\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 143, 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} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/143\hfil Existence of positive solutions] {Existence of positive solutions for some nonlinear parabolic equations in the half space} \author[A. Ghanmi\hfil EJDE-2010/143\hfilneg] {Abdeljabbar Ghanmi} \address{D\'epartement de Math\'ematiques, Facult\'e des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia} \email{Abdeljabbar.ghanmi@lamsin.rnu.tn} \thanks{Submitted June 18, 2010. Published October 12, 2010.} \subjclass[2000]{35J55, 35J60, 35J65} \keywords{Parabolic Kato class; parabolic equation; positive solutions} \begin{abstract} We prove the existence of positive solutions to the nonlinear parabolic equation $$\Delta u - \frac{\partial u}{\partial t}=p(x,t)f(u)$$ in the half space $\mathbb{R}^n_{+}$, $n\geq 2$, subject to Dirichlet boundary conditions. The function $f$ is nonnegative continuous non-increasing, and the potential $p$ is nonnegative and satisfies some hypotheses related to the parabolic Kato class. We use potential theory arguments to prove our main result. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \section{Introduction} In this article, we study the existence and asymptotic behaviour of continuous positive solution, in the sense of distributions, for the nonlinear parabolic equation $$\label{eP} \begin{gathered} \Delta u-\frac{\partial u}{\partial t}=p(x,t)f(u)\quad \text{in } \mathbb{R}_{+}^n\times (0,\infty ) \\ u(x,0)=u_0(x)\quad \text{in }\mathbb{R}_{+}^n \\ u(z,t)=0\quad \text{on }\partial \mathbb{R}_{+}^n\times (0,\infty ), \end{gathered}$$ where $u_0$ is a nonnegative measurable function in $\mathbb{R}_{+}^n$, the function $f:(0,\infty )\to [ 0,\infty )$ is non-increasing and continuous and the potential $p:\mathbb{R} _{+}^n\times (0,\infty )\to [ 0,\infty )$ is measurable and satisfies some hypotheses related to the parabolic Kato class $P^{\infty}(\mathbb{R}_{+}^n)$ studied in \cite{hm2,lm}. In this article, we denote $\mathbb{R}_{+}^n=\{x=(x_{1},x_{2}, \ldots ,x_n)\in \mathbb{R}^n:x_n>0\}$, $n\geq 2$, we denote by $\partial \mathbb{R}_{+}^n$ the boundary of $\mathbb{R}_{+}^n$ and by $C(\mathbb{R} _{+}^n\times (0,\infty ))$ the set of continuous functions in $\mathbb{R}_{+}^n\times (0,\infty )$. Note that $x\to \partial \mathbb{R} _{+}^n$ means that $x=(x',x_n)$ tends to a point $(\xi ,0)$ of $\partial \mathbb{R}_{+}^n$. For each nonnegative measurable function $f$ on $\mathbb{R}_{+}^n$, we denoted $P_tf(x)=Pf(x,t)=\int_{\mathbb{R}_{+}^n}\Gamma (x,t,y,0)f(y)dy,\quad t>0,\;x\in \mathbb{R}_{+}^n,$ where $\Gamma (x,t,y,s)$ is the heat kernel in $\mathbb{R}_{+}^n\times (0,\infty )$ with Dirichlet boundary conditions $u=0$ on $\partial \mathbb{R}_{+}^n\times (0,\infty )$ given by $\Gamma (x,t,y,s)=(4\pi )^{-n/2}\frac{1}{(t-s)^{n/2}} \exp\big(-\frac{|x-y|}{4(t-s)}\big)(1-\exp\big(-\frac{x_ny_n}{t-s} \big))$ for $t>s$, $x, y \in \mathbb{R}_{+}^n$. We note that the family of kernels $(P_t)_{t>0}$ is sub-Markov semi-group, that is $P_{t+s}=P_tP_{s}$ for all $s,t>0$ and $P_t1\leq 1$. We mention that for each nonnegative $f$ on $\mathbb{R}_{+}^n$, the map $(x,t)\to P_tf(x)$ is lower semi-continuous on $\mathbb{R}_{+}^n$ and becomes continuous if $f$ is further bounded. Moreover, let $w$ be a nonnegative superharmonic function on $\mathbb{R}_{+}^n$, then for every $t>0$, $P_tw\leq w$ and consequently the mapping $t\to P_tw$ is non-increasing. The motivation for our study are the results presented in \cite{dgr,mrjde,mruk,mr,kr,hm2,hm1,lm,Zg1} and their references. Zhang \cite{Zg1} gave an existence result of the parabolic problem $$\label{e1} \begin{gathered} \Delta u-\frac{\partial u}{\partial t}+q(x,t)u^{p}=0,\quad \text{in }D\times (0,\infty ) \\ u(x,0)=u_0(x), \quad x\in D, \end{gathered}$$ where $D=\mathbb{R}^n(n\geq 3)$, $u_0$ is a bounded function of class $C^2(\mathbb{R}^n)$ and $q(x,t)$ is in the parabolic Kato class $P^{\infty }(\mathbb{R}^n)$ which was introduced in \cite{zg2}. Inspired by the papers by Zhang \cite{Zg1} and Zhang and Zhao \cite{zgz}, Maatoug and Riahi introduced for the case of the half space a parabolic Kato class $P^{\infty }(\mathbb{R}_{+}^n)$ and gave an existence result for \eqref{e1} where $D=\mathbb{R}_{+}^n$. Maagli et al \cite{hm2} studied the problem $$\label{eQ} \begin{gathered} \Delta u-u\varphi (.,u)-\frac{\partial u}{\partial t}=0 \quad \text{in }\mathbb{R}_{+}^n\times (0,\infty ) \\ u(x,0)=u_0(x), \quad x\in \mathbb{R}_{+}^n \\ u=0 \quad \text{in }\partial \mathbb{R}_{+}^n\times (0,\infty ), \end{gathered}$$ where $u_0$ is a nonnegative measurable function defined on $\mathbb{R}_{+}^n$ and satisfies some properties which allows $u_0$ to be not bounded, the perturbed nonlinear term $u\varphi (.,u)$ satisfies some hypotheses related to the parabolic Kato class $P^{\infty }(\mathbb{R} _{+}^n)$. Under some conditions imposed on the initial value $u_0$ and the nonlinear term $\varphi$, the authors proved in \cite{hm2} the following result. \begin{theorem} \label{thm1} Problem \eqref{eQ} has a positive continuous solution $u$ in ${\mathbb{R}}^n_{+}\times (0,\infty)$ satisfying $cP_tu_0(x)\leq u(x,t)\leq P_tu_0(x),$ for each $t>0$ and $x\in {\mathbb{R}}^n_{+}$, where $c\in (0,1)$. \end{theorem} The elliptic counterpart of the problem \eqref{eP} was studied in \cite{a2}. There the author proved existence and nonexistence results for the semilinear elliptic equation $$\label{e2} \begin{gathered} \Delta u=g(u) \quad \text{in }D \\ u=\varphi \quad \text{on }\partial D, \end{gathered}$$ where $D$ is a simply connected bounded $C^2$-domain in $\mathbb{R}^d$ $(d\geq 3)$, $g$ is a continuous function on $(0,\infty )$ such that $0\leq g(u)\leq \max (1,u^{-\alpha })$, for $0<\alpha <1$ and $\varphi$ is a nontrivial nonnegative continuous function on $\partial D$. More precisely, Athreya \cite{a2} proved the following result. \begin{theorem} \label{thm2} There exists $00$. \end{definition} To illustrate the above definition, we consider the following examples of functions satisfying (H0); see \cite{hm2}. $\bullet$ Every bounded nonnegative superharmonic function $\omega$ in $\mathbb{R}^n_{+}$ satisfies (H0). $\bullet$ $\omega (x)=x_n^{\beta }$, $0<\beta \leq 1$. Indeed, $\Delta \omega =\beta (1-\beta )\omega ^{\frac{\beta -2}{\beta }}$, then $\omega$ is a superharmonic function. Moreover, by a simple calculation we obtain $\omega (x)-P_t\omega (x)=\int_0^{t}P_{s}\omega ^{\frac{\beta -2}{\beta } }(x)ds,(x,t)\in \mathbb{R}_{+}^n\times (0,\infty ).$ Hence, $P\omega \leq \omega$ and so $\lim_{x\to \partial \mathbb{R} _{+}^n}P_t\omega (x)=0$. Furthermore, the function $(x,t)\to \omega (x)-P_t\omega (x)$ is upper semicontinuous, which ensures the continuity of the function $(x,t)\to P_t\omega (x)$. $\bullet$ $\omega (x)=K\nu (x)$, where $\nu$ is a nonnegative measure on $\partial \mathbb{R}_{+}^n$ satisfying for $0<\alpha \leq n/2$ $\sup_{x\in \mathbb{R}_{+}^n}\int_{\partial \mathbb{R}_{+}^n} \frac{x_n}{|x-z|^{n-2\alpha }}\nu (dz)<\infty .$ $\bullet$ $\omega (x)=\Sigma _{p=1}^{\infty }\min(p,\alpha _{p}\mathcal{G} (x,e_{p}))$, where $\mathcal{G}$ is the Green's function of $\Delta$ in $\mathbb{R}_{+}^n$ with zero boundary condition, $e_{p}=(0,\dots,0,p)$ and $\alpha _{p}>0$ is chosen such that $\alpha _{p}\mathcal{G}(x,e_{p})\leq 2^{-p}$ for $x\in B^{c}(e_{p},\frac{1}{2})\cap \mathbb{R}_{+}^n$. This last example is studied in \cite{hm2}, where the authors proved that the function $\omega$ is an unbounded potential satisfying condition (H0). For the rest of this article, we fix a nonnegative superhahmonic function $\omega$ satisfying the condition (H0), and we assume the following hypotheses: \begin{itemize} \item[(H1)] The function $f:(0,\infty )\to [0,\infty )$ is nonincreasing and continuous. \item[(H2)] For all $x\in \mathbb{R}_{+}^n$, we have $\lim_{t\to 0}P_tu_0(x)=u_0(x)$ and $$Pu_0\in C(\mathbb{R}_{+}^n\times (0,\infty ))\;\text{and}\; \lim_{x\to \xi \in \partial \mathbb{R}_{+}^n}P_tu_0(x)=0. \label{e1b}$$ We note that if there exists $c>0$ such that $0\leq u_0\leq c\omega$, then \eqref{e1b} is satisfied. \item[(H3)] $p:\mathbb{R}_{+}^n\times (0,\infty )\to [ 0,\infty )$ is measurable such that the function $\widetilde{p}:=\frac{pf(P\omega )}{P\omega }$ belongs to the parabolic Kato class $P^{\infty }(\mathbb{R}_{+}^n)$. \end{itemize} Before stating our main result, we give an example where (H3) is satisfied. \begin{example} \label{exa1} \rm Let $f$ be a non-increasing continuous function such that there exists $\eta>0$ satisfying $0\leq f(t)\leq \eta (t+1) \quad \forall t>0.$ Let $\omega(x)=x_n$, $x\in{\mathbb{R}}^n_{+}$ and let $p$ be a nonnegative function such that $p\leq \frac{\omega}{1+\omega}q$ for some $q\in P^{\infty}({\mathbb{R}}^n_{+})$. Then we have $\widetilde{p}=\frac{pf(P\omega )}{P\omega} =\frac{pf(\omega )}{\omega}\leq \eta\frac{1+\omega}{1+\omega} q =\eta q$ which belongs to $P^{\infty}({\mathbb{R}}^n_{+})$. \end{example} More examples where (H3) is satisfied will be developed in section 4. Now, we give our main result. \begin{theorem} \label{thm3} Under the assumptions {\rm (H1)--(H3)}, there exist a constant $c>1$ such that if $u_0\geq c\omega$ on ${\mathbb{R}}^n_{+}$, then \eqref{eP} has a positive continuous solution $u$ satisfying, for each $x\in{\mathbb{R}}^n_{+}$ and $t>0$, $P_t\omega(x) \leq u(x,t) \leq P_tu_0(x).$ \end{theorem} The outline of this article is as follows. In section 2, we give some notations and we recall some properties of the parabolic Kato class $P^{\infty }(\mathbb{R}_{+}^n)$. Section 3 concerns the proof of Theorem \ref{thm3} by using a potential theory approach. The last section is reserved for examples. \section{Preliminary results} In this section we collect some useful results concerning the parabolic Kato class $P^{\infty}(\mathbb{R}^n_{+})$, which is stated in \cite{hm2} and \cite{lm}. \begin{definition}[\cite{hm2}] \label{def2} \rm A Borel measurable function $q$ in ${\mathbb{R}} ^n_{+}\times{\mathbb{R}}$ belongs to the class $P^{\infty}({\mathbb{R}}^n_{+})$ if for all $c>0$, $\lim_{h\to0}\sup_{(x,t)\in\mathbb{R} _+^n\times{\mathbb{R}}} \int_{t-h}^{t+h}\int_{B(x,\sqrt{h})\cap{\mathbb{R}} ^n_{+}} \min(1,\frac{y_n^2}{|t-s|})G_{c}(x,|t-s|,y,0)|q(y,s)|dyds=0$ and $\sup_{(x,t)\in{\mathbb{R}}^n_{+}\times{\mathbb{R}}} \int_{-\infty}^{+\infty}\int_{{\mathbb{R}}^n_{+}} \min(1,\frac{y_n^2}{ |t-s|})G_{c}(x,|t-s|,y,0)|q(y,s)|dyds<\infty,$ where $G_{c}(x,t,y,s):=\frac{1}{(t-s)^{n/2}}exp(-c\frac{|x-y|^2}{t-s} ), \quad t>s, x, y\in \mathbb{R}_+^n.$ \end{definition} \begin{remark} \label{rmk1} \rm The parabolic Kato class $P^{\infty}({\mathbb{R}}^n_{+})$ is quite rich. In particular, it contains the time independent Kato class $K^{\infty}({\mathbb{R}}^n_{+})$ used in the study of elliptic equations (See \cite{IB,IBLM} for definition and properties). \end{remark} Other examples of functions belonging to $P^{\infty}(\mathbb{R}^n_{+})$ are given by the following proposition. \begin{proposition}[\cite{hm2}] \label{prop1} \begin{itemize} \item[(i)] $L^{\infty}({\mathbb{R}}^n_{+})\otimes L^{1}({\mathbb{R}} )\subset P^{\infty}({\mathbb{R}}^n_{+})$. \item[(ii)] $K^{\infty}({\mathbb{R}}^n_{+})\otimes L^{\infty}({\mathbb{R}} )\subset P^{\infty}({\mathbb{R}}^n_{+})$. \item[(iii)] For $1\frac{np}{2}$ and $\delta<\frac{2}{p}-\frac{n}{s}<\nu$ we have $\frac{L^{s}({\mathbb{R}}^n_{+})}{\theta (.)^{\delta}(1+|.|)^{\nu-\delta}} \otimes L^{q}({\mathbb{R}}^n_{+})\subset P^{\infty}({\mathbb{R}}^n_{+}),$ where $\theta$ is defined on $\mathbb{R}^n_{+}$ by $\theta(x)=x_n$. \end{itemize} \end{proposition} We state now an elementary inclusion of the class $P^{\infty}(\mathbb{R} ^n_{+})$ as follows. \begin{proposition}[\cite{hm2}] \label{prop2} Let $q\in P^{\infty}({\mathbb{R}}^n_{+})$, then the function $(y,s)\mapsto y_n^2q(y,s)$ is in $L^{1}_{\rm loc}(\overline{{\mathbb{R}}^n_{+}}\times{\mathbb{R}})$. In particular, we have $P^{\infty}({\mathbb{R}}^n_{+})\subset L^{1}_{\rm loc} ({\mathbb{R}}^n_{+}\times{\mathbb{R}})$. \end{proposition} For any nonnegative measurable function $f$ in $\mathbb{R}^n_{+}\times(0,\infty)$, we denote $Vf(x,t):=\int_0^{t}\int_{\mathbb{R}^n_{+}}\Gamma(x,t,y,s)f(y,s)dyds= \int_0^{t}P_{t-s}(f(.,s))(x)ds$ and we give the following propositions that will be useful in proving the existence and continuity of solutions to \eqref{eP}. \begin{proposition}[\cite{hm2}] \label{prop3} Let $q$ be a nonnegative function in $P^{\infty}({\mathbb{R}}^n_{+})$ then there exists a positive constant $\alpha_{q}$ such that for each nonnegative superharmonic function $v$ in ${\mathbb{R}}^n_{+}$, $V(qPv)(x,t)=\int_0^{t}\int_{{\mathbb{R}}^n_{+}}\Gamma (x,t,y,s)f(y,s)P_tv(y)dyds\leq \alpha_{q}P_tv(x),$ for every $(x,t)\in {\mathbb{R}}^n_{+}\times (0,\infty)$. \end{proposition} \begin{proposition}[\cite{hm2}] \label{prop4} Let $w$ be a nonnegative superharmonic function in ${\mathbb{R}}^n_{+}$ satisfying (H0) and $q$ be a nonnegative function in $P^{\infty}({\mathbb{R}}^n_{+})$ then the family of functions $\Big\{(x,t)\to\,Vf(x,t)=\int_0^{t}\int_{{\mathbb{R}} ^n_{+}}\Gamma(x,t,y,s)f(y,s)dyds, \,|f|\leq qPw\Big\}$ is equicontinuous in ${\mathbb{R}}^n_{+}\times (0,\infty)$. Moreover, for each $(x,t)\in {\mathbb{R}}^n_{+}\times (0,\infty)$, we have $$\lim_{s\to 0}Vf(x,s)=\lim_{y\to \partial{\mathbb{R}}^n_{+}}Vf(y,t)=0,$$ uniformly on $f$. \end{proposition} We will apply the following auxiliary result, several times in this article. \begin{proposition} \label{prop5} Let $\omega$ be a nonnegative superharmonic function satisfying condition {\rm (H0)} and $\varphi$ be a nonnegative measurable function such that $\varphi \leq\omega$ on ${\mathbb{R}}^n_{+}$, then the function $(x,t)\to P_t\varphi (x)$ is continuous on ${\mathbb{R}}^n_{+}\times (0,\infty)$ and $\lim_{x\to\partial{\mathbb{R}}^n_{+}}P_t\varphi(x)=0$, for every $t>0$. \end{proposition} \begin{proof} For each $(x,t)\in \mathbb{R}_{+}^n\times (0,\infty )$, we write $P_t\omega (x)=P_t\varphi (x)+P_t(\omega -\varphi )(x).$ So, from (H0) we have $(x,t)\to P_t\omega (x)$ is continuous in $\mathbb{R}_{+}^n\times (0,\infty )$ and from the fact that $(x,t)\to P_t\varphi (x)$ and $(x,t)\to P_t(\omega -\varphi )(x)$ are lower semicontinuous in $\mathbb{R}_{+}^n\times (0,\infty )$, we deduce that $(x,t)\to P_t\varphi (x)$ is continuous in $\mathbb{R}_{+}^n\times (0,\infty )$. On the other hand since $0\leq P_t\varphi \leq P_t\omega$ and $\lim_{x\to \partial \mathbb{R}_{+}^n}P_t\omega (x)=0$, then we have $\lim_{x\to \partial \mathbb{R}_{+}^n}P_t\varphi (x)=0$, for every $t>0$. \end{proof} \section{Proof of theorem \ref{thm3}} Let $\widetilde{p}$ be the function given in hypothesis (H3) and let $\alpha _{\widetilde{p}}$ be the constant defined in Proposition \ref{prop3}. We put $c:=1+\alpha _{\widetilde{p}}$ and we consider a nonnegative continuous function $u_0$ on $\mathbb{R}_{+}^n$ such that $u_0\geq c\omega$. Let $\Lambda$ be the non-empty closed convex set given by $\Lambda =\{v\, \text{measurable function in }\mathbb{R}_{+}^n\times (0,\infty ):P\omega \leq v\leq Pu_0\}.$ We define the integral operator $T$ on $\Lambda$ by $T(v)=Pu_0-V(pf(v)).$ We aim to prove that $T$ has a fixed point $u$ in $\Lambda$. First, we prove that $T$ maps $\Lambda$ into itself. Let $v\in \Lambda$, since $v\geq P\omega \geq 0$, we have $Tv\leq Pu_0.$ Furthermore, by the monotonicity of the function $f$ we have \begin{align*} Tv &= Pu_0-V(pf(v)) \\ &\geq Pu_0-V(\widetilde{p}P\omega ) \\ &\geq c_{1}P\omega -\alpha _{\widetilde{p}}P\omega \\ &\geq (c_{1}-\alpha _{\widetilde{p}})P\omega \geq P\omega . \end{align*} Secondly, we claim that $T$ is nondecreasing on $\Lambda$. Indeed, let $u,v\in \Lambda$ such that $u\leq v$. Then it follows from the monotonicity of the function $f$ that $Tv-Tu=V(p(f(u)-f(v)))\geq 0.$ Now, we define the sequence $(v_{k})_{k\in \mathbb{N}}$ by $v_0=P\omega \quad \text{and}\quad v_{k+1}=Tv_{k}, \quad \text{for } k\in\mathbb{N}.$ Since $T\Lambda \subset \Lambda$, then from the monotonicity of $T$, we obtain for all $k\in \mathbb{N}$ $P\omega \leq v_{k}\leq v_{k+1}\leq Pu_0.$ So, the sequence $(v_{k})_{k\in \mathbb{N}}$ converge to a function $u\in \Lambda$. Moreover, using hypothesis (H3) and the monotonicity of the function $f$ we obtain for each $k\in \mathbb{N}$ $pf(v_{k})\leq pf(Pw)=\widetilde{p}P\omega .$ So, by Proposition \ref{prop3} and Lebesgue's theorem we deduce that $V(pf(v_{k})$ converges to $V(pf(u))$ as $k$ tends to infinity. Then, on $\mathbb{R}_{+}^n\times (0,\infty )$, $u$ satisfies $$u=Pu_0-V(pf(u)). \label{e2b}$$ At the remainder of the proof, we aim to show that $u$ is a desired solution of \eqref{eP}. It is obvious that $$pf(u)\leq \widetilde{p}Pw. \label{e3}$$ So, from the hypothesis (H0) and Proposition \ref{prop2}, we deduce that $pf(u)\in L_{\rm loc}^{1}(\mathbb{R}_{+}^n\times (0,\infty ))$ moreover, by \eqref{e3} and Proposition \ref{prop4}, we obtain $V(pf(u))\in C(\mathbb{R}_{+}^n\times (0,\infty )) \subset L_{\rm loc}^{1}( \mathbb{R}_{+}^n\times (0,\infty )).$ In addition, using \eqref{e1b} and Proposition \ref{prop5} we obtain $Pu_0\in C(\mathbb{R}_{+}^n\times (0,\infty )).$ Thus, by \eqref{e2b} it follows that $u\in C(\mathbb{R}_{+}^n\times (0,\infty ))$. Now, applying the heat operator $\Delta -\frac{\partial }{\partial t}$ in \eqref{e2b}, we obtain clearly that $u$ is a positive continuous solution (in the distributional sense) of $\Delta u-\frac{\partial u}{\partial t}=p(x,t)f(u)\, \,\text{in}\, \mathbb{R }_{+}^n\times (0,\infty ).$ Next, using \eqref{e1b} and hypothesis (H2), it follows that $\lim_{t\to 0}u(x,t)=\lim_{t\to 0}P_tu_0(x)=u_0(x)\quad\text{and}\quad \lim_{x\to \xi \in \partial \mathbb{R}_{+}^n}P_tu_0(x)=0.$ Finally, from \eqref{e3} and Proposition \ref{prop4}, we conclude that for each $x\in \mathbb{R}_{+}^n$ we have $\lim_{t\to 0}V(pf(u))(x,t)=0.$ Hence, $u$ is a positive continuous solution in $\mathbb{R}_{+}^n\times (0,\infty )$ of the problem \eqref{eP}. This completes the Proof. \section{Examples} In this section we give some examples. The first one concerns functions satisfying the hypothesis (H3), the second is an application of Theorem \ref{thm3}. \begin{example} \label{exa2} \rm Let $f$ be a nonnegative bounded continuous function on $(0,\infty)$ and $\sigma$ be a nonnegative measure on $\partial{\mathbb{R}}^n_{+}$ satisfying $\sup_{x\in{\mathbb{R}}^n_{+}}\int_{\partial{\mathbb{R}} ^n_{+}}\frac{x_n}{|x-z|^{n-2\alpha}}\sigma(dz)<\infty,$ for some $0<\alpha\leq n/2$. Then, it was shown in \cite{hm2}, that the harmonic function defined on ${\mathbb{R}}^n_{+}$ by $K\sigma(x):=\Gamma(\frac{n}{2})\pi^{-n/2} \int_{\partial\mathbb{R} ^n_{+}}\frac{x_n}{|x-z|^n}\sigma(dz)$ satisfies condition (H0). Now, let $\omega=K\sigma$ and let $p$ be a nonnegative function such that $p\leq qP(\omega)$ for some $q\in P^{\infty}({\mathbb{R}}^n_{+})$, then $\widetilde{p}=\frac{pf(P\omega)}{P\omega}\leq ||f||_{\infty}q\in P^{\infty}({ \mathbb{R}}^n_{+}).$ Hence, hypothesis (H3) is satisfied. \end{example} \begin{example} \label{exa3} \rm Let $1\leq s <\infty$ and $r\geq 1$ such that $\frac{1}{s}+\frac{1}{r}=1$. Let $\sigma\geq \frac{ns}{2}$ and $\rho < \frac{2}{s}-\frac{n}{\sigma}<\mu$. For each $(x,t)\in {\mathbb{R}}^n_{+}\times (0,\infty)$, We put $p(x,t)=\frac{|g(x)|}{x_n^{\rho-(\gamma+1)}(1+|x|)^{\mu-\rho}}|h(t)|,$ where $\gamma>0$, $g\in L^{\sigma}({\mathbb{R}}^n_{+})$ and $h\in L^{r}({\mathbb{R}})$. Let $u_0$ be a nonnegative continuous function on ${\mathbb{R}}^n_{+}$ satisfying hypothesis (H2). Then, there exist a constant $c>1$ such that if $u_0(x)\geq cx_n$, for all $x\in {\mathbb{R}}^n_{+}$, the problem \begin{gather*} \Delta u - \frac{\partial u}{\partial t} =p(x,t)u^{-\gamma} \quad \text{in } {\mathbb{R}}^n_{+}\times (0,\infty) \\ u(x,0)=u_0(x)\quad \text{in } {\mathbb{R}}^n_{+} \\ u(z,t)=0\quad \text{on } \partial{\mathbb{R}}^n_{+}\times (0,\infty), \end{gather*} has a positive continuous solution $u$ satisfying, for each $(x,t)\in{\mathbb{R}}^n_{+}\times (0,\infty)$, $x_n \leq u (x,t)\leq P_tu_0(x).$ \end{example} \subsection*{Acknowledgments} The author wants to thank Professor Habib M\^aagli for his guidance and useful discussions, and the anonymumous referees for their suggestions. \begin{thebibliography}{00} \bibitem{a1} D. Armitage, S. Gardiner; \emph{Classical Potential Theory}, Springer-Verlag, Berlin, 2001. \bibitem{a2} S. 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