\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{Existence of stable periodic solutions } { Abderrahmane El Hachimi \& Abdelilah Lamrani Alaoui } \begin{document} \setcounter{page}{117} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2002-Fez conference on Partial Differential Equations,\newline Electronic Journal of Differential Equations, Conference 09, 2002, pp 117--126. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Existence of stable periodic solutions for quasilinear parabolic problems in the presence of well-ordered lower and upper-solutions % \thanks{ {\em Mathematics Subject Classifications:} 35K55, 35B20, 35B40. \hfil\break\indent {\em Key words:} Leray-lions operator, penalization, lower and upper-solutions, \hfil\break\indent monotone process, periodic solutions, stabilization. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published December 28, 2002. } } \date{} \author{Abderrahmane El Hachimi \& Abdelilah Lamrani Alaoui} \maketitle \begin{abstract} We present existence and stability results for periodic solutions of quasilinear parabolic equation related to Leray-Lions's type operators. To prove existence and localization, we use the penalty method; while for stability we use an approximation scheme. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction} In the last few years, many works have been devoted to the existence and stability of periodic solutions of problem \begin{equation} \label{P} \begin{gathered} \frac{\partial u}{\partial t}+A(u)+F(u,\nabla u) =0 \quad\mbox{in } \Omega \times \mathbb{R}^{+}, \\ u = 0 \quad \mbox{on }\partial \Omega \times \mathbb{R}^{+}, \\ u(0) = u(T) \quad \mbox{in }\Omega , \end{gathered} \end{equation} where $\Omega $ is a bounded and open subset of $\mathbb{R}^N$, $N\geq 1$. For the usual Leray-Lions's operator $A$, Deuel and Hess \cite{DH} obtained existence of periodic solutions under the presence of well-ordered lower and upper-solutions. Unfortunately, uniqueness and therefore stability, can not be derived from the definition they used for solutions of \eqref{P}. For $A(u)=-\Delta g(u)$ and $F$ depending only on $(x,t)$, Harraux and Kenmochi \cite{HK} proved both existence and stability results by using subdifferential theory on Hilbert spaces. Recently, Boldrini and Crema \cite{BC} considered the case where $A(u)$ is the p-laplacian operator, with $p\geq 2$, and $F$ is independent of $\nabla u.$ They obtained an existence result via Shauder's fixed point theorem. More recently, De coster and Omari \cite{CO} considered problem \eqref{P} with a linear uniformly elliptic operator $A$ $$Au := -\sum_{i,j=1}^N \partial _{x_i}(a_{i,j}\partial _{x_j}u)+\sum_{i=1}^Na_{i}\partial_{x_i}u+a_{0}u$$ and $F$ independent of $\nabla u$. These last authors obtained stability in a suitable sense of the maximal and minimal solutions in the presence of well-ordered lower and upper-solutions. The aim of this paper is to show that the result of De Coster and Omari still holds for general problem \eqref{P}. Our existence result is obtained by using a classical method of penalization as it was done by Grenon in \cite{DH}; while the stability one follows essentially the principal arguments of De Coster and Omari with some changes imposed by the nonlinear character of the equation in \eqref{P}. This paper is organized as follows: In section 2 we recall some known results related to the initial boundary value problem associated with \eqref{P}, and give hypotheses and definitions of solutions. In section 3, we give existence and uniqueness results concerning periodic solutions of problem \eqref{P}, while section 4 is devoted to the stability result of periodic solutions. Finally, in section 5 we give an application to a periodic-parabolic problem associated to the $p$-laplacian operator. \section{Hypotheses, definitions, and known results} Let $\Omega $ be an open bounded subset of $\mathbb{R}^N$ with boundary $\partial \Omega$ and $T>0$ a fixed real. We shall denote $$ Q_T:=\Omega \times ]0,T[,\quad \Sigma_T:=\partial \Omega \times ]0,T[, $$ and for a real $p$ with $10$, and $k_{i}\in L^{p'}(Q_T)$, so that for all $s\in \mathbb{R}$ and all $\xi \in \mathbb{R}^N$: \[ |A_{i}(x,t,s,\xi )|\leq \beta _{i}(|s|^{p-1}+|\xi|^{p-1}+k_{i}(x,t)),\forall i=1,\dots,N, \] \item[(A2)] For all $s\in \mathbb{R}$ and all $\xi,\xi^{*}\in \mathbb{R}^N$, with $\xi \neq \xi^{*}$, we have \[ \sum_{i=1}^N[A_{i}(x,t,s,\xi )-A_{i}(x,t,s,\xi ^{*})](\xi -\xi ^{*})>0 \quad\mbox {a.e. in }Q_T. \] \item[(A3)] There exists $\alpha >0$, so that for all $s\in \mathbb{R}$ and all $\xi \in \mathbb{R}^N$, we have \[ \sum_{i=1}^NA_{i}(x,t,s,\xi )\xi _{i}\geq \alpha |\xi|^{p} \quad\mbox {a.e. in }Q_T. \] \item[(A4)] The function $f$ is of caratheodory type on $Q_T\times \mathbb{R}\times \mathbb{R}^N$, and there exist functions $b :{\mathbb{R}^{+} \to \mathbb{R}^{+}}$ increasing, and $h\in L^{1}(Q_T),\; h\geq 0$, such that \[ |f(x,t,s,\xi )| \leq b(|s|)(h(x,t)+|\xi |^{p}),\quad\mbox{for } (x,t,s,\xi)\in Q_T\times \mathbb{R}\times \mathbb{R}^N. \] \end{itemize} We denote by $F$ the Nemyskii operator related to $f$ and defined by $$ F(u,\nabla u)(x,t):=f(x,t,u,\nabla u). $$ To obtain (among other results) global existence for initial boundary-value problems associated with \eqref{P}, we shall assume the following. \begin{itemize} \item[(A5)] There exists $c_{i}>0$ and $l_{i}\in L^{p'}(Q_T)$ with $l_{i}\geq 0$, such that for all $s,s^{*}\in \mathbb{R}$ and all $\xi \in \mathbb{R}^N$, \[ |A_{i}(x,t,s,\xi )-A_{i}(x,t,s ^{*},\xi )| \leq c_{i}|s-s^{*}|[l_{i}(x,t)+|\xi |^{p-1}] \quad\mbox{a.e. in }Q_T. \] \item[(A6)] All data (coefficients and second member) are periodic in time with period $T$. \end{itemize} % We are interested in the existence and stability of the solutions of problem \begin{equation} \label{POT} \begin{gathered} \frac{\partial u}{\partial t}+A(u)+F(u,\nabla u) =0 \quad \mbox{ in }Q _T, \\ u = 0 \quad \mbox{on }\Sigma_T, \\ u(0) = u(T) \quad\mbox{in }\Omega . \end{gathered} \end{equation} To this end, we consider the problem $(\mathcal{P}_{t_1,t_2;u_{0}})$: \begin{equation} \label{Pt1} %{P}_{t_1,t_2;u_{0}}) \begin{gathered} \frac{\partial u}{\partial t}+A(u)+F(u,\nabla u) =0 \quad \mbox{in }\Omega\times ]t_1,t_2[,\\ u = 0 \quad \mbox{on }\partial\Omega \times ]t_1,t_2[,\\ u(t_1) = u_{0} \quad \mbox{in }\Omega , \end{gathered} \end{equation} where $0\leq t_10$ and $u_{0}\in L^{\infty}(\Omega)$. We refer the reader to \cite{grenon} for proofs. \begin{lemma} \label{lm2.1} Assume (A1)--(A5) and let $(\alpha_1,\alpha_2)$ and $(\beta_1,\beta_2)$ be respectively pairs of lower and upper-solutions of $(\mathcal{P}_{0,T';u_0})$ such that $$ \sup (\alpha_1 ,\alpha_2 )\leq \inf (\beta_1 ,\beta_2) \quad \mbox{a.e. in } Q_{T'}. $$ Then, there exists a solution $u\in C([0,T' ]; L^{q}(\Omega) )$ for any $q\geq 1,$ of $(\mathcal{P}_{t_1,t_2;u_{0}})$ such that $\sup (\alpha_1 ,\alpha_2 )\leq u \leq \inf (\beta_1 ,\beta_2)$ a.e. in $Q_{T'}$. Moreover, when $\alpha_1 = \alpha_2 $ and $\beta_1 = \beta_2 $, the Hypothesis (A5) can be removed. \end{lemma} \begin{lemma} \label{lm2.2} Assume (A1)--(A5) and let $\alpha$ and $\beta$ be respectively lower and upper-solutions of $(\mathcal{P}_{0,T';u_{0}})$, for any $T'>0$. Then, there exists $u$ (resp. $v$) $\in C([0,+\infty[;L^{q}(\Omega ))$, $\forall q\geq 1$ such that for any $T'> 0$, the restriction of $u$ (resp. $v$) on $[0,T']$ is the minimal (resp. maximal) solution of $(\mathcal{P}_{0,T';u_0})$ located between $\alpha $ and $\beta $. Moreover, if $u_0$ and $v_0$ are in $L^{\infty}(\Omega)$ and satisfy $$ \alpha(0)\leq u_{0}\leq v_{0}\leq \beta (0) \quad\mbox{a.e. in }\Omega $$ and $u_{\rm min } (u_0 )$ (resp. $u_{\rm min } (v_0 )$) is the minimal solution of $(\mathcal{P}_{0,T';u_{0}})$ with $u_0$ (resp. $(\mathcal{P}_{0,T';v_{0}})$ with $v_0$) laying between $\alpha $ and $\beta $, then $$ u_{\rm min}(u_0)\leq u_{\rm min}(v_0 ), \quad\mbox{ a.e. in } Q_{T'} . $$ Furthermore, the same holds for maximal solutions. \end{lemma} \begin{lemma} \label{lm2.3} Assume (A1)--(A5). Let $0< T_1 < T_2$ and $\alpha$ and $\beta $ be respectively lower and upper-solution of $(\mathcal{P}_{0,T';u_{0}})$ with $T' > 0$. Let $u_1 $ (resp. $u_2$) be the minimal solution of $(\mathcal{P}_{0,T_1; u_0})$ (resp. $(\mathcal{P}_{0,T_2;u_0})$) located between $\alpha $ and $\beta $. Then $u_1$ is the restriction of $u_2$ on $[0,T_2]$ and the same holds for maximal solutions. \end{lemma} \section{Existence and uniqueness of periodic solutions} The first result of this section is the following. \begin{theorem} \label{thm3.1} Assume (A1)--(A4) and let $\alpha $ and $\beta $ be respectively lower and upper-solutions of \eqref{POT} with $\alpha \leq \beta $ a.e. in $Q_T.$ Then, problem \eqref{POT} has a weak solution $u$ satisfying $u\in C([0,T];L^{q}(\Omega ))$, for any $q\geq 1$, and $\alpha \leq u \leq \beta $ a.e. in $Q_T$. \end{theorem} \paragraph{Proof} The proof is similar to that of the corresponding initial boundary-value problem treated in \cite{grenon}. We shall give here only a sketch.\\ (i) We regularize \eqref{POT} by taking \begin{gather*} A_{i}^{*}(u,\nabla u):=A_{i}(Su,\nabla u),\ i\in \{1,\dots,N\} \\ F^{*}_{\epsilon}(u,\nabla u):=\frac{F(Su,\nabla Su)}{1+\epsilon|F(Su,\nabla Su)|} \end{gather*} where $Su:=u+(\alpha-u)^{+}-(u-\beta)^{-}$, $\epsilon > 0$, and using the penalization operator $\theta _{\eta}$ related to the convex $$ {\cal{K}}:=\{v\in V{\mbox{ such that }}-k \leq v \leq k \mbox{ a.e. in }Q_T\}, $$ where $k$ is such that $-k \leq\alpha -1\leq\beta + 1\leq k$. For $\eta$ and $\epsilon > 0$ fixed, consider the problem \begin{equation} \label{Pee} %%$(\mathcal{P}_{\eta,\epsilon})$ \begin{gathered} u_{\eta,\epsilon}\in V, \quad \frac{\partial u_{\eta,\epsilon}}{\partial t}\in V',\\ \frac{\partial u_{\eta,\epsilon}}{\partial t}+ \sum_{i=1}^N\frac{\partial}{\partial x_{i}}A_{i}^{*}(u_{\eta,\epsilon},\nabla u_{\eta,\epsilon})+F^{*}_{\epsilon}(u_{\eta,\epsilon},\nabla u_{\eta,\epsilon})+\theta _{\eta}(u_{\eta,\epsilon})= 0\quad\mbox{in } Q_T,\\ u_{\eta,\epsilon}(0)=u_{\eta,\epsilon}(T)\quad \mbox{in }\Omega . \end{gathered} \end{equation} By \cite[Theorem 1.1]{Lions} (see also section 2.2 of chapter 3, p. 328), this problem has a solution $u_{\eta,\epsilon}$. moreover the estimates of \cite[lemmas 3.6, 39]{grenon} still apply and eventually after extracting a subsequence, we get $$ \lim _{\eta \to 0^{+}} u_{\eta,\epsilon}=u_{\epsilon}\mbox{ in } V, $$ with $ u_{\epsilon}$ a solution of the variational inequality \begin{equation} \label{Ie} % (I_{\epsilon }) \begin{gathered} \langle\frac{\partial u_{\epsilon }}{\partial t},v-u_{\epsilon }\rangle + \int_{Q_T}A^{*}(u_{\epsilon },\nabla u_{\epsilon })\nabla (v-u_{\epsilon }) +\int_{Q_T}F^{*}_{\epsilon}(u_{\epsilon }, \nabla u_{\epsilon })(v-u_{\epsilon })\geq 0 \\ u_{\epsilon }\in {\cal{K}} ,\quad\mbox{for } v\in \cal{K}. \end{gathered} \end{equation} and of the system of equations \begin{equation} \label{Ee} %E_\epsilon \begin{gathered} \frac{\partial u_{\epsilon }}{\partial t} -\mathop{\rm div}(A^{*}(u_{\epsilon},\nabla u_{\epsilon })) +F^{*}_{\epsilon }(u_{\epsilon },\nabla u_{\epsilon })+ g_{\epsilon } = 0 \quad \mbox{in } Q_T\\ u_{\epsilon } = 0 \quad\mbox{on } \Sigma _T,\\ u_{\epsilon}(0) = u_{\epsilon }(T)\quad \mbox{in } \Omega , \end{gathered} \end{equation} where $$\lim_{\eta \to 0^{+}}{\theta _{\eta}(u_{\eta,\epsilon })}=g_{\epsilon} \quad\mbox{in } L^{p'}(Q_T) \mbox{ weak }. $$ As in \cite[p.\ 93]{grenon}, there exists $u\in V$ such that $\lim_{\epsilon \to 0^{+}}u_{\epsilon }=u$ in $V$ and $ \lim _{\epsilon \to 0^{+}}\frac{\partial u_{\epsilon}}{\partial t}=\frac{\partial u}{\partial t}\mbox{ in }V'+L^{1}(Q_T),$ with $u$ satisfying $$ \frac{\partial u}{\partial t}=\mathop{\rm div} (A(Su,\nabla u))+F(Su,\nabla Su). $$ To conclude that $u\in C([0,T];L^{q}(\Omega ))$ for any $q\geq 1$, it suffices to show that $u(0)\in L^{\infty }(\Omega )$ and then use \cite[Lemma 3.2]{grenon}. In fact, $u_{\epsilon} \in L^{p}(0,T;W_{0}^{1,p}(\Omega )\cap L^{\infty }(\Omega ))$, and $\frac{\partial u_{\epsilon }}{\partial t}\in V'$ so that $u_{\epsilon} \in C([0,T];L^{2}(\Omega ))$ by Lions's lemma \cite[p.\ 156]{Lions}. But $u_{\epsilon}\in {\cal{K}}$, so the following claim gives $-k \leq u_{\epsilon}(0) \leq k $ a.e. in $\Omega $. \paragraph{Claim.} Let $u,v \in C([0,T];L^{1}(\Omega ))$ with $u\geq v$ a.e. in $Q_T$. Then $u(t)\geq v(t)$ a.e. in $\Omega $ for all $t\in [0,T]$.\\ To prove this claim take $w:=(v-u)^{+}$, so that $w=0$ a.e. in $Q_T$. The continuity and the non negativity of $t\to \int_{\Omega }w(x,t)dx$ on $[0,T]$ gives the result. \noindent (ii) A careful application of \cite[Lemma 3.1]{grenon} shows that $$ \langle\langle\frac{\partial \alpha}{\partial t} - \frac{\partial u_{\epsilon }}{\partial t},(\alpha - u_{\epsilon })^{+} \rangle\rangle \geq 0. $$ Where $\langle\langle . ,. \rangle \rangle$ is the duality between $V\cap L^{\infty }(Q_T)$ and $V'+L^{1}(Q_T)$. So we get: $\alpha \leq u$ a.e. in $Q_T$ and by similar arguments, we also obtain $u\leq \beta$ a.e. in $Q_T$. \smallskip Now we state a uniqueness result concerning maximal and minimal solutions. \begin{theorem} \label{thm3.2} Assume (A1)--(A5) and let $\alpha $ and $\beta $ be respectively lower and upper-solutions of \eqref{POT} such that $\alpha \leq \beta $. Then, there exist a minimal solution $v$ and a maximal solution $w$ of \eqref{POT} such that $\alpha \leq v\leq w \leq \beta $ a.e. in $Q_T$. \end{theorem} The proof is based on the following lemma. \begin{lemma} \label{lm3.1} Assume (A1)--(A5) and let $\alpha _1 , \alpha _2$ be two lower-solutions and $\beta $ be an upper-solutions of \eqref{POT} such that $sup(\alpha_1 ,\alpha_2)\leq \beta_1 $ a.e. in $Q_T$. Then, there exists at least one weak solution of \eqref{POT} such that $\sup(\alpha_1,\alpha_2)\leq u \leq \beta $ a.e. in $Q_T$ \end{lemma} The proof of this lemma is the same as that in \cite[Theorem 3.2]{grenon}, except for what concerns the inequality of \cite[Lemma 3.18]{grenon}, which must be replaced by $$ \langle \langle \frac{\partial \alpha _1}{\partial t },[1-\beta_{\delta}(\alpha_2-\alpha_1)]\omega_{\delta }\rangle\rangle + \langle\langle \frac{\partial \alpha _2}{\partial t },\beta_{\delta}(\alpha_2-\alpha_1)\omega_{\delta }\rangle\rangle + \langle- \frac{\partial u_{\epsilon}}{\partial t },\omega _{\delta}\rangle \geq \varphi(\delta) $$ where $\gamma _\delta,\ \beta _\delta $ and $\omega _\delta$ are defined as in \cite[p.~31]{grenon}, $\varphi$ is given by the uniform continuity of the function $s\to s^{+}$ on some compact set associated to $\cal{K}$ and is such that $\varphi(\delta)\to 0 $ as $\delta \to 0^{+}$, and where $\langle.,.\rangle$ designates the duality between $V$ an $V'$. \section{Stability result} The aim of this section is to prove the following theorem. \begin{theorem} \label{thm4.1} Assume (A1)--(A6) and let $\alpha $ and $\beta $ be respectively lower and upper-solution of \eqref{POT} with $\alpha \leq \beta $ a.e. in $Q_T$ and $\alpha (0),\beta (0)\in L^{\infty }(\Omega)$. Denote by $v$ (resp. $\omega $) the minimal (resp. maximal) solution of \eqref{POT} located between $\alpha $ and $\beta $. Then, for all $u_{0}\in L^{\infty }(\Omega )$ satisfying $\alpha (0)\leq u_{0} \leq v(0)$ (resp. $\omega (0)\leq u_{0} \leq \beta (0)$), the set $\mathcal{U}(u_{0},\alpha ,v )$ (resp. $\mathcal{U}(u_{0},\beta ,\omega )$) of all solutions $u$ of $(\mathcal{P}_{0,+\infty;u_{0}})$ satisfying $\alpha \leq u \leq v $ (resp. $\omega \leq u \leq \beta $)in $\Omega \times (0,+\infty )$, is nonempty and is such that for any $q \geq 1$, we have \begin{equation} \label{4.1} \begin{gathered} { \lim _{t\to +\infty }}\| u(.,t)-v(.,t)\|_{L^{q}(\Omega )}=0 \\ (\mbox{resp. } \lim _{t\to +\infty }\| u(.,t)-\omega (.,t)\|_{L^{q}(\Omega )}=0), \end{gathered} \end{equation} \end{theorem} This theorem is a consequence of the following lemma. \begin{lemma} \label{lm4.1} Assume (A1)--(A6) and let $Z$ be a solution of \eqref{POT} such that $Z(0)\in L^{\infty }(\Omega )$. Then, we have: \noindent (a) If $\alpha $ is a lower-solution of \eqref{POT} with $\alpha (0)\in L^{\infty }(\Omega )$ such that $\alpha \leq Z $ a.e. in $Q_T,$ with strict inequality in a subset of positive measure, and such that every solution $v$ of \eqref{POT} satisfying $\alpha \leq v \leq Z $ is equal to $Z$. Then the minimal solution ${\tilde{\alpha }}$ of $(\mathcal{P}_{0,+\infty;\alpha (0)})$ is such that $\alpha \leq {\tilde{\alpha }}\leq Z$, and \begin{equation} \label{4.2} \lim _{t\to +\infty}\| {\tilde{\alpha}}(.,t)-Z(.,t)\|_{L^{q}(\Omega )}=0,\ \forall q\geq 1. \end{equation} (b) If $\beta $ is an upper-solution of \eqref{POT} with $\beta(0)\in L^{\infty }(\Omega )$ such that $ Z \leq \beta $ a.e. in $Q_T,$ with strict inequality in a subset of positive measure, and such that every solution $v$ of \eqref{POT} satisfying $ Z\leq v \leq \beta $ is equal to $Z$. Then the maximal solution ${\tilde{\beta }}$ of $(\mathcal{P}_{0,+\infty;\beta (0)})$ is such that $Z \leq {\tilde{\beta }}\leq \beta,$ and $$ \lim _{t\to +\infty }\| {\tilde{\beta}}(.,t)-Z(.,t)\|_{L^{q}(\Omega )}=0,\ \forall q\geq 1 . $$ \end{lemma} \paragraph{Proof.} With the help of the lemmas in section 2, we apply the method of De coster and Omari \cite{CO}. First we show (a), and then (b) can be obtained by similar way. The proof is divided into three steps. \noindent (i) We construct a sequence of lower-solutions of \eqref{POT} converging to $Z$: Let $\alpha $ be a lower-solution of $(\mathcal{P}_{0,T;\alpha(0)})$, and $Z$ verify $Z(0)\geq \alpha (0)$. Then $Z$ is an upper-solution of $(\mathcal{P}_{0,T;\alpha(0)})$. By lemma \ref{lm2.2}, there exists a minimal solution ${\tilde{\alpha}}_{0}$ of $(\mathcal{P}_{0,T;\alpha(0)})$ such that $\alpha \leq {\tilde{\alpha}}_{0} \leq Z$ a.e. in $Q_T$. So ${\tilde{\alpha}}_{0}(T)\geq \alpha (T) \geq \alpha (0) ={{\tilde{\alpha}}_{0}}(0)$. Now, we define by induction, the sequence $({\tilde{\alpha}}_{n})_{n}$ such that ${{\tilde{\alpha}}_{n}}$ is the minimal solution $u$ of \begin{equation*} %\leqno(\mathcal{P}_{n}) \begin{gathered} \frac{\partial u}{\partial t}+A(u)+F(u,\nabla u) =0 \quad \mbox{in } Q_T,\\ u = 0 \quad \mbox{on } \Sigma _T,\\ u(0) ={{\tilde{\alpha}}_{n-1}}(T) \quad \mbox{in }\Omega , \end{gathered} \end{equation*} satisfying $${{\tilde{\alpha}}_{n-1}}\leq u \leq Z \quad \mbox{a.e. in }Q_T. $$ Hence ${{\tilde{\alpha}}_{n}}$ is a lower-solution of \eqref{POT}. Consequently, \begin{equation} \label{4.3} \alpha \leq {{\tilde{\alpha}}_{n-1}} \leq {{\tilde{\alpha}}_{n}} \leq Z, \mbox{ for all } n. \end{equation} and \begin{equation} \label{4.4} {{\tilde{\alpha}}_{n-1}}(T)={{\tilde{\alpha}}_{n}}(0), \quad \mbox{for all }n. \end{equation} By Lebesgue dominated convergence theorem, there exists $u\in L^{\infty}(Q_T)$ such that $\alpha \leq u \leq Z$ a.e. in $Q_T$ and $ \lim_{n\to +\infty }{{\tilde{\alpha}}_{n}}=u$ in $L^{q}(Q_T)$, for any $q\geq 1$. Moreover, ${{\tilde{\alpha}}_{n}}$, $u\in C([0,T];L^{q}(\Omega))$. By (4.3) and this claim, we get \begin{equation} \label{4.5} {\lim _{n\to +\infty }}{{\tilde{\alpha}}_{n}}(t) =u(t)\quad\mbox{in } L^{q}(\Omega ), \forall q\geq 1. \end{equation} Let $f_{n}(t):=\int_{\Omega }(u-{{\tilde{\alpha}}_{n}})^{q}(x,t)dx$, for any $n\geq 1$. We have, $(f_{n})_{n}\subset C([0,T];\mathbb{R})$ and converges simply to zero. By Dini's theorem one has $$ {\lim _{n\to +\infty }} \sup _{[0,T]} \|{{\tilde{\alpha}}_{n}}(t)-u(t)\|_{q}=0. $$ (ii) Using \cite[Theorem 3.6]{grenon}, we deduce that $u$ satisfies the first two equations in \eqref{POT}. The third equation, the periodicity condition, is a consequence of (4.4). Then $u$ is a solution of \eqref{POT} with $\alpha \leq u \leq \beta$. Therefore, we have $u=Z$ a.e. in $Q_T$ and \begin{equation} \label{4.6} \lim _{n\to +\infty}\sup_{[0,T]}\|{{\tilde{\alpha}}_{n}}(t)-Z(t) \|_{q}=0. \end{equation} Let ${{\tilde{\alpha}}}(x,t):={{\tilde{\alpha}}_{n}}(x,t-nT)$ for $(x,t)\in \Omega \times [nT,(n+1)T[$. Then ${{\tilde{\alpha}}}$ is a solution of $(\mathcal{P}_{0,\alpha(0)})$ satisfying (4.2). Indeed, we have $$ \|{{\tilde{\alpha}}}(.,t)-Z(.,t)\|_{L^{q}(\Omega)}\leq \sup_{\theta \in [0,T]}\|{{\tilde{\alpha}}_{n_{t}}}(.,\theta)-Z(.,\theta)\|_{L^{q}(\Omega)}, $$ where $n_{t}=[t/T]$ is the integer part of $t/T$. Now, (4.2) is a consequence of (4.6). \noindent (iii)The minimality of ${{\tilde{\alpha}}}$ as a solution of $(\mathcal{P}_{0,\alpha(0)})$ satisfying $\alpha \leq {\tilde{\alpha}} \leq Z$ is obtained exactly as in \cite{CO}. \hfill$\square$ \paragraph{Remark} % 4.1 In the sequel we shall identify a lower or an upper-solution $\phi$ defined on $\Omega \times [0,T)$ to its prolongment on $\Omega \times [0,+\infty)$ defined by ${\tilde{\phi}}(x,t):=\phi (x,t-nT)$ $\forall (x,t)\in \Omega \times [nT,(n+1)T[$. \paragraph{Proof of Theorem \ref{thm4.1}} We prove the result concerning the minimal solution, the one corresponding to the maximal solution is obtained in a similar way. Let $u_{0}$ be such that $\alpha (0)\leq u_{0}\leq v(0)$. We first show that: ${\cal{U}}(u_{0},\alpha,v)\neq \emptyset $. $v$ (resp. $\alpha $) is an upper (resp. lower) solution of $(\mathcal{P}_{0,T';u_{0}})$, for any $T'>0$. By lemma \ref{lm2.3} the maximal and minimal solutions of $\mathcal{P}_{0,+\infty;u_{0}}$ are defined globally. Let $u\in {\cal{U}}(u_{0},\alpha,v)$ and $u_{\rm min}$ the minimal solution of $(\mathcal{P}_{0,T';u_{0}})$, $T'>0$. We have $\alpha \leq u_{\rm min}\leq u\leq v$ on $\Omega \times (0,+\infty )$. And from lemma \ref{lm2.2}, we get \begin{equation} \label{4.7} \alpha \leq {\tilde{\alpha}}\leq u_{\rm min}\leq u \leq v, \end{equation} where ${\tilde{\alpha}}$ is the minimal solution of $(\mathcal{P}_{0,\alpha (0)})$ and $u_0$ satisfying $\alpha (0)\leq u_{0}$. Hence the proof is completed \hfill$\square$ \section{Applications} In this section we give some sufficient conditions on the data in order to obtain existence of lower and upper-solutions for a periodic-parabolic problem associated with the $p$-laplacian operator. Consider the problem \begin{equation} \label{5P} %\mathcal{P} \begin{gathered} \frac{\partial u}{\partial t}-{\Delta_ p} u+g(u) =h(x,t) \quad \mbox{in }\Omega \times \mathbb{R}^{+} ,\\ u = 0 \quad \mbox{on }\partial \Omega \times \mathbb{R}^{+} , \\ u(0) = u(T) \quad \mbox{in }\Omega , \end{gathered} \end{equation} where $\Delta_{p}u=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$, with $p$ such that $1