nabla_{y} w|^2+\Big(\frac{1}{4}(|\nabla_{y} \varphi|^2+2\triangle_{y}
 \varphi)\Big)w^2\Big)\mathrm{d}y\,\mathrm{d}z\\
 &\quad +\int_{\Sigma}e^{-z+\varphi(y)}
 V(e^{-\frac{\varphi(y)}{2}}w,y)
 \mathrm{d}y\,\mathrm{d}z.
 \end{split}
 \end{equation}
Let $\tilde{J}_2,\tilde{\Gamma}_2;
\bar{J}_2,\bar{\Gamma}_2$ be corresponding functionals
to $\tilde{\varphi}$ and $\bar{\varphi}$ respectively. Thus,
by virtue of \eqref{shun1} and  \eqref{shun2},  we have 
$\tilde{J}_2\geq\bar{J}_2$
 if \eqref{ping1} and \eqref{fa} hold.
And by \eqref{mp4}, then $\tilde{\lambda}\leq\bar{\lambda}$; that is,
$\tilde{c}\leq\bar{c}$.

Similarly, we can obtain the remaining results.
\end{proof}

\begin{theorem}\label{shun7}
 Assume {\rm (A1)--(A3)} hold. Then if
 \begin{equation}\label{ping3}
0\geq e^{\varphi(y)}
 V(e^{-\frac{\varphi(y)}{2}}w,y)\geq
 V(w,y)
\end{equation}
for
 \begin{equation}\label{fa2}
|\nabla_{y}\varphi|^2+2\triangle_{y}
 \varphi\geq0,
\end{equation}
then we have $\tilde{\lambda}\leq\bar{\lambda}$; that is,
$\tilde{c}\leq\bar{c}$.
If
\begin{equation}\label{ping4}
e^{\varphi(y)}  V(e^{-\frac{\varphi(y)}{2}}w,y)\leq
 V(w,y)
\end{equation}
for
\begin{equation}\label{fay1}
|\nabla_{y}\varphi|^2+2\triangle_{y}
 \varphi\leq0,
\end{equation}
then $\tilde{\lambda}\geq\bar{\lambda}$, that is
$\tilde{c}\geq\bar{c}$, where the wave velocity of the advection
equation \eqref{rdap} and equation without advection (that is,
$\varphi=0$ in Equation \eqref{rdap}) be $\tilde{c}$ and $\bar{c}$
respectively.
\end{theorem}

\begin{proof}
 Let $\tilde{J}_2,\;\tilde{\Gamma}_2$ and $\bar{J}_2,\;\bar{\Gamma}_2$ be 
the corresponding functionals to the advection equation \eqref{rdap} 
and equation without advection (that is, $\varphi=0$ in Equation \eqref{rdap})
 respectively.
By \eqref{shun1} and  \eqref{shun2},  if \eqref{ping3}
and \eqref{fa2} hold, we have $\tilde{J}_2\geq\bar{J}_2$, and by
\eqref{mp4}, then $\tilde{\lambda}\leq\bar{\lambda}$, that is
$\tilde{c}\leq\bar{c}$.
Similarly, we also can obtain the remaining results.
\end{proof}

\begin{remark} \label{rmk2.14} \rm
 We can obtain some specific conditions such that
\eqref{ping1} (\eqref{ping3}) and \eqref{ping2} (\eqref{ping4}) hold
in Theorem \ref{shun3} (Theorem \ref{shun7}) respectively.
For example, if
\[
\left\{
\begin{array}{ll}
V(u,y)\geq0\;\textrm{and}\;
V(e^{-\frac{\tilde{\varphi}(y)}{2}}w,y)\geq
 V(e^{-\frac{\bar{\varphi}(y)}{2}}w,y)\;(\textrm{or}\;V_{u}(u,y)\leq0) \;\textrm{for}\;\tilde{\varphi}\geq\bar{\varphi}\;\;
\\\textrm{or}\\
V(u,y)\leq0\;\textrm{and}\;
V(e^{-\frac{\tilde{\varphi}(y)}{2}}w,y)\geq
 V(e^{-\frac{\bar{\varphi}(y)}{2}}w,y)\;(\textrm{or}\;V_{u}(u,y)\geq0)
  \;\textrm{for}\;\tilde{\varphi}\leq\bar{\varphi},
\end{array}\right .
\]
\[
\left\{
\begin{array}{ll}
V(u,y)\geq0\;\textrm{and}\;
V(e^{-\frac{\tilde{\varphi}(y)}{2}}w,y)\leq
 V(e^{-\frac{\bar{\varphi}(y)}{2}}w,y)\;(\textrm{or}\;V_{u}(u,y)\leq0) \;\textrm{for}\;\tilde{\varphi}\leq\bar{\varphi}\;\;
\\\textrm{or}\\
V(u,y)\leq0\;\textrm{and}\;
V(e^{-\frac{\tilde{\varphi}(y)}{2}}w,y)\leq
 V(e^{-\frac{\bar{\varphi}(y)}{2}}w,y)\;(\textrm{or}\;V_{u}(u,y)\geq0)
 \;\textrm{for}\;\tilde{\varphi}\geq\bar{\varphi},
\end{array}\right .
\]
\[
V(u,y)\leq0\;\textrm{and}\;V(e^{-\frac{\varphi(y)}{2}}w,y)\geq
 V(w,y)\;(\textrm{or}\;V_{u}(u,y)\geq0)
\;\textrm{for}\;\varphi(y)\leq0
\]
and
\[
\left\{
\begin{array}{ll}
V(u,y)\leq0\;\textrm{and}\;V(e^{-\frac{\varphi(y)}{2}}w,y)\leq
 V(w,y)\;(\textrm{or}\;V_{u}(u,y)\geq0)
\;\textrm{for}\;\varphi(y)\geq0\;\;
\\\textrm{or}\\
V(u,y)\geq0\;\textrm{and}\;V(e^{-\frac{\varphi(y)}{2}}w,y)\leq
 V(w,y)\;(\textrm{or}\;V_{u}(u,y)\leq0)
\;\textrm{for}\;\varphi(y)\leq0
\end{array}\right .
\]
hold, then \eqref{ping1}, \eqref{ping2}, \eqref{ping3} and
\eqref{ping4} corresponding hold.
\end{remark}

 \subsection*{Acknowledgments}
This work was supported by the Natural Science Foundation of Gansu
 Province (213232), by the Youth Science Foundation of
Lanzhou Jiaotong University (2013027), and by the NSF of China (11361032).


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