\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{epic} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 86, pp. 1--9.\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 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/86\hfil Travelling wave solutions] {Travelling wave for absorption-convection-diffusion equations} \author[A. Hamydy\hfil EJDE-2006/86\hfilneg] {Ahmed Hamydy} \address {Ahmed Hamydy \newline Universite Abdelmalek Essaadi, Faculte des Sciences Departement \\ de Mathematiques et informatique B. P. 2121 Tetouan, Maroc} \email{hamydy@caramail.com} \date{} \thanks{Submitted April 5, 2006. Published August 2,2006.} \subjclass[2000]{35K55, 35K65} \keywords{Travelling wave; phase-plane; diffusion; convection; absorption; \hfill\break\indent asymptotic behavior} \begin{abstract} In this paper, we use the phase plane method for finding finite travelling waves solutions for the diffusion-absorption-convection equation $u_t=A(|u_x|^{p-2}u_{x})_x+B(u^n)_x-Cu^q, \quad (x,t)\in \mathbb{R}\times \mathbb{R}^{+}.$ We show that the existence of solutions which depends on the parameters $p$, $q$ and $n$. Also we study the asymptotic behavior of these solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} This work concerns the nonlinear parabolic equation $$U_t=A\Delta _pU+B(U^n)_x-CU^q, \quad (x,t)\in \mathbb{R}\times \mathbb{R}^{+}\label{E11}$$ with $\Delta _pU=(|U_x|^{p-2}U_x) _x$, $A,B,C,q>0$, $p>2$ and $1\neq n>0$. Equation \eqref{E11} is usually called a convection absorption diffusion equation. It is a simple and widely used model for various physical, chemical problems involving diffusion with absorption and with convection. Note that the operator of diffusion $\Delta_p$ (the $p$-laplacian) arises from a variety of physical phenomena. It is used in Non-Newtonian fluids and also appears in nonlinear elasticity, glaciology and petroleum extraction (see \cite{AD,F,AE,LA}). Our main interest is finite travelling waves solutions of \eqref{E11}. For $p=2$, the travelling waves have been widely studied since the paper \cite{KPP} was published (see also \cite{H}). It is worth mentioning that, if we change $\Delta _pU$ by $(U^m) _{xx}$, we get the porous medium equation, for which the travelling waves have been studied with a term of diffusion, or convection or both at the same time (\cite{PS,PV,HV}). We know that for $p>2$ the operator $\Delta _p$ have a finite propagation property. The introduction of absorption term and convection term may have a deep influence on the qualitative behavior of the solutions of \eqref{E11}. To clear up this phenomena we will study a particular family of solutions. More precisely, we investigate the existence and the asymptotic behavior of finite travelling waves for variants values of $A>0,B>0$ and $C>0$. By a simple rescaling we may put $A=1$ and $B=\frac1{n}$. The formulae will be much simpler. In fact we take $$V(x,t)=\alpha U(\beta x,\gamma t) \label{E12}$$ with $$\alpha =1,\quad \beta =(nB)^{-\frac 1{p-1}},\quad \gamma =A^{-\frac 1{p-1}}(nB)^{-\frac p{p-1}} \label{E13}$$ Hence $V$ satisfies the diffusion absorption convection equation $$V_t=(|V_x|^{p-2}V_x) _x+\frac 1n(V^n)_x-dV^q, \label{E14}$$ where $d=\gamma C.$ For simplicity of notation we take $d=1$. By a local finite travelling wave solution with velocity $c\in\mathbb{R}$, we mean a solution $w(x,t)$ of \eqref{E14} in $I\times\mathbb{R}^{+}$ (for some interval $I=]-\infty,\xi_\infty[, \xi_\infty>0$) of the form $$w(x,t)=\varphi (ct-x)=\varphi (\xi ) \label{E15}$$ with $\varphi :I\to \mathbb{R}^{+}$ such that $\varphi$ is strictly positive in $] 0,\infty[ \cap I$ and vanishing in $] -\infty ,0]$. If $I=\mathbb{R}$, we say that $\varphi$ is a finite travelling wave solution (finite travelling wave ). Before stating the main results of this paper, let us introduce some useful notations. For any real $r$ we set $M_{r}$ (resp. $A_r$) the solution of the algebra equation $M^{-\frac 1{p-1}}-M+r=0$ (resp $. M^{-\frac 1{p-1}}-rM+1=0$). Define, \begin{gather} Q=\frac{(q+1)(p-1)}p, \label{1.6} \\ L=\max\{n,Q\},\quad l=\min\{n,Q\}, \label{1.7} \\ M_0=\begin{cases} M_c &\text{if } l=Q=1,\\ \frac 1n, &\text{if } l=n\neq Q, \\ A_n, &\text{if } n=Q=l,\\ Q^{(1-p)/p} &\text{if } l=Q\neq n,1,\\ \end{cases} \label{1.8} \\ M_\infty=\begin{cases} M_c &\text{if } L=Q=1,\\ \frac 1n,&\text{if } L=n\neq Q, \\ A_n, &\text{if } L=n=Q,\\ Q^{(1-p)/p} &\text{if } L=Q\neq n,1.\\ \end{cases} \label{1.9} \end{gather} The main results are follows. \begin{theorem}\label{th11} Let $c\in\mathbb{R}^\star$ $(c\neq 0)$. Equation \eqref{E14} admits a finite travelling wave solution with velocity $c$ if and only if one of the following conditions is satisfied : \begin{itemize} \item[(i)] $c>0$, $L\leq{ p-1}$; \item[(ii)] $c<0$, $l\leq {1}\leq {L}\leq {p-1}$; \item[(iii)] $c<0$, $L<1$, $q\leq{1}$; \item[(iv)] $c<0$, $l>1$, $q<1$. \end{itemize} \end{theorem} \begin{theorem}\label{th12} Let $\varphi$ a finite travelling wave solution at some velocity $c\neq 0$ of \eqref{E14}. Then $\xi ^{-\alpha }\varphi (\xi )$ converges to constant $K=\alpha ^{-\alpha }M^{\alpha /p-1}$, when $\xi$ approaches $0$, with : \begin{itemize} \item[(i)] For $c>0 \;\;or \;\; c (l-1)\geq 0$, $\alpha =\frac{p-1}{p-inf\{n,Q,1\}-1}$ and $M=M_0$. \item[(ii)] For $c<0$ \;and \;\; $l>1$, $\alpha =\frac {1}{1-q}$ and $M=(-c)^{1-p}$. \end{itemize} \end{theorem} For the asymptotic behavior when $\xi$ approaches $\infty$, there is the following result: \begin{theorem}\label{th13} Let $\varphi$ a finite travelling wave solution of \eqref{E14} at some velocity $c\neq 0$. \begin{itemize} \item[(i)] For $c>0 \;\;or \;\; c \;(L-1)\geq 0$, we have: If $L2$, $q>0$ and $1\neq n>0$. Substituting $u(x,t)=\varphi(ct-x)$ in \eqref{E14} we get $$\big(|\varphi'|^{p-2}\varphi'\big)'-c\varphi'-\frac{1}{n} (\varphi ^n)' -\varphi ^q=0 \quad\text{in } \mathbb{R}, \label{E21}$$ If $\varphi$ vanishes for $\xi\leq \xi_0$, by translation we may put $\xi_0=0$. Integrating (\ref{E21}) we obtain $\varphi^\prime(0)=0$. Consequently, our problem can reformulated as finding a real $c$ and a function $\varphi$ such that $$\big(|\varphi'|^{p-2}\varphi'\big)'-c\varphi'-\frac{1}{n} (\varphi ^n)' -\varphi ^q=0 \quad\text{in } \mathbb{R}, \label{E22}$$ $$\varphi (0)=0, \varphi'(0)=0.\label{E23}$$ We start with the following lemma which gives us a monotonicity property of a finite travelling wave solutions . \begin{lemma}\label{lem21} Let $c\in \mathbb{R}^{*}$ and $\varphi$ a local finite travelling wave solution of \eqref{E14} with velocity $c$. Then $\varphi$ is increasing on $] 0,+\infty[$. \end{lemma} \begin{proof} The proof is divided into two steps.\\ {\bf Step 1.} $c>0.$ Let $E$ the energy function defined by $$E(\xi )=\frac{p-1}p|\varphi'(\xi )|^p-\frac 1{q+1}\varphi ^{q+1}(\xi ) \label{E24}$$ Then $E$ satisfies $$E'(\xi )=[c+\varphi ^{n-1}(\xi )] \varphi ^{\prime 2}(\xi ) \label{E25}$$ Hence $E$ is increasing. Assume $\varphi$ is not increasing on $% ] 0,+\infty[$ and let $\xi _1$ the first zero of $\varphi'$. Therefore $$E(\xi _1)=-\frac 1{q+1}\varphi ^{q+1}(\xi _1)\geq E(0)=0 \label{E26}$$ which a contradiction and then $\varphi'(\xi )>0$, for any $\xi >0$. \noindent{\bf Step 2.} $c<0$. From the definition of finite travelling wave, there exists some $\xi _0>0$ such that $\varphi$ is strictly increasing on $[ 0,\xi _0[$. Assume that $\xi _0$ is a local maximum, then $(|\varphi'|^{p-2}\varphi')'(\xi _0)\leq 0$. But from the equation satisfied by $\varphi$ we get $(|\varphi'|^{p-2}\varphi')'(\xi _0)>0$, which a contradiction. Consequently for any $c\in \mathbb{R}^{*}, \varphi'(\xi )\geq 0$. \end{proof} In the sequel we analyze the corresponding phase portrait of the O.D.E system associated to problem \eqref{E22}--\eqref{E23}. Hence we introduce the following change variables $$X=\varphi \text{\;\;and\;\;} Y=(\varphi')^{p-1} \label{E27}$$ and then \eqref{E22}-\eqref{E23} is equivalent to the system of O.D.E $$\begin{gathered} X'=Y^{\frac 1{p-1}} \\ Y'=X^q+(c+X^{n-1})Y^{\frac 1{p-1}} \\ (X(0), Y(0))=(0,0) \end{gathered}\label{E28}$$ In order to solve the above, we write the first O.D.E for the trajectories: $$\frac{dY}{dX}=c+X^{n-1}+X^qY^{-\frac 1{p-1}}. \label{E29}$$ and consequently, we consider the problem: Finding the non trivial trajectories $(X,Y)$ solutions of $$\begin{gathered} \frac{dY}{dX}=c+X^{n-1}+X^qY^{-\frac 1{p-1}} \\ Y(0)=0\,. \end{gathered}\label{E210}$$ We start with the following result. \begin{proposition}\label{prop21} For any $c\in \mathbb{R}^{*}$, problem \eqref{E210} has a unique global solution. \end{proposition} Here the fixed-point does not work. So we use perturbation methods. We consider the following approximation problem $$%(Q_\varepsilon ) \begin{gathered} \frac{dY}{dX}=c+X^{n-1}+X^qY^{-\frac 1{p-1}}=F(X,Y) \\ Y(0)=\varepsilon . \end{gathered} \label{E211}$$ for any $\varepsilon >0$. \begin{lemma}\label{lem22} For any $\varepsilon >0$, problem \eqref{E211} has a unique global solution. \end{lemma} \begin{proof} As the function $(X,Y)\to F(X,Y)$ is locally Lipschitzienne continuous function in $\mathbb{R}^{+}\times [\varepsilon ,+\infty[$, we deduce from the theory of O.D.E (see for example \cite{A}) the existence of unique local solution $Y_\varepsilon$ of $(Q_\varepsilon )$. First, we remark that if $c>0$, the function $X\to Y_\varepsilon (X)$ is strictly increasing and satisfies the following inequality $$\frac{dY_\varepsilon }{dX}\leq c+X^{n-1}+X^q\varepsilon ^{-\frac 1{p-1}} \label{E212}$$ and thereby $Y_\varepsilon$ is global solution. On the other hand if $c<0$, introduce the curve $(C)$ solution of the equation $F(X,Y)=0$. It is given explicitly by $$\widetilde{Y}(X)=(\frac{-X^q}{c+X^{n-1}})^{1/(p-1)},\label{E213}$$ with $0\leq X\leq (-c)^{\frac 1{n-1}} \; if \; n>1$ and $X>(-c)^{1/(n-1)} if n<1$. The curve $(C)$ divide the plane into two regions: in the first $R_1$ we have $F(X,Y)<0$ while in the second part (say $R_2)$, $F(X,Y)>0$; see figure \ref{fig1}). \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(93,43)(-2,6) \drawline(0,40)(0,10)(38,10) \qbezier(0,10)(20,9)(25,40) \dashline{1}(30,10)(30,12) \dashline{1}(30,15)(30,40) \put(2,23){$F(x,y)<0$} \put(18,12){$F(x,y)>0$} \put(19,37){($C$)} \put(20,3){$(-c)^{1/(n-1)}$} \put(37,7){$x$} \put(-3,40){$y$} \dashline{1}(54,10)(54,40) \drawline(47,40)(47,10)(90,10) \qbezier(59,40)(62,5)(87,20) \put(60,37){($C$)} \put(64,23){$F(x,y)<0$} \put(64,12){$F(x,y)>0$} \put(47,3){$(-c)^{1/(n-1)}$} \put(88,7){$x$} \put(44,40){$y$} \end{picture} \end{center} \caption{Curve $C$: Case $n>1$ (left). Case $n<1$ (right) \label{fig1}} \end{figure} For $n>1,\;Y_\varepsilon$ starts in the region $R_1$ and $Y_\varepsilon (0)=\varepsilon >0$, then $Y_\varepsilon$ must cross the curve $(C)$ in some point with horizontal tangent and after $Y_\varepsilon$ lies in the region $R_2$, where $Y_\varepsilon$ is strictly increasing. Hence the minimum $m_\varepsilon$ of $Y_\varepsilon$ reaches on $(C)$ and strictly positive. But, for $n<1$, $Y_\varepsilon$ starts in the region $R_2$ and $Y_\varepsilon (0)=\varepsilon >0$, then $Y_\varepsilon$ remains always increasing or it must cross the curve $(C)$ in some point with horizontal tangent and after $Y_\varepsilon$ lies in the region $R_1$, where $Y_\varepsilon$ is strictly decreasing, but if it leaves $R_1$, it becomes increasing. As $lim_{X\to\infty}\widetilde{Y}(X)=\infty$ then also in this case the minimum $m_\varepsilon$ of $Y_\varepsilon$ is strictly positive. So $$\frac{dY_\varepsilon }{dX}\leq c+X^{n-1}+X^qm_\varepsilon ^{-\frac 1{p-1}} \label{E214}$$ and thereby $Y_\varepsilon$ is a global solution. This completes the proof of the lemma. \end{proof} Now we consider the Cauchy problem $$%(D_\varepsilon ) \begin{gathered} \frac{dZ_\varepsilon }{dX}=c+X^{n-1}+X^qZ_\varepsilon ^{-\frac 1{p-1}} \\ Z_\varepsilon (\varepsilon )=0\,. \end{gathered} \label{E215}$$ \begin{lemma}\label{lem23} For any $\varepsilon >0$, problem \eqref{E215} has a global solution. \end{lemma} \begin{proof} We consider the Cauchy problem $$%(L_\varepsilon ) \begin{gathered} \frac{du}{dt}=\frac{t^{\frac 1{p-1}}}{u^q(t)+[ c+u^{n-1}(t)]t ^{\frac 1{p-1}}}=\frac{1}{F(u,t)}=g(t,u) \\ u(0)=\varepsilon \end{gathered}\label{E216}$$ It is easy to see that the above problem has a unique local solution $u_\varepsilon$. In fact the local existence of $u_\varepsilon$ follows easily from the theory of O.D.E \cite{A}. In a first place, we suppose that $c>0$ or $c<0$ and $n>1$. If $c>0$, we have $$0\leq \frac{du_\varepsilon }{dt}\leq \frac 1{c+u^{n-1}}\leq \frac 1{c+\varepsilon ^{n-1}} \label{E217}$$ and thereby $u_\varepsilon$is global. But if $c<0$ and $n>1$, we note by $C'$ the curve $F(u,t)=0$, which is exactly symmetrical with $C$ (introduce in the proof of lemma \ref{lem22}) compared to axis, $t=u$ (see Figure \ref{fig2}). \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(67,41)(-13,3) \drawline(0,38)(0,5)(52,5) \qbezier(0,5)(16,26)(52,28) \dashline{1}(0,30)(52,30) \put(5,35){$g(t,u)=1/F(u,t)>0$} \put(17,12){$g(t,u)=1/F(u,t)<0$} \put(-15,28){$(-c)^{\frac{1}{n-1}}$} \put(46,23.5){($C'$)} \put(51,2){$t$} \put(-3,37){$u$} \end{picture} \end{center} \caption{Curve $C'$: Case $n>1$ \label{fig2}} \end{figure} Then $C'$ divide the plane into two parts: In the first part $\frac{du_\varepsilon }{dt}$ is strictly positive and approaches $+\infty$ when $F(t,u_\varepsilon )$ approaches $0$; that it is $(t,u_\varepsilon (t))$ draw near to the curve $C'$. Consequently $u_\varepsilon$ is strictly increasing and does never touch the curve $C'$. Therefore, $u_\varepsilon$ is global. On another side, as $u_\epsilon$ is increasing then $\lim_{t\to+\infty}u_\varepsilon (t)=l$ exists in $]0,+\infty[$. If $l$ is finite then $\lim_{t\to+\infty}\frac{du_\varepsilon }{dt }=0$, but from \eqref{E216}, we get $\lim_{t\to+\infty}\frac{du_\varepsilon }{dt}=\frac 1{l^{n-1}+c}(>0)$ , which a contradiction and thereby $\lim_{t\to +\infty }u_\varepsilon (t)=+\infty$. Consequently $u_\varepsilon$ is a one to one from $[0,+\infty[$ to $[ \varepsilon ,+\infty[$. Now set $Z_\varepsilon (=u_\varepsilon ^{-1})$ the inverse function of $u_\varepsilon$ defined from $[ \varepsilon ,+\infty [$ to $[ 0,+\infty[$. By a simple computation we see that $Z_\varepsilon$ satisfies the following Cauchy problem $$\begin{gathered} \frac{dZ_\varepsilon }{dX}=c+X^{n-1}+X^qZ_\varepsilon ^{-\frac 1{p-1}} \\ Z_\varepsilon (\varepsilon )=0 \end{gathered} \label{E218}$$ On the other hand, we suppose that $c<0$ and $n<1$. Since $\frac{du_\varepsilon}{dt} \simeq \varepsilon^{-q} t^{1/(p-1)}$, at neighborhood $0$, then there is $t_\varepsilon >0$ such as $u_\varepsilon$ is a one to one from $[0,t_\varepsilon[$ to $[\varepsilon ,u_\varepsilon (t_\varepsilon)[$. We take $Z_\varepsilon (=u_\varepsilon ^{-1})$ the inverse function of $u_\varepsilon$ defined from $[\varepsilon ,u_\varepsilon(t_\varepsilon)[$ to $[0,t_\varepsilon[$. By a simple calculation, we obtain $Z_\varepsilon$ satisfies, in $[\varepsilon ,u_\varepsilon(t_\varepsilon)[$, the Cauchy problem $$\begin{gathered} \frac{dZ_\varepsilon }{dX}=c+X^{n-1}+X^qZ_\varepsilon ^{-\frac 1{p-1}} \\ Z_\varepsilon (\varepsilon )=0 \end{gathered} \label{E219}$$ With an aim of prolonging solution $Z$, one considers $$\begin{gathered} \frac{dZ_\varepsilon }{dX}=c+X^{n-1}+X^qZ_\varepsilon ^{-\frac 1{p-1}} \\ Z_\varepsilon (u(t_\varepsilon) )=t_\varepsilon>0 \end{gathered} \label{E220}$$ By employing the same technique that we used in lemma \ref{lem22} one obtains that (\ref{E220}) admits a solution on $[u_\varepsilon(t_\varepsilon),\infty[$. It is deduced that the problem $(D_\varepsilon)$ admits a global solution. \end{proof} \begin{proof}[Proof of proposition (\ref{prop21})] The proof is divided into two steps\\ {\bf Step 1: Uniqueness.} Assume that there exists two solutions $Y$ and $Z$ of \eqref{E210} such that $Y\neq Z$. Define the real $R$ by R=\sup \{ r>0; Z(X)=Y(X), \text{ for }0\leq XR$. Then without loss of generality we can assume $$\begin{gathered} Y(X)=Z(X) \quad \text{in }[0,R[ \\ Y(X_0)>Z(X_0) \end{gathered} \label{E222}$$ Set$f(X)=(Y-Z)(X)$, then there exists some real$\theta \in ] R,X_0[ $such that $$0\leq f(X_0)-f(R)=\theta ^q[Y^{-\frac 1{p-1}}(\theta )-Z^{-\frac 1{p-1}}(\theta )] <0 \label{E223}$$ which gives a contradiction. Consequently$Y=Z$. \noindent{\bf Step 2: Existence.} Let$(Y_\varepsilon )$the solution of \eqref{E211}. As$\varepsilon \to Y_\varepsilon $is increasing and positive, then$Y_\varepsilon (x)$converges to some function$Y(X)=lim_{\varepsilon\to 0}Y_{\varepsilon} (X)\geq0, X \in] 0,+\infty[ $with$Y_\varepsilon(0)=\varepsilon\to Y(0)=0$. In order to prove that$Y$is the solution of \eqref{E210}, we start with the followings claims. \noindent{\bf Claim 1.}$Y(x)$is strictly positive for any$x>0$. In fact, Since$Z_\varepsilon $and$Y_\varepsilon $satisfy the same equation on$] \varepsilon ,+\infty[$and$Y_\varepsilon (\varepsilon )>Z_\varepsilon(\varepsilon )=0$,$Y_\varepsilon (X)>Z_\varepsilon (X)>0$for any$X\in ]\varepsilon ,+\infty[ $. Now, take some$X_0\in ] 0,+\infty[ $and using the fact that$(Z_\varepsilon)_{\varepsilon >0}$is a decreasing sequence we get $$\lim_{\varepsilon \to 0}Y_\varepsilon (X_0)\geq \lim_{\varepsilon \to 0}Z_\varepsilon (X_0)\geq Z_{\frac{X_0}2}(X_0)>0 \label{E224}$$ and consequently$Y$is strictly positive on$] 0,+\infty[$. \noindent{\bf Claim 2.} The function$Y$is a solution of problem \eqref{E210}. In fact, since$ Y_\varepsilon $is the solution of \eqref{E211}, then for any test function$\Phi \in D(]0,+\infty[ )$, $$\int_0^{+\infty }\Phi (X)\left\{ c+X^{n-1}+X^qY_\varepsilon ^{-\frac 1{p-1}}(X)\right\} dX+\int_0^{+\infty }Y_\varepsilon (X)\Phi '(X)dX=0 \label{E225}$$ When$\varepsilon $approaches$0$, $$\frac{dY}{dX}=c+X^{n-1}+X^qY^{-\frac 1{p-1}} \text{ in } D'(] 0,+\infty[ ) \label{E226}$$ Then for any$00$, the problem $$\begin{gathered} \frac{d\varphi }{d\xi }(\xi )=Y^{\frac 1{p-1}}(\varphi (\xi )) \\ \varphi (0)=A \end{gathered} \label{E227}$$ has a unique maximal solution defined in$] -\infty ,\beta[ $, where$\beta\in \overline{\mathbb{R}}$. Moreover $$\lim_{\xi \to -\infty }\varphi (\xi )=0\text{ and }\lim_{\xi \to \beta^{-}}\varphi (\xi )=+\infty \label{E228}$$ \end{lemma} \begin{proof} Since$Y$is regular and non-negative in$] 0,+\infty[ $, there exists a maximal solution$\varphi $on some interval$] \alpha ,\beta[ $. Moreover since$\varphi $is positive and increasing in$] \alpha ,\beta[ $,$\lim_{\xi \to \alpha ^{+}}\varphi (\xi )=l$exists and$l\geq 0$. If$l=0$we get$\lim_{\xi \to \alpha ^{+}} \frac{d\varphi }{d\xi }(\xi )=0$and thereby if$\alpha $is finite we can prolong solution$\varphi $by$0$on$] -\infty ,\alpha ] $, what contradicts the fact that$] \alpha ,\beta [ $is a maximum interval, thereby$\alpha=-\infty$. While if$l>0$we remark that the Cauchy problem $$\begin{gathered} \frac{d\varphi (\xi )}{d\xi }=Y^{\frac 1{p-1}}(\varphi (\xi )) \\ \varphi (\alpha )=l>0 \end{gathered} \label{E229}$$ has a unique local solution around$\beta $and then inevitably$\alpha =-\infty$. We put$l=lim_{\xi\to-\infty}\varphi(\xi)$, employing \eqref{E227} we have$l=0$. On the other hand, as$\varphi $is strictly increasing we obtain$\lim_{\xi \to \beta ^{-}}\varphi (\xi )=+\infty $if$\beta $is finite; while if$\beta=+\infty $, it is easy enough to use \eqref{E227} to get also$\lim_{\xi \to \beta ^{-}}\varphi (\xi )=+\infty $. \end{proof} \begin{remark}\label{rem21} \rm Any solution$\varphi $of \eqref{E227} defined in$]-\infty ,\beta[ $satisfies $$\big(|\varphi'|^{p-2}\varphi'\big)'(\xi )-c\varphi'-\frac 1n(\varphi ^n)'-\varphi ^q(\xi )=0\quad \text{in }] -\infty ,\beta[ . \label{E230}$$ Among the solutions of \eqref{E227} one will seek the global solutions which satisfied$\varphi (0)=0$. For that one needs the study of the asymptotic behavior of solutions$Y(X)$of the problem \eqref{E210} checked by the vector field. \end{remark} \begin{proposition}\label{prop22} Assume$c\in \mathbb{R}^\ast, q> 0$and$1\neq n>0$. Let$Y$the solution of \eqref{E210}. Then$Y$have the following behavior: \begin{itemize} \item[(a)] When$X$approaches$0$:\\ (i) If$c>0 $or$ c (l-1)\geq 0$then$Y(X)\approx MX^{l}$, with$M=M_0$.\\ (ii) If$ c<0$and$ l>1 $then$ Y(X)\approx (-c)^{1-p}X^{q(p-1)}$\item[(b)] When$X$approaches$\infty$:\\ (i) If$c>0 $or$c (L-1)\leq 0$then$Y(X)\approx MX^{L}$, with$M=M_\infty$.\\ (ii) If$ c<0$and$ L<1 $then$Y(X)\approx(-c)^{1-p}X^{q(p-1)}$. \end{itemize} \end{proposition} \begin{proof} We consider$H(X)=MX^\theta $, where$M>0$and$\theta>0 $. It is easy to see that$H$is a super-solution of \eqref{E210} (resp. sub-solution) if and only if $$\theta M\geq cX^{1-\theta }+X^{n-\theta }+M^{-1/(p-1)}X^{p(Q-\theta )/(p-1)}, \label{E231}$$ respectively $$\theta M\leq cX^{1-\theta }+X^{n-\theta }+M^{\frac{-1}{p-1}}X^{p(Q-\theta )/(p-1)}, \label{E232}$$ We start with the asymptotic behavior at the neighborhood of$0$, in fact we have two cases: \noindent(a)$c>0$or$(l-1)c\geq 0$. In order to have$H$satisfied (\ref{E231}) (resp.(\ref{E232} )) at the neighborhood$0$, we must have$ l\geq \theta $(resp.$l\leq \theta$). Let$\theta =l $. Then$H$is a super-solution of \eqref{E210} (resp. sub-solution) for all$M>M_0$(resp.$M1$. We take in this case$\theta =q(p-1)$then (\ref{E231}) (resp.(\ref{E232})) becomes $$\theta M\geq [c+M^{-1/(p-1)}+X^{n-1}] X^{1-q(p-1)}, \label{E233}$$ respectively $$\theta M\leq [c+M^{-1/(p-1)}+X^{n-1}] X^{1-q(p-1)}, \label{E234}$$ Since$Q>1$, we have$1-q(p-1)<0$, this gives that (\ref{E233}) (resp. (\ref{E234})) is checked for all$M\geq (-c)^{1-p}$(resp.$M<(-c)^{1-p}$), from where$Y(X)\approx (-c)^{1-p}X^{q(p-1)}$. Now, we pass to the behavior at neighborhood of$\infty$, we distinguish two cases: \noindent (a)$c>0$or$(L-1)c<0$. We take$\theta=L$then$H$super-solution (resp. sub-solution) for all$M>M_\infty $(resp.$M0$(because$Q<1$) then$Y(X)\approx (-c)^{1-p}X^{q(p-1)}$. \end{proof} \begin{proof}[Proof of Theorems \ref{th11}, \ref{th12} and \ref{th13})] That is to say$\varphi $a solution of the problem (\ref{E227}) whose maximum interval of existence is$]-\infty, \beta[ $. Then, as long as$\varphi (\xi)\neq 0$(consequently$Y(\varphi(\xi))\neq 0 )$one has $$Y^{-1/(p-1)}(\varphi (\xi))\varphi '(\xi)=1. \label{E235}$$ While integrating (\ref{E235}) on$(\xi, \xi _ 1) \subset] -\infty, \beta[ $one obtains $$\xi _ 1-\xi = \int_{\varphi (\xi)}^{\varphi (\xi _ 1)}Y^{-1/(p-1)}(s)ds, \label{E236}$$ for all$\xi \in ] -\infty, \beta[$, such as$\varphi (\xi)>0$. If$\varphi $never vanishes on$] -\infty, \beta[ $, we can make tending$\xi $to$-\infty $in the formula (\ref{E236}), thereby we have$\int_0^{\varphi (\xi _ 1)}Y^{-1/(p-1)}(s)ds=\infty. $Thus$\varphi $vanish in a point if and only if $$\int_0Y^{-1/(p-1)}(s)ds<\infty \label{337}$$ In addition, by tending$\xi _ 1$to$\beta $in the formula (\ref{E236}) we obtain$\beta = \infty $if and only if $$\int^{+\infty}Y^{-1/(p-1)}(s)ds=\infty. \label{238}$$ Let us call the asymptotic behavior$Y$(solution of the problem (\ref{E210}) and remark \ref{rem21}, the theorem \ref{th11} rises immediately. One combines again the results of the behavior asymptotic of the solution$Y$and relations$(\ref{E228})$and$(\ref{E231})\$, one obtains the results concerning the asymptotic behavior (theorems \ref{th12} and \ref{th13}). \end{proof} \subsection*{Acknowledgments} The author would like to express his gratitude to professor A. Gmira, also to the anonymous referee for his/her helpful comments. This work was supported financially by Centre National de Coordination et de Planification de La Recherche Scientifique et Technique PARS MI 29. \begin{thebibliography}{00} \bibitem{A} H. Amann, \emph{Ordinary Differential Equations}, Walter de Gruyter. Berlin, New York, 1996. \bibitem{AD} D. Arcoya, J. Diaz, L. Tello; \emph{S-Shaped biffurcation branch in quasilinear multivalued model arising in climatology}, J. DIff Equ., 150 (1998), 215-225. \bibitem{F} A. C. Fowler, \emph{Mathematical models in the Applied sciences}, Cambridge texts in Applied Mathematics, Cambridge University Press 1997. \bibitem{AE} N. O. Alikakos and L. C. Evans, \emph{Continuity of the gradient for weak solutions of a degenerate prabolique equation}. J. Math pures appl. T. 62 (1983), 253-268. \bibitem{LA} S. V. Lee and W. F. Ames, \emph{Similarity solutions for Non-Newtonian fluids}, A.T.CH.E. Journal, 12, 4(1966), 700-708. \bibitem{PS} A. De Pablo and A. Sanchez, \emph{Global Travelling Waves in Reaction- Convection- Diffusion Equations}. J. Diff. Eq. 165, (2000), 377-413. \bibitem{PV} A. De Pablo and J. L. Vazquez, \emph{Travelling waves and finite propagation in reaction-Diffusion equation}, J. Diff . Eq. 93, 1 (1991), 19-61. \bibitem{H} K. P. Hadeler, \emph{Travelling Front and Free Boundary value problems}, Numerical Treatment of Free Boundary Problems, Birkh\"auser verlag 1981. \bibitem{HV} M. A. Herrero and J. L. Vazquez, \emph{Thermal waves in absorption media}, J. Diff. Eq. 74, (1988),218-233. \bibitem{KPP} A. Kolmogorov, I. Petrovsky and N. Piskunov, \emph{2tude de l'\'equation de la diffusion avec croissance de la quantit\'e de mati\ere et son application \a un probl\`eme biologique}. Bull. Univ. Moskov, ser. Internat. Sec A 1,6 (1937) ,1-25. \bibitem{SG} F. Sanchez-Garduno, \emph{Travelling Wave Phenomena in Some degenerate Reaction-diffusion Equations}, J.Dif. Eq. 177, (1995) 281-319. \bibitem{VG} C. J. Van Duin and J. M. De Graef, \emph{Large Time Behavior of Solutions of the Porous Medium Equation with Convection}, J. Diff. Eq. 84, (1990), 183-203. \end{thebibliography} \end{document}