## 3 Positive Global Solutions for Second Problem
\begin{array}{l}\mathcal{F}\left(u^{(n)}\right)_{t}+\sigma\left(u^ {(n)}\right)_{t}=c_{U}\left(y,U\right)U_{t},\quad\text{in}\ \ \Theta,\\ \left(u^{(n)}\right)_{t}=0,\qquad\qquad\qquad\qquad\qquad\qquad\text{on }\ \partial\Theta,\end{array} \right.\]  
in which \(U=u^{(n-1)}\). Since \(c_{U}(y,U)U_{t}\) is located in \(\Theta\), it is bounded. We set  
\[w_{n}=\frac{u^{(n)}\left(y,\delta\right)-u^{(n)}\left(y,0\right)}{\delta},\,\, y\in\Omega.\,\,\delta>0,\]  
As a result by (12) and (13), we get \(w_{n}\leq 0\). Hence, \(\left(u^{(n)}\left(y,0\right)\right)_{t}\leq 0\), \(y\in\Omega\). Furthermore, we have \(\left(u^{(n)}\right)_{t}\leq 0\). We can apply the same proof of Theorem 1 to get  
\[u^{(n)}\mapsto u\in C^{1+\eta}\ \ \text{on }\ t,\ \ \text{for}\ \ 0<\eta<1.\]  
Thus, \(u_{t}\left(y,t\right)\leq 0\) in \(\Theta\). Herewith, the proof is complete. Let us assume \(c\left(y,t,u\right)\) is a \(C^{1}\)-mapping for \(u\) and satisfies the inequalities:  
\[\left\{ \begin{array}{l}c\left(y,t,c_{1}\right)\geq 0,\ \ \ c\left(y,t,c_{2}\right)\leq 0, \ \ \text{in}\ \ \Omega\cap\mathbb{R}^{+},\\ c_{1}\mu\left(y,t\right)\leq h\left(y,t\right)\leq c_{2}\mu\left(y,t\right), \ \ \text{on}\,\partial\Omega\cap\mathbb{R}^{+},\end{array} \right. \tag{14}\]  
where \(c_{1}>0\), \(c_{2}>0\) are constants with \(c_{1}<c_{2}\). **Theorem 8**. : _Assume that (14) holds. If there exists constants \(c_{1}>0\), \(c_{2}>0\) with \(c_{1}<c_{2}\). Then, for all \(u\in[c_{1},c_{2}]\), (2) has a unique positive and bounded global solution._  
Proof. : Let \(\alpha=c_{2}\), \(\beta=c_{1}\); then, by (14), we get  
\[\left\{ \begin{array}{l}\mathcal{F}\alpha=0\geq c\left(y,t,c_{2}\right)=c \left(y,t,\alpha\right),\qquad\quad\text{in }\ \ \Omega\times\mathbb{R}^{+},\\ \mathcal{N}\alpha=\lambda\alpha_{y}+\mu\alpha=c_{2}\mu\left(y,t\right)\geq h \left(y,t\right),\ \ \text{on}\,\partial\Omega\times\mathbb{R}^{+},\\ \alpha=c_{2},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{ in }\ \ \Omega.\end{array} \right.\]This allows to conclude that \(\alpha=c_{2}\), is a US. Similarly, \(\beta=c_{1}\) is an LS. Consequently, thanks to the Theorem 6, we conclude the result. We are now ready to prove the uniqueness result of global positive solution. To this purpose, we assume that:  
\((a)\) For every \(y\in\Omega\), \(y^{\prime}\in\partial\Omega\): \(c\left(y,t\right)\geq 0\), \(h\left(y^{\prime},t\right)\geq 0\) and \(u_{0}\left(y\right)\geq 0\). **Theorem 9**. : _Suppose that \((a)\), \((5)\) and \((6)\) hold. If there exists a mapping \(M\) with_  
\[c_{u}\left(y,t,u\right)u\leq M\left(y,t\right)u,\text{ for every }T<\infty,\text{ }u\geq 0, \text{ in }\Theta, \tag{15}\]  
_then, \((2)\) has a unique global positive solution._  
Proof. : Using the mean-value theorem, \((5)\) and \((6)\), we have  
\[c\left(y,t,u\right) =c_{u}\left(y,t,\xi\right)u+c\left(y,t,0\right),\] \[\geq c_{u}\left(y,t,\xi\right)u,\text{ in }\Theta, \tag{16}\]  
in which \(\xi=\xi\left(y,t\right)\) is intermediate value between \(u\) and \(0\). By Lemma 3 and \(\mathcal{F}\), we write  
\[c_{3}\left(y,t\right)=-c_{u}\left(y,t,\xi\right).\]  
Hence, \(u=0,\text{ }or\text{ }0<u\) in \(\Theta\). \(u\) is positive because, if it is not, \(u=0\) only if all of the maps in \((a)\) is equal to \(0\). This implies that \(u=\beta\) is an LS of \((2)\). Let \(w\) be a solution of:  
\[\left\{ \begin{array}{ll}\mathcal{F}w=Mw+c\left(y,t,0\right)&\text{in }\ \ \Theta,\\ \mathcal{N}w=h\left(y,t\right)&\text{on}\ \partial\Theta,\\ w\left(y,0\right)=u_{0}\left(y\right)&\text{in }\ \Omega,\end{array} \right.\]  
Therefore, \(w\) is a upper positive solution of \((2)\). As a matter of fact, for \(\alpha=w\) and again applying the mean-value theorem,  
\[c\left(y,t,\alpha\right)=c_{\alpha}\left(y,t,\xi\right)\left(\alpha\right)+c \left(y,t,0\right),\]  
where \(\xi=\xi\left(y,t\right)\) is located between \(\alpha\) and \(0\).