## 3 Results
#### 3.2.1 Construction of the Upper and Lower Solutions
 \tag{34}\]  
Here, \(v^{(\mp)}(\varrho^{(\mp)}(\tau,\mathbf{y}),\mathbf{y})\) are the functions defined by the expressions in Equation (11) and  
\[\tilde{G}^{(\mp)}(\tau,\mathbf{y}):=q^{(\mp)}g(\tau,\mathbf{y})+\frac{|\tilde{ g}_{u}^{(\mp)}(\tau,\mathbf{y})|}{\tilde{f}_{u}^{(\mp)}(\tau,\mathbf{y})}\Big{(}q^{( \mp)}f(\tau,\mathbf{y})-d^{u}e^{-\varkappa^{u}|\tau|}\Big{)}. \tag{35}\]  
We choose the constant \(\delta^{v}\) so that for all \(\mathbf{y}\in\gamma\), the inequalities \(\delta^{v}-\beta^{(\mp)}(\,\mathbf{y})>0\) hold; the value \(d^{u}\) in the expression in Equation (35) is chosen large enough, and \(\varkappa^{u}\) is chosen small enough to provide the negativeness of the functions \(\tilde{G}^{(\mp)}(\tau,\mathbf{y})\) in the corresponding domains \(\tau\leq 0,\mathbf{y}\in\gamma\) and \(\tau\geq 0,\mathbf{y}\in\gamma\). This guaranties that the solutions to Problems (33) are strictly positive, and we can obtain these functions explicitly:  
\[q^{(\mp)}v(\tau,\mathbf{y})=\Big{(}\delta^{v}-\beta^{(\mp)}( \mathbf{y})\Big{)}\frac{\Phi^{(\mp)}\Big{(}\tilde{\varrho}^{(\mp)}(\tau, \mathbf{y}),\mathbf{y}\Big{)}}{\Phi^{(\mp)}(q(\mathbf{y}),\mathbf{y})}+\] \[\quad+\Phi^{(\mp)}\Big{(}\varrho^{(\mp)}(\tau,\mathbf{y}), \mathbf{y}\Big{)}\int\limits_{0}^{\tau}\!\!\frac{d\tau_{1}}{\big{(}\Phi^{(\mp)} \big{(}\tilde{\varrho}^{(\mp)}(\tau_{1},\mathbf{y}),\mathbf{y}\big{)}\big{)}^{2 }}\!\!  
\[q^{(\mp)}v(\tau,\mathbf{y})=\Big{(}\delta^{v}-\beta^{(\mp)}( \mathbf{y})\Big{)}W^{(\mp)}(\tau,\mathbf{y})+\\ +W^{(\mp)}(\tau,\mathbf{y})\int\limits_{0}^{\tau}\frac{d\tau_{1}}{ \big{(}W^{(\mp)}(\tau_{1},\mathbf{y})\big{)}^{2}}\int\limits_{\mp\infty}^{\tau_ {1}}W^{(\mp)}(\tau_{2},\mathbf{y})\tilde{G}^{(\mp)}(\tau_{2},\mathbf{y})\,d \tau_{2},\]  
where \(W^{(\mp)}(\tau,\mathbf{y})\) are the solutions to  
\[W^{(\mp)}_{\tau\tau}-v^{(\mp)}(\tilde{v}^{(\mp)}(\tau,\mathbf{y}),\mathbf{y}) \cdot W^{(\mp)}=0,\ \ \tau\in\mathbb{R}^{\mp},\ \mathbf{y}\in\gamma,\ \ W^{(\mp)}(0,\mathbf{y})=1,\ \ W^{(\mp)}(\mp \infty,\mathbf{y})=0.\]  
These functions are strictly positive and decrease exponentially to zero as \(|\tau|\to\infty\) (see [15] and [14] (T. 9.3 p. 354)), and for all \(\mathbf{y}\in\gamma\), the inequalities hold [15]:  
\[\frac{\partial W^{(-)}}{\partial\tau}(0,\mathbf{y})>0,\ \ \ \ \frac{\partial W^{(+)}}{\partial\tau}(0,\mathbf{y})<0. \tag{36}\]  
The functions \(q^{(\mp)}u(\tau,\mathbf{y})\), expressed with the equalities in the first line of Equation (32), also turn out to be positive since Proposition 1 is satisfied. They have estimates like that in Equation (24) as well as the functions \(q^{(\mp)}v(\tau,\mathbf{y})\). The \(m^{(\mp)}u(\sigma,\mathbf{y})\) functions remove residuals in the inequalities in Condition (A\({}_{2}\)) (the first line), introduced by the functions \(\alpha^{(\mp)}(\mathbf{x})\), \(\beta^{(\mp)}(\mathbf{x})\), \(q^{(\mp)}u(\tau,\mathbf{y})\), \(q^{(\mp)}v(\tau,\mathbf{y})\) in the transition layer. We set them as the solutions to  
\[ \begin{split}\frac{\partial^{2}m^{(\mp)}u}{\partial\sigma^{2}}= \tilde{f}^{(\mp)}_{u}(\sigma,\mathbf{y})\cdot m^{(\mp)}u(\sigma,\mathbf{y})+ \hat{F}^{(\mp)}(\sigma,\mathbf{y}),\ \ \sigma\in\mathbb{R}^{\mp},\ \mathbf{y}\in \gamma,\\ m^{(\mp)}u(0,\mathbf{y})=\delta^{u}-\alpha^{(\mp)}(\mathbf{y})-q ^{(\mp)}u(0,\mathbf{y}),\ \ m^{(\mp)}u(\mp\infty,\mathbf{y})=0,\end{split}  \tag{37}\]  
where \(\delta^{u}\) is a positive value and \(\hat{f}^{(\mp)}_{u}(\sigma,\mathbf{y})\) has the sense of the notation from Equation (18):  
\[ \begin{split}\hat{F}^{(\mp)}(\sigma,\mathbf{y}):=& \ \Big{[}\tilde{f}^{(\mp)}_{u}(\sigma,\mathbf{y})-\tilde{f}^{(\mp)}_{u}(0, \mathbf{y})\Big{]}\Big{(}\alpha^{(\mp)}(\mathbf{y})+q^{(\mp)}u(0,\mathbf{y}) \Big{)}+\\ +&\ \Big{[}\tilde{f}^{(\mp)}_{v}(\sigma,\mathbf{y})- \tilde{f}^{(\mp)}_{v}(0,\mathbf{y})\Big{]}\delta^{v}-D^{u}e^{-K^{u}|\sigma|}. \end{split}