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\begin{document}

\title{Existence and stability of the solution to the system of two diffusion
equations in medium with discontinuous characteristics}



\author[1]{Natalia LEVASHOVA}%
\author[1]{Bogdan Tishchenko}%
\affil[1]{Moskovskij gosudarstvennyj universitet imeni M V Lomonosova}%


\vspace{-1em}



  
  \date{May 04, 2024}


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\sloppy


\section*{Introduction}\label{sec0}
The article considers a system of two second order nonlinear differential equations  with discontinuous functions on the right sides. The aim of the work is to obtain sufficient conditions for the existence, local uniqueness, and asymptotic stability of a stationary solution of a parabolic system with a large gradient in the vicinity of the discontinuity points of the right-hand sides. The area where the function undergoes large gradient is called the internal transition layer.

The authors arrived at this formulation of the problem during the development of the autowave model for the development of megacities \hyperref[csl:1]{(Sidorova et al., 2018}; \hyperref[csl:2]{Levashova et al., 2019)}. This model is based on the activator-inhibitor system of two equations where the urban area acts as the activator, and the inhibitor is determined by environmental or economic factors due to urban planning policies of a country. The presence of barriers that prevent the propagation of the front of the activator, for example, large bodies of water, is taken into account in the model as a jump in the functions on the right-hand sides. Obviously, the numerical solution of such a problem should be preceded by an analytical study of the existence of the mentioned solution, which was done in the present work.

The proof of the existence and asymptotic stability of the stationary solution of the initial-boundary-value problem here is carried out using the asymptotic method of differential inequalities \hyperref[csl:3]{(Nefedov, 1995}; \hyperref[csl:4]{Butuzov et al., 2012)}, based on the method of super- and subsolutions.The latter was extended to problems with a single discontinuity point of the first kind on the right-hand sides of the equations based on a modified proof of the corresponding theorem from \hyperref[csl:5]{(Pao, 1992)}, where it was carried out for the case of $C^2$ continuous right-hand sides.

\section*{Problem statement}\label{sec1}
We consider the following initial-boundary-value problem:

\begin{eqnarray}
\label{par_syst} 
&\varepsilon^{4}y_{xx}-y_t=f(y,z,x,\varepsilon),\;\; x \in (0,1),\; t>0, \quad y_x(0,t) =y_x(1,t)=0, \;\;y(x,0) = u^0(x),\\
&\varepsilon^{2}z_{xx}-z_t=g(y,z,x,\varepsilon),\;\; x \in (0,1),\; t>0, \quad  z_x(0,t) =z_x(1,t)=0, \;\;z(x,0) = v^0(x),

\end{eqnarray}
where $u^0(x)$, $v^0(x)\in C([0,1])$ and $u^0_x(0)=u^0_x(1)=v^0_x(0)=v^0_x(1)=0$, $\varepsilon \in (0,\varepsilon_0]$ is a small parameter.

The functions $f(u,v,x,\varepsilon)$ and $g(u,v,x,\varepsilon)$ have the first kind discontinuities  across the surface $\{ u \in I_u, v\in I_v,\,x=x_{0} \in (0,1)\}$, where $I_u$ and $I_v$ are respectively permissible $u$ and $v$ change intervals:

\begin{equation*}
f(u,v,x,\varepsilon) =

\begin{cases}
 f^{(-)}(u,v,x,\varepsilon),\\
 f^{(+)}(u,v,x,\varepsilon),

\end{cases}
\quad
g(u,v,x,\varepsilon) =

\begin{cases}
g^{(-)}(u,v,x,\varepsilon),& u \in I_u, v\in I_v,\,0<x\leq x_{0}\,,\\
g^{(+)}(u,v,x,\varepsilon),& u \in I_u, v\in I_v,\,x_{0}<x\leq 1\,,

\end{cases}

\end{equation*}
$f^{(-)}(u,v,x,\varepsilon)$ and $g^{(-)}(u,v,x,\varepsilon)$ are of class $C^{4}(I_{u}\times I_{v}\times [0,x_{0}]\times[0,\varepsilon_0])$,  $f^{(+)}(u,v,x,\varepsilon)$ and $g^{(+)}(u,v,x,\varepsilon)$ are of class \linebreak
$C^{4}(I_{u}\times I_{v}\times [x_{0},1]\times[0,\varepsilon_0])$.

Denote $D_T := (0,1)\times \mathbb{R^+}$, $D_T^{(-)} := (0,x_0)\times \mathbb{R^+}$, $ D_T^{(+)} := (x_0,1)\times \mathbb{R^+}$.

\begin{definition}
\label{par_sol_def} 
A pair of functions $(y_\varepsilon(x,t), z_\varepsilon(x,t))$ in  $C^{1,0}\left(\overline{D_T}\right)\cap C^{2,1}\left(D_T^{(-)}\cup D_T^{(+)}\right)$ is called the solution to problem \eqref{par_syst} if it satisfies
equations \eqref{par_syst} in $D_T^{(-)}\cup D_T^{(+)}$, the boundary and initial conditions.

\end{definition}

\begin{proposition}
\label{H1} 
 Each of the equations $f^{(\mp)}(u,v,x,0)=0$ is solvable with respect to $u$ and the functions
$u=\varphi^{(\mp)}(v,x)$ are the isolated solutions to these equations, respectively, in domains $I_v\times [0, x_0]$ and $I_v\times [x_0, 1]$, the inequality $\varphi^{(-)}(v,x_0)<\varphi^{(+)}(v,x_0)$ holds for all $v\in I_v$ and $f_{u}^{(\mp)}(\varphi^{(\mp)}(v,x),v,x,0) >0$ in respective domains.

\end{proposition}

Denote $h^{(\mp)}(v,x)= g^{(\mp)}(\varphi^{(\mp)}(v,x),v,x,0)$.

\begin{proposition}
\label{H2} 
 Each of the equations $h^{(\mp)}(v,x)=0$ is solvable with respect to $v$ and the functions
$v=\psi^{(\mp)}(x)$ are the isolated solutions to these equations, respectively, in the segments $[0, x_0]$ and $[x_0, 1]$, and the inequalities $h_{v}^{(\mp)}(\psi^{(\mp)}(x),x)>0$ hold in respective segments.

\end{proposition}

The main aim of this article is to obtain the existence and stability conditions for the stationary solution to problem\ref{par_syst} that is close to functions $(\varphi^{(-)},\psi^{(-)})$ to the left of point $x_0$ close to functions $(\varphi^{(+)},\psi^{(+)})$ to the right of this point and has a large gradient in the vicinity of point $x_0,$ changing rapidly from values $(\varphi^{(-)},\psi^{(-)})$ to $(\varphi^{(+)},\psi^{(+)})$. Obviously the stable stationary solution of problem\ref{par_syst} is a solution to the following problem

\begin{equation}
\label{ell_syst} 
\varepsilon^{4}u''=f(u,v,x,\varepsilon),\;\; \varepsilon^{2}v''=g(u,v,x,\varepsilon),\;\;  x \in (0,1), \quad  u'(0) =u'(1)=0,\;\; v'(0) =u'(1)=0.

\end{equation}

\begin{definition}
A pair of functions $(u_\varepsilon(x), v_\varepsilon(x))$ in  $C^1([0,1])\cap C^2((0,1)\backslash x_0)$ is called the solution to problem \eqref{ell_syst} if it satisfies
equations \eqref{ell_syst}  in $x\in(0,x_0)\cup(x_0,1)$ and the boundary conditions.

\end{definition}

\begin{proposition}
\label{H3} 
(Quasi-monotonicity) Let the inequalities hold: $f_{v}^{(\mp)}(u,v,x,\varepsilon)>0$,
$g_{u}^{(\mp)}(u,v,x,\varepsilon)<0$ for all $(u,v,x)\in I_u\times I_v\times [0,1]$.

\end{proposition}

Let's consider so-called associated equations for problem\ref{ell_syst}:

\begin{eqnarray}
&
\dfrac{d^{2}\tilde{v}}{d\tau^2}=h^{(-)}(\tilde{v},x_{0}),\;\; \tau<0,\quad
\dfrac{d^{2}\tilde{v}}{d\tau^2}=h^{(+)}(\tilde{v},x_{0}),\;\; \tau>0,
\quad\tau := \dfrac{x-x_0}{\varepsilon}
\label{PrisEqV} \\
&
\dfrac{d^{2}\hat{u}}{d\sigma^{2}}=f^{(-)}\left(\hat{u},v,x_{0},0\right),\;\; \sigma<0,\quad
\dfrac{d^{2}\hat{u}}{d\sigma^{2}}=f^{(+)}\left(\hat{u},v,x_{0},0\right),\;\; \sigma>0,
\quad\sigma := \dfrac{x-x_0}{\varepsilon^2}.
\label{PrisEqU} 

\end{eqnarray}
Each of the associated equations is equivalent to related associated system

\begin{equation*}
\frac{d\tilde{v}}{d\tau}= \Phi^{(\mp)},\;\; \frac{d\Phi^{(\mp)}}{d\tau}=h^{(\mp)}(\tilde{v},x_{0});\quad
\frac{d\hat{u}}{d\sigma}= \Psi^{(\mp)},\;\; \frac{d\Psi^{(\mp)}}{d\sigma}=f^{(\mp)}\left(\hat{u},v,x_{0},0\right).

\end{equation*}
By Propositions\ref{H1} and \ref{H2} the points $(\psi^{(\mp)},0)$ are the saddle-type rest points respectively for the first pair of systems on the phase plain $(\tilde v, \Phi)$ and the points $(\varphi^{(\mp)}(v,x_0),0)$ for each parameter $v\in I_v$ are respectively the saddles of the second pair of associated systems on phase plane $(\hat u, \Psi)$.

The functions

\begin{equation*}
\Phi^{(\mp)}(v)= \sqrt{\;\; 2\mkern-16mu\int\limits_{\psi^{(\mp)}(x_0)}^{v} \!\mkern-14mu h^{(\mp)}(s,x_0)ds}\;,\quad\Psi^{(\mp)}(u,v) = \sqrt{\;\; 2\mkern-28mu\int\limits_{\phantom{v^{*},}\varphi^{(\mp)}(v,x_0)}^{u} \!\mkern-30mu f^{(\mp)}(s,v,x_0,0)ds}

\end{equation*}
are the separatrixes of respective saddle points. If the function $\tilde v\to \psi^{(-)}(x_0)$ as $\tau\to-\infty$ and $\tilde v\to \psi^{(+)}(x_0)$ as $\tau\to+\infty$ then the separatrixes $\Phi^{(-)}$ and $\Phi^{(+)}$ intersect. If the function $\hat u\to \varphi^{(-)}(v,x_0)$ as $\sigma\to-\infty$ and $\hat u\to \varphi^{(+)}(v,x_0)$ as $\sigma\to+\infty$ then the separatrixes $\Psi^{(-)}$ and $\Psi^{(+)}$ intersect.
We denote

\begin{equation}
\label{H-Func} 
H^{v}(\tilde v) :=  \Phi^{(-)}(\tilde v)- \Phi^{(+)}(\tilde v),\;\;
H^{u}(\hat u,v) :=  \Psi^{(-)}(\hat u,v)- \Psi^{(+)}(\hat u,v).

\end{equation}

\begin{proposition}
\label{H4} 
 Let there exist the values $q_0$ in the interval $\left( \psi^{(-)}(x_0), \psi^{(+)}(x_0) \right)$
and $p_0$ in the interval $\left(\varphi^{(-)}\left( \psi^{(-)}(x_0),x_0\right), \varphi^{(+)}\left( \psi^{(+)}(x_0),x_0\right) \right)$ such that $q_0$ is the unique solution of the equation $H^{v}(\tilde v) = 0$, $p_0$ is the unique solution of $H^{u}(\hat u,q_0) = 0$ in the respective intervals and

\begin{equation*}
\frac{dH^{v}}{dv}(q_{0}) = h^{(-)}(q_0,x_0)-h^{(+)}(q_0,x_0)  > 0,\quad
\frac{\partial H^{u}}{\partial u}(p_{0},q_{0})=f^{(-)}(p_0,q_0,x_0,0)-f^{(+)}(p_0,q_0,x_0,0) > 0.

\end{equation*}

\end{proposition}
We introduce the functions

\begin{equation}
\label{nu} 
\nu^{(\mp)}(v,x) := g_{v}^{(\mp)}(\varphi^{(\mp)}(v,x ),v,x,0) \,+\frac{f_{v}^{(\mp)}(\varphi^{(\mp)}(v,x ),v,x,0)}{f_{u}^{(\mp)}(\varphi^{(\mp)}(v,x ),v,x,0)}\mkern -1.5mu \cdot \mkern -1.5mu g_{u}^{(\mp)}(\varphi^{(\mp)}(v,x ),v,x,0),\quad
\bar{\nu}^{(\mp)}(x) := \nu^{(\mp)}(\psi^{(\mp)}(x),x).

\end{equation}

\begin{proposition}
\label{H5} 
 In the respective segments $\bar{\nu}^{(\mp)}(x)>0$. Also, $\tilde{\nu}^{(\mp)}(\tilde{v}^{(\mp)}(\tau))$ are such that

\begin{equation*}
\int\limits_{\psi^{(-)}(x_0)}^{\tilde{v}} \mkern-10mu \tilde{\nu}^{(-)}(s)ds >0
,\;\;\tilde v\in (\psi^{(-)}(x_0),\psi^{(+)}(x_0)], \quad
 \int\limits_{\psi^{(+)}(x_0)}^{\tilde v} \mkern-10mu \tilde{\nu}^{(+)}(s)ds >0,\;\;\tilde v\in [\psi^{(-)}(x_0),\psi^{(+)}(x_0)).

\end{equation*}

\end{proposition}

\section*{Asymptotic approximation}\label{sec2}
Further to prove the existence and stability theorems we will use the method of differential inequalities\hyperref[csl:3]{(Nefedov, 1995}; \hyperref[csl:4]{Butuzov et al., 2012)}. The method is valid for problems with internal transition layers and it is based on the method of upper and lower solutions\hyperref[csl:5]{(Pao, 1992)}. It implies construction of the upper and lower solutions as modifications of it's asymptotic approximations.

The asymptotic approximation of problem\ref{ell_syst} here is quite similar to that constructer in paper\hyperref[csl:4]{(Butuzov et al., 2012)}, where a similar system with continuous right-hand sides was considered.
We define an  asymptotic approximation of \eqref{ell_syst} as

\begin{equation}
\label{LeftRightUV} 
U_1(x,\varepsilon) =

\begin{cases}
 U^{(-)}_1(x,\varepsilon), \,\, & 0 \leq x \leq x_{0} \,,\\
 U^{(+)}_1(x,\varepsilon), \,\, & x_{0} \leq x\leq 1 \,,

\end{cases}
\quad
V_1(x,\varepsilon) =

\begin{cases}
 V^{(-)}_1(x,\varepsilon), \,\, & 0 \leq x \leq x_{0} \,,\\
 V^{(+)}_1(x,\varepsilon), \,\, & x_{0} \leq x\leq 1 \,.

\end{cases}

\end{equation}

The functions $U^{(\mp)}$ and $V^{(\mp)}$ are the sums of the following terms:

\begin{equation}
\label{asimpt_expl} 
U^{(\mp)}_1=\bar{u}^{(\mp)}(x,\varepsilon)+Q^{(\mp)}u(\tau, \varepsilon)+M^{(\mp)}u(\sigma, \varepsilon)+
P^{(\mp)}_1u(\zeta^{(\mp)},\varepsilon),
\;\;
V^{(\mp)}_1=\bar{v}^{(\mp)}(x,\varepsilon)+Q^{(\mp)}v(\tau, \varepsilon)+
P^{(\mp)}_1v(\zeta^{(\mp)}),

\end{equation}

\begin{itemize}
\item $\bar{u}^{(\mp)}(x,\varepsilon)=\bar{u}^{(\mp)}_{0}(x)+\varepsilon\bar{u}^{(\mp)}_{1}(x),$ $\bar{v}^{(\mp)}(x,\varepsilon)=\bar{v}^{(\mp)}_{0}(x)+\varepsilon\bar{v}^{(\mp)}_{1}(x)$ are the regular part. These functions define the solution behavior far from borders $x=0,$ $x=1,$ $x=x_0.$
\item $Q^{(\mp)}u(\tau, \varepsilon)=Q^{(\mp)}_{0}u(\tau)+\varepsilon Q^{(\mp)}_{1}u(\tau),$ $Q^{(\mp)}v(\tau, \varepsilon)=Q^{(\mp)}_{0}v(\tau)+\varepsilon Q^{(\mp)}_{1}v(\tau),$ $M^{(\mp)}u(\sigma, \varepsilon)=M^{(\mp)}_{0}u(\sigma)+\varepsilon M^{(\mp)}_{1}u(\sigma)$ are the functions describing the two-scaled transition layer,
    \item $P^{(\mp)}_1u(\zeta^{(\mp)}),$ $P^{(\mp)}_1v(\zeta^{(\mp)})$ are the boundary layer functions, where $\zeta^{(-)} = x/\varepsilon , \zeta^{(+)} = (1-x)/\varepsilon$.

\end{itemize}

The demand the equality holds

\begin{equation}
\label{C0} 
U_{1}^{(-)}(x_0,\varepsilon)=U_{1}^{(+)}(x_0,\varepsilon)=p^*;\quad
V_{1}^{(-)}(x_0,\varepsilon)=V_{1}^{(+)}(x_0,\varepsilon)=q^{*}.

\end{equation}
that provides functions $U_1$ and $V_1$ continuity.

The systems of equations for regular part functions are obtained by aggregating the coefficients with the same $\varepsilon$ exponents in Taylor expansion of equalities

\begin{equation*}
f^{(\mp)}(\bar{u}^{(\mp)}(x,\varepsilon),\bar{v}^{(\mp)}(x,\varepsilon),x,\varepsilon)-\varepsilon^{4}\dfrac{d\bar{u}^{(\mp)}}{dx}(x,\varepsilon)=0,\;\;
g^{(\mp)}(\bar{u}^{(\mp)}(x,\varepsilon),\bar{v}^{(\mp)}(x,\varepsilon),x,\varepsilon)-\varepsilon^{2}\dfrac{d\bar{v}^{(\mp)}}{dx}(x,\varepsilon)=0.

\end{equation*}

Particularly for the 0-th order we have
$
\bar{v}^{(\mp)}_0(x)= \psi^{(\mp)}(x),  \;\; \bar{u}^{(\mp)}_0(x)=\varphi^{(\mp)}( \psi^{(\mp)}(x),x).
$

We obtain the equations for the transitional layer functions by aggregating the coefficients for the same
exponents of $\varepsilon$  in Taylor expansions of equalities:

\begin{equation}
\label{transQ} 
\varepsilon^4 \frac{d^2Q^{(\mp)}u}{d\tau^2}=Q^{(\mp)}f, \quad \varepsilon^2 \frac{d^2Q^{(\mp)}v}{d\tau^2}=Q^{(\mp)}g, \\

\end{equation}
where

\begin{eqnarray}
\label{Qf} 
&Q^{(\mp)}f(\tau,\varepsilon):= f^{(\mp)}(\bar{u}^{(\mp)}(x_0+\varepsilon \tau,\varepsilon)+Q^{(\mp)}u(\tau,\varepsilon),\bar{v}^{(\mp)}(x_0+\varepsilon \tau,\varepsilon)+Q^{(\mp)}v(\tau,\varepsilon),x_0+\varepsilon \tau,\varepsilon)-\\
&-f^{(\mp)}(\bar{u}^{(\mp)}(x_0+\varepsilon \tau,\varepsilon),\bar{v}^{(\mp)}(x_0+\varepsilon \tau,\varepsilon),x_0+\varepsilon \tau,\varepsilon)\notag

\end{eqnarray}
and $Q^{(\mp)}g(\tau,\varepsilon)$ have similar meaning;

\begin{equation}
\label{transM} 
\varepsilon^4 \frac{d^2M^{(\mp)}u}{d\sigma^2}=M^{(\mp)}f,

\end{equation}
where

\begin{multline}
\label{Mf} 
M^{(\mp)}f^{(\mp)}(\sigma,\varepsilon):= \\
:=f^{(\mp)}(\bar{u}^{(\mp)}(x_0+\varepsilon^2 \sigma,\varepsilon)+Q^{(\mp)}u(\varepsilon\sigma,\varepsilon)+M^{(\mp)}u(\sigma,\varepsilon),\bar{v}^{(\mp)}(x_0+\varepsilon^2 \sigma,\varepsilon)+Q^{(\mp)}v(\varepsilon\sigma,\varepsilon),x_0+\varepsilon^2 \sigma,\varepsilon)-\\
-f^{(\mp)}(\bar{u}^{(\mp)}(x_0+\varepsilon^2 \sigma,\varepsilon)+Q^{(\mp)}u(\varepsilon\sigma,\varepsilon),\bar{v}^{(\mp)}(x_0+\varepsilon^2 \sigma,\varepsilon)+Q^{(\mp)}v(\varepsilon\sigma,\varepsilon),x_0+\varepsilon^2 \sigma,\varepsilon),

\end{multline}
and $M^{(\mp)}g(\sigma,\varepsilon)$ have similar meaning.
Additionally we demand $Q_{i}^{(\mp)}u(\tau) \rightarrow 0, Q_{i}^{(\mp)}v(\tau)\rightarrow 0$ when $\tau \rightarrow \mp \infty$,  $M_{i}^{(\mp)}u(\sigma) \rightarrow 0$ when $\sigma \rightarrow \mp \infty$ for $i=0,1.$

\subsection*{0-th order transition layer functions}

Denote

\begin{equation}
\label{tilde-Phi-notation} 
\tilde{u}^{(\mp)}(\tau):= \varphi^{(\mp)}(\psi^{(\mp)}(x_0),x_0)+Q^{(\mp)}_{0}u(\tau),\;\;
\tilde{v}^{(\mp)}(\tau):= \psi^{(\mp)}(x_0)+Q^{(\mp)}_{0}u(\tau),\;\;\Phi^{(\mp)}(\tau)=\dfrac{d\tilde v}{d\tau}.

\end{equation}
From equalities\ref{transQ} in 0-th order we obtain equation.
$f^{(\mp)}\left(\tilde{u}^{(\mp)}(\tau),\tilde{v}^{(\mp)}(\tau),x_{0},0\right)=0$
from which it comes
$
\tilde{u}^{(\mp)}(\tau)=\varphi^{(\mp)}(\tilde{v}^{(\mp)}(\tau),x_0).
$
Using this from the second equation\ref{transQ} in 0-th order and the joining condition\ref{C0} we obtain problems to determine functions $\tilde{u}^{(\mp)}(\tau)$

\begin{equation}
\label{tildeV} 
\dfrac{d^{2}\tilde{v}^{(\mp)}(\tau)}{d\tau^2}=h^{(\mp)}(\tilde{v}^{(\mp)}(\tau),x_{0}),\;
\tilde{v}^{(\mp)}(0)=q^*,\;
\tilde{v}^{(\mp)}(\mp\infty)=\psi^{(\mp)}(x_0).

\end{equation}
These equations are similar to associated equations\ref{PrisEqV} which have solutions with exponential estimates\hyperref[csl:6]{(Fife \& McLeod, 1977)}

\begin{equation*}
\left|\tilde{v}^{(\mp)}(\tau) - \psi^{(\mp)}(x_0) \right| \leq \tilde{C_0}e^{-\kappa_0|\tau|},\;\;
\left|\tilde{u}^{(\mp)}(\tau) - \varphi^{(\mp)}\left(\psi^{(\mp)}(x_0),x_0\right) \right| \leq \tilde{C_0}e^{-\kappa_0|\tau|}.

\end{equation*}
Analogously we denote functions

\begin{equation}
\label{hat-Psi-notation} 
\hat{u}^{(\mp)}(\sigma) := \varphi^{(\mp)}(q^*,x_{0})+M^{(\mp)}_{0}u(\sigma),\;\;\Psi(\sigma,q^*)=\dfrac{d\hat u}{d\sigma}.

\end{equation}

To determine functions $\hat{u}^{(\mp)}(\sigma)$ we obtain problems

\begin{equation}
\label{tildeU} 
\dfrac{d^{2}\hat{u}^{(\mp)}(\sigma)}{d\sigma^{2}}=f^{(\mp)}\left(\hat{u}^{(\mp)}(\sigma),q^*,x_{0},0\right),\;
\hat{u}^{(\mp)}(0)=p^*,\;
\hat{u}^{(\mp)}(\mp\infty)=\varphi^{(\mp)}(q^*,x_{0}).

\end{equation}
These equations are similar to associated equations\ref{PrisEqU} which have exponentially bounded solutions \hyperref[csl:6]{(Fife \& McLeod, 1977)}:

\begin{equation*}
\left|\hat{u}^{(\mp)}(\sigma) - \varphi^{(\mp)}\left(q^*,x_0\right) \right| \leq \hat{C_0}e^{-K_0|\sigma|}.

\end{equation*}

\subsection*{1-th order transition layer functions}
Denote

\begin{eqnarray*}
&\tilde{f}^{(\mp)}(\tau) := f^{(\mp)}( \varphi^{(\mp)}(\tilde{v}^{(\mp)}(\tau),x_0), \tilde{v}^{(\mp)}(\tau),x_0,0),\;\;
\tilde{g}^{(\mp)}(\tau) := g^{(\mp)}( \varphi^{(\mp)}(\tilde{v}^{(\mp)}(\tau),x_0), \tilde{v}^{(\mp)}(\tau),x_0,0),
\\
&\hat{f}^{(\mp)}(\sigma):= f^{(\mp)}\left(\hat{u}^{(\mp)}(\sigma),q_0,x_{0},0\right),\;\;
\hat{g}^{(\mp)}(\sigma):= g^{(\mp)}\left(\hat{u}^{(\mp)}(\sigma),q_0,x_{0},0\right),
\\
&\tilde{\varphi}^{(\mp)}(\tau) := \varphi^{(\mp)}(\tilde{v}^{(\mp)}(\tau),x_0),\;\;
\tilde{h}^{(\mp)}(\tau) := h^{(\mp)}(\tilde{v}^{(\mp)}(\tau),x_0),

\end{eqnarray*}
and the same meaning have the derivatives.

For the functions $Q^{(\mp)}_{1}u(\tau)$ and $Q^{(\mp)}_{1}v(\tau)$ we obtain the following systems of equations from\ref{transQ} with boundary conditions from\ref{C0}:

\begin{eqnarray}
\label{Q1U_Q1V_rel} 
&0=\tilde{f}_{u}^{(\mp)}(\tau)Q^{(\mp)}_{1}u(\tau)+\tilde{f}_{v}^{(\mp)}(\tau)Q^{(\mp)}_{1}v(\tau)+ Q^{(\mp)}_1\tilde{f}(\tau),\notag\\
&\dfrac{d^{2}Q^{(\mp)}_{1}v}{d\tau^{2}}=\tilde{h}_{v}^{(\mp)}(\tau)Q_{1}^{(\mp)}v(\tau)+Q^{(\mp)}_1\tilde{g}(\tau),\;
Q^{(\mp)}_{1}v(0)=-\bar{v}_{1}^{(\mp)}(x_{0}),\;
Q^{(\mp)}_{1}v(\mp \infty)=0.

\end{eqnarray}

For the functions $M^{(\mp)}_{1}u(\sigma)$ we obtain the following problems from\ref{transM} with boundary conditions from\ref{C0}:

\begin{equation}
\label{M1U} 
\frac{d^{2}M^{(\mp)}_{1}u}{d\sigma^{2}}= \hat{f}^{(\mp)}_{u}(\sigma)M_{1}^{(\mp)}u(\sigma)+ M^{(\mp)}_1\hat{f}(\sigma),\;\;
M_{1}^{(\mp)}u(0)=-\bar{u}_{1}^{(\mp)}(x_{0})-Q^{(\mp)}_{1}u(0),\;\;
M_{1}^{(\mp)}u(\mp\infty)=0,

\end{equation}

The functions $ Q^{(\mp)}_1\tilde{f}(\tau),$ $Q^{(\mp)}_1\tilde{g}(\tau)$ and $M^{(\mp)}_1\hat{f}(\sigma)$ in\ref{Q1U_Q1V_rel} and \ref{M1U} are known and they exponentially decrease to zero as $\tau\to\mp\infty$, $\sigma\to\mp\infty$ respectively. The problems\ref{Q1U_Q1V_rel} and \ref{M1U} are linear and thus solvable and the following exponential estimates are  valid:

\begin{equation*}
\left| Q_1^{(\mp)}u(\tau) \right| \leq \tilde{C_1}e^{-\kappa_1|\tau|},\;\;
\left| Q_1^{(\mp)}v(\tau) \right| \leq \tilde{C_1}e^{-\kappa_1|\tau|},\;\;
\left| M_1^{(\mp)}u(\sigma) \right| \leq \hat{C_1}e^{-K_1|\sigma|}.

\end{equation*}

\subsection*{The higher order functions}
As it is mentioned in\hyperref[csl:4]{(Butuzov et al., 2012)} to construct the upper and lower solutions we have to define the boundary layer functions  $P^{(\mp)}_{i}u(\zeta^{(\mp)})$ and $P^{(\mp)}_{i}v(\zeta^{(\mp)})$ for $i=1,2$ $R^{(\mp)}_{3}u(\eta^{(\mp)}$, where $\eta^{(-)}=x/\varepsilon^2,$ $\eta^{(+)}=(x-1)/\varepsilon^2,$ and also the transition layer functions $M^{(\mp)}_iv(\sigma)$ for $i=2,3.$

The boundary layer functions are standardly defined as in\hyperref[csl:7]{(Butuzov \& Nedelko, 2000)}.

The problems for functions $M^{(\mp)}_iv(\sigma)$ for $i=2,3.$ can be determined analogously to problems for functions $M^{(\mp)}_iu(\sigma)$ from equalities $\varepsilon^2 \frac{d^2M^{(\mp)}v}{d\sigma^2}=M^{(\mp)}g,$ where $M^{(\mp)}g$ has the similar sense as\ref{Mf}.

\subsection*{The derivatives joining condition}
Let's assume the following conditions for derivatives

\begin{equation*}
\left. \left( \frac{dM^{(-)}_{0}u}{d\sigma} - \frac{dM^{(+)}_{0}u}{d\sigma} \right)\right|_{\sigma=0}
+\varepsilon \Bigg{[}
\left. \left(  \frac{dQ^{(-)}_{0}u}{d\tau} - \frac{dQ^{(+)}_{0}u}{d\tau}  \right)\right|_{\tau=0}+
\left. \left(  \frac{dM^{(-)}_{1}u}{d\sigma} - \frac{dM^{(+)}_{1}u}{d\sigma}  \right)\right|_{\sigma=0} \Bigg{]}
+ O(\varepsilon^{2})= 0

\end{equation*}

\begin{multline}
\label{C1} 
\left. \left( \frac{dQ^{(-)}_{0}v}{d\tau} - \frac{dQ^{(+)}_{0}v}{d\tau} \right)\right|_{\tau=0}
+\varepsilon \Bigg{[}
\left. \left(  \frac{dQ^{(-)}_{1}v}{d\tau} - \frac{dQ^{(+)}_{1}v}{d\tau}  \right)\right|_{\tau=0} +\\
+\left. \left(  \frac{dM^{(-)}_{2}v}{d\sigma} - \frac{dM^{(+)}_{2}v}{d\sigma}  \right)\right|_{\sigma=0}+
\left. \left(  \frac{d\psi^{(-)}}{dx} - \frac{d\psi^{(+)}}{dx}  \right)\right|_{x=x_{0}}
\Bigg{]}
+ O(\varepsilon^{2})=0.

\end{multline}
We also assume the following representation for values $q^*$ and $p^*$ that are parameters of $Q-$ and $M-$ functions (see settlements \ref{tildeV} and \ref{tildeU}: $q^*=q_0+\varepsilon q_1$ and $p^*=p_0+\varepsilon p_1.$

Zeroth order in $\varepsilon$ exponents of\ref{C1} yields
$H^{v}(q_0)=0$, $H^{u}(p_0,q_0)=0$, (see notations\ref{H-Func},\ref{tilde-Phi-notation} and\ref{hat-Psi-notation}). The values $q_0$ and $p_0$ exist due to Proposition\ref{H4}.
Denote
$\Phi(0) := \Phi^{(-)}(q_0) = \Phi^{(+)}(q_0),\;
\Psi(0) := \Psi^{(-)}(p_0,q_0) = \Psi^{(+)}(p_0,q_0)$.
First order, in\ref{C1}, yields

\begin{equation*}
\frac{1}{\Phi(0)}\frac{dH^{v}}{dv}(q_0)\cdot q_1= H_1^v(p_0,q_0), \;\;  \frac{1}{\Psi(0)}\frac{\partial H^{u}}{\partial u}(p_0,q_0)\cdot p_1  = H_1^u(p_0,q_0),

\end{equation*}
where $H_1^v(p_0,q_0)$, $H_1^u(p_0,q_0)$ are known functions.

\section*{UPPER AND LOWER SOLUTIONS}\label{sec3}
Denote

\begin{equation*}
L_{u,\varepsilon}(u,v):= \varepsilon^4\frac{d^{2}u}{dx^2}-f(u,v,x,\varepsilon),\;\;
L_{v,\varepsilon}(u,v):= \varepsilon^2\frac{d^{2}v}{dx^2}-g(u,v,x,\varepsilon).

\end{equation*}

\begin{definition}
Pairs of functions $\left(\bar{U}, \bar{V}\right)$ and $\left(\tilde{U}, \tilde{V}\right)$  in  $C([0,1])\cap C^2((0,1)\backslash x_0)$ are called respectively upper and lower
solutions of the problem \eqref{ell_syst} if

\begin{itemize}
\item[(A$_1$).] $\tilde{U}(x) \leq \bar{U}(x)$,\; $\tilde{V}(x) \leq \bar{V}(x)$,\; $ x\in[0,1] $;
\item[(A$_2$).]
$L_{1,\varepsilon}(\bar{U},v)\leq  0 \leq L_{1,\varepsilon}(\tilde{U},v)  \;\; \tilde{V}\leq v \leq \bar{V},\; x \in (0,1) \backslash x_0, \;\;
L_{2,\varepsilon}(u,\bar{V})\leq  0 \leq L_{1,\varepsilon}(v,\tilde{V})  \;\; \tilde{U}\leq u \leq \bar{U},\; x \in (0,1) \backslash x_0$;
\item[(A$_3$).]
$\bar{U}_x(0)\leq 0 \leq \tilde{U}_x(0),\; \bar{U}_x(1)\geq 0 \geq \tilde{U}_x(1),\;
\bar{V}_x(0)\leq 0 \leq \tilde{V}_x(0),\; \bar{V}_x(1)\geq 0 \geq \tilde{V}_x(1)$;
\item[(A$_4$).]
$\left.\left(\bar{U}^{(-)}_x-\bar{U}^{(+)}_x\right)\right|_{x=x_0}\geq 0,\;\left.\left(\tilde{U}^{(-)}_x-\tilde{U}^{(+)}_x\right)\right|_{x=x_0}\leq0,
\left.\left(\bar{V}^{(-)}_x-\bar{V}^{(+)}_x\right)\right|_{x=x_0}\geq 0,\;\left.\left(\tilde{V}^{(-)}_x-\tilde{V}^{(+)}_x\right)\right|_{x=x_0}\leq0$.

\end{itemize}

\end{definition}
In case of Proposition\ref{H3} the inequality (A$_2$) will hold if

\begin{eqnarray}
\label{operFPQM} 
L_{u,\varepsilon}(\bar{U},\tilde{V})<0<L_{u,\varepsilon}(\tilde{U},\bar{V}),\quad
L_{v,\varepsilon}(\bar{U},\bar{V})  <0<L_{v,\varepsilon}(\tilde{U},\tilde{V}),\quad
x\in (0,1)\backslash x_0

\end{eqnarray}
In this article the upper and lower solutions are of the analogous structure to \eqref{LeftRightUV}-\eqref{asimpt_expl} with two separate parts ~--- left and right relative to $x_0$.
Functions $\bar{U}^{(\mp)}$, $\tilde{U}^{(\mp)}$, $\bar{V}^{(\mp)}$, $\tilde{V}^{(\mp)}$ are the modifications of asymptotic approximation of the solution in respective regions:

\begin{eqnarray*}
&\bar{U}^{(\mp)}= U_{1}^{(\mp)}+\varepsilon\left(\alpha^{(\mp)}(x)+q^{(\mp)}U(\tau)+m^{(\mp)}U(\sigma)\right) +\varepsilon\bar\Omega_u(x,\varepsilon),\\
&\tilde{U}^{(\mp)}= U_{1}^{(\mp)}-\varepsilon\left(\alpha^{(\mp)}(x)+q^{(\mp)}U(\tau)+m^{(\mp)}U(\sigma)\right) +\varepsilon\tilde{\Omega}_u(x,\varepsilon),\\
&\bar{V}^{(\mp)}= V_{1}^{(\mp)}+\varepsilon\left(\beta^{(\mp)}(x)+q^{(\mp)}V(\tau)\right)+
\varepsilon\bar\Omega_v(x,\varepsilon),\\
&\tilde{V}^{(\mp)}= V_{1}^{(\mp)}-\varepsilon\left(\beta^{(\mp)}(x)+q^{(\mp)}V(\tau)\right)+
\varepsilon\tilde\Omega_v(x,\varepsilon),

\end{eqnarray*}
where functions $\bar\Omega_u,$ $\tilde\Omega_u,$ $\bar\Omega_v,$ $\tilde\Omega_v$ are additions to provide the inequalities (A$_3$) (A$_2$) in the vicinities of boundary points $x=0$ and $x=1$ and are similar to\hyperref[csl:4]{(Butuzov et al., 2012)}.
The continuity condition in $x_0$ yields

\begin{equation*}
\bar{U}^{(-)}(x_0,\varepsilon)=\bar{U}^{(+)}(x_0,\varepsilon),\;\tilde{U}^{(-)}(x_0,\varepsilon)=\tilde{U}^{(+)}(x_0,\varepsilon),\;\;
\bar{V}^{(-)}(x_0,\varepsilon)=\bar{V}^{(+)}(x_0,\varepsilon),\;\tilde{V}^{(-)}(x_0,\varepsilon)=\tilde{V}^{(+)}(x_0,\varepsilon).

\end{equation*}
Assuming $A$, $B$ to be arbitrary positive constants we define the systems for $\alpha^{(\mp)}(x)$ and
$\beta^{(\mp)}(x)$ as

\begin{equation}
\label{PM_a_b_syst} 
\bar{f}_{u}^{(\mp)}(x)\alpha^{(\mp)}(x)- \bar{f}_{v}^{(\mp)}(x)\beta^{(\mp)}(x)=A,\quad
\bar{g}_{u}^{(\mp)}(x)\alpha^{(\mp)}(x)+ \bar{g}_{v}^{(\mp)}(x)\beta^{(\mp)}(x)=B,

\end{equation}
The systems are solvable due to Propositions\ref{H1}-\ref{H3} and\ref{H5} and the functions $\alpha^{(\mp)}(x)$ and
$\beta^{(\mp)}(x)$ have positive values for sufficient $A$ and $B$.

Moving forward, for sufficiently big values $d^{u}$ and $D^u$ and also sufficiently small values $k^{u}$ and $K^u$ we determine the following problems for functions $qU^{(\mp)}(\tau)$ and $qV^{(\mp)}(\tau)$.

\begin{equation}
\label{PMqUeq} 
\tilde{f}_{u}^{(\mp)}(\tau)qU^{(\mp)}(\tau)- \tilde{f}_{v}^{(\mp)}(x)qV^{(\mp)}(\tau)+
\left[\tilde{f}_{u}^{(\mp)}(\tau)- \bar{f}_{u}^{(\mp)}(x_0)\right]\alpha^{(\mp)}(x_0) -
\left[\tilde{f}_{v}^{(\mp)}(\tau)- \bar{f}_{v}^{(\mp)}(x_0)\right]\beta^{(\mp)}(x_0)=d^{u}e^{-k^{u}|\tau|},

\end{equation}

\begin{equation}
\label{PMqVsyst} 
\frac{d^{2}qV^{(\mp)}(\tau)}{d\tau^{2}}=\tilde{\nu}^{(\mp)}(\tau)\cdot q^{(\mp)}V(\tau)+\tilde{G}^{(\mp)}(\tau),\;\;
q^{(\mp)}V(0)=\delta_{v}-\beta^{(\mp)}(x_0),\;\;
q^{(\mp)}V(\mp\infty)=0.

\end{equation}
where functions $\tilde{\nu}^{(\mp)}(\tau)$ are defined in\ref{nu},

\begin{multline*}
\tilde{G}^{(\mp)}(\tau) := \left[\left(\tilde{g}_{u}^{(\mp)}(\tau)- \bar{g}_{u}^{(\mp)}(x_0)\right)-\frac{\tilde{g}_{u}^{(\mp)}(\tau)}{\tilde{f}_{u}^{(\mp)}(\tau)}\left( \tilde{f}_{u}^{(\mp)}(\tau)- \bar{f}_{u}^{(\mp)}(x_0)\right) \right]\alpha^{(\mp)}(x_0)+\\
+\left[\left(\tilde{g}_{v}^{(\mp)}(\tau)- \bar{g}_{v}^{(\mp)}(x_0)\right)+\frac{\tilde{g}_{u}^{(\mp)}(\tau)}{\tilde{f}_{u}^{(\mp)}(\tau)}\left( \tilde{f}_{v}^{(\mp)}(\tau)- \bar{f}_{v}^{(\mp)}(x_0)\right) \right]\beta^{(\mp)}(x_0)-
\frac{\tilde{g}_{u}^{(\mp)}(\tau)}{\tilde{f}_{u}^{(\mp)}(\tau)}d^{u}e^{-k^{u}|\tau|}-D^{v}e^{-K^{v}|\tau|}.

\end{multline*}

The following lemma is true:

\textbf{Lemma 1} .
Assume $\tilde{\nu}^{(\mp)}(\tilde{v}^{(\mp)}(\tau))$ satisfy Proposition\ref{H5}. Then on $\mathbb{R}_{\mp}$ for the equations
$W^{(\mp)}_{\tau \tau}(\tau)-\tilde{\nu}^{(\mp)}(\tilde{v}^{(\mp)}(\tau))W^{(\mp)}(\tau)=0$ there exist  positive fundamental solutions $W^{(\mp)}(\tau)$ such that
$W^{(\mp)}(\tau)\leq C_{\gamma} e^{-\gamma |\tau|}$ and $W^{(-)}_{\tau}(0,q_0)/W^{(-)}(0,q_0)>0$, $W^{(+)}_{\tau}(0,q_0)/W^{(+)}(0,q_0)<0$.

The proof of Lemma 1 is included in Appendix.

If we choose the value $D^v$ sufficiently large then the functions $\tilde{G}^{(\mp)}(\tau)$ are negative and
explicit solutions to problems\ref{PMqVsyst} are positive:

\begin{equation*}
q^{(\mp)}V(\tau) =\left(\delta_{v}-\beta^{(\mp)}(x_0)\right)\frac{W^{(\mp)}(\tau,q_0)}{W^{(\mp)}(0,q_0)} \,+
W^{(\mp)}(\tau,q_0)\int\limits_{0}^{\tau} \frac{d\tau_1}{\left[W^{(\mp)}(\tau_1,q_0)\right]^{2}}\int\limits_{\mp \infty}^{\tau_1} W^{(\mp)}(\tau_2,q_0)\tilde{G}^{(\mp)}(\tau_2)
d\tau_2.

\end{equation*}

The functions $m^{(\mp)}U(\sigma)$ we define as solutions to problems

\begin{equation*}
\frac{d^{2}m^{(\mp)}U}{d\sigma^{2}}=\hat{f}^{(\mp)}_{v}(\sigma)\cdot m^{(\mp)}U(\sigma)+\hat{F}^{(\mp)}(\sigma),\;\;
m^{(\mp)}U(0)=\delta_{u}-\alpha^{(\mp)}(x_0)-q^{(\mp)}U(0),\;\;
m^{(\mp)}U(\mp\infty)=0.

\end{equation*}
where

\begin{equation*}
\hat{F}^{(\mp)}(\sigma) := \left[\hat{f}_{u}^{(\mp)}(\sigma)- \tilde{f}_{u}^{(\mp)}(0)\right]\left(\alpha^{(\mp)}(x_0)+q^{(\mp)}U(0)\right)+\left[\hat{f}_{v}^{(\mp)}(\sigma)- \tilde{f}_{v}^{(\mp)}(0)\right]\delta_{v}-D^{u}e^{-K^{u}|\sigma|}.

\end{equation*}
For sufficiently large coefficient $D^{u}$ these problems have positive solutions

\begin{multline*}
mU^{(\mp)}(\sigma) = \left(\delta_{u}-\alpha^{(\mp)}(x_0)-q^{(\mp)}U(0)\right)\frac{\Psi^{(\mp)}\left(\hat{u}^{(\mp)}(\sigma),q_0\right)}{\Psi^{(\mp)}(p_0,q_0)} +
\\
+\Psi^{(\mp)}\left(\hat{u}^{(\mp)}(\sigma),q_0\right)\!\!\int\limits_{0}^{\sigma} \!\!\frac{d\sigma_1}{\left[\Psi^{(\mp)}\left(\hat{u}^{(\mp)}(\sigma_1),q_0\right)\right]^{2}}\int\limits_{\mp \infty}^{\sigma_1}\!\!\Psi^{(\mp)}\left(\hat{u}^{(\mp)}(\sigma_2),q_0\right)\underbrace{\hat{F}^{(\mp)}(\sigma_2)}_{\text{$<\!0$}}d\sigma_2.

\end{multline*}

Due to the choice of functions $\alpha^{(\mp)}(x),\;\beta^{(\mp)}(x),\;q^{(\mp)}U(\tau),\;q^{(\mp)}V(\tau),$ and $m^{(\mp)}U(\sigma)$ the inequalities
(A$_1$) and (A$_2$) are satisfied.

To satisfy conditions in $x_0$ we consider for the upper solutions the following equations (in case of lower solutions right-hand sides have negative sign):

\begin{eqnarray}
\label{derdisc1} 
\left.\left(\dfrac{d\hat{U}^{(-)}}{dx}-\dfrac{d\hat{U}^{(+)}}{dx}\right)\right|_{x=x_0} &= \varepsilon^2\left.\left(\dfrac{dm^{(-)}U}{d\sigma}-\dfrac{dm^{(+)}U}{d\sigma}\right)\right|_{\sigma=0}+O(\varepsilon^3),\\
\left.\left(\frac{d\hat{V}^{(-)}}{dx}-\dfrac{d\hat{V}^{(+)}}{dx}\right)\right|_{x=x_0} &= \varepsilon\left.\left(\dfrac{dq^{(-)}V}{d\tau}-\dfrac{dq^{(+)}V^{(+)}}{d\tau}\right)\right|_{\xi=0}+O(\varepsilon^2).\notag

\end{eqnarray}

As

\begin{multline*}
\left.\left(\frac{dq^{(-)}V}{d\sigma}-\frac{dq^{(+)}V}{d\tau}\right)\right|_{\tau=0}=\left(
\frac{W^{(-)}_{\tau}(0,q_0)}{W^{(-)}(0,q_0)} -\frac{W^{(+)}_{\tau}(0,q_0)}{W^{(+)}(0,q_0)}\right)\cdot\delta_{v} -\frac{W^{(-)}_{\tau}(0,q_0)}{W^{(-)}(0,q_0)}\cdot\beta^{(-)}(x_0) +\frac{W^{(+)}_{\tau}(0,q_0)}{W^{(+)}(0,q_0)}\cdot\beta^{(+)}(x_0)+
\\ + \frac{1}{W^{(-)}(0,q_0)}
\!\; \int\limits_{-\infty}^{0}\!\! W^{(-)}(\tau,q_0)\tilde{G}^{(-)}(\tau)
d\tau+
\frac{1}{W^{(+)}(0,q_0)}\!\int\limits_{0}^{+\infty}\!\!W^{(+)}(\tau,q_0)\tilde{G}^{(+)}(\tau)d\tau .

\end{multline*}
and

\begin{multline*}
\left.\left(\frac{dm^{(-)}U}{d\sigma}-\frac{dm^{(+)}U}{d\sigma}\right)\right|_{\sigma=0} = \\ =
\frac{1}{\Psi(0)}
\left(
\frac{\partial H^{u}}{\partial u}(q_0,p_0)\cdot\delta_u -
\left[\alpha^{(-)}(x_0)+q^{(-)}U(0)\right]\cdot f^{(-)}\left(q_0,p_0,x_0,0\right)+
\left[\alpha^{(+)}(x_0)+q^{(+)}U(0)\right]\cdot f^{(+)}\left(q_0,p_0,x_0,0\right)
\right) +\\
+\frac{1}{\Psi(0)}
\left(\;
\int\limits_{-\infty}^{0}\!\! \Psi^{(-)}\!\left(\hat{u}^{(-)}(\sigma),q_0\right)\!\hat{F}^{(-)}(\sigma)d\sigma -
\int\limits_{+\infty}^{0}\!\! \Psi^{(+)}\!\left(\hat{u}^{(+)}(\sigma),q_0\right)\!\hat{F}^{(+)}(\sigma)d\sigma
\right).

\end{multline*}
The right-hand side of\ref{derdisc1} can be made positive by choosing sufficiently big $\delta_v$, $\delta_u$ and  sufficiently small $\varepsilon$ due to Lemma 1 and Proposition\ref{H4}.

\section*{The existence of stationary solution}\label{sec4}

\begin{theorem}
\label{theorem_1} 
Suppose Propositions\ref{H1}-\ref{H5} hold. Then for sufficiently small $\varepsilon > 0$ there exists a solution $(u_\varepsilon(x),u_\varepsilon(x))$ of the
problem \eqref{ell_syst}, for which the pair of functions $\left( U(x,\varepsilon),V(x,\varepsilon) \right) $ is a uniform asymptotic approximation with the accuracy of $O(\varepsilon^2)$, that is,
for all $x\in[0,1]$, the inequality holds

\begin{equation*}
|U(x,\varepsilon) - u_\varepsilon(x)| + |V(x,\varepsilon) - v_\varepsilon(x)| \leq C\varepsilon^2, \; x \in [0,1],

\end{equation*}
where C is positive constant independent on $\varepsilon$.

\end{theorem}

Proof of the theorem is  based on the  proof of Pao \hyperref[csl:5]{(Pao, 1992)} with slight modifications concerning presence of simple discontinuity in $x_0$. We define the iterative process as

\begin{equation}
\label{ProblemPao} 
-\varepsilon^{4}\frac{d^{2}u^{(k)}}{dx^2}+cu^{(k)}=\mathcal{F}_1^{(k-1)}(x),\;\;
-\varepsilon^{2}\frac{d^{2}v^{(k)}}{dx^2}+cv^{(k)}=\mathcal{F}_2^{(k-1)}(x),\;\;
\left.\frac{du^{(k)}}{dx} \right|_{x=0} \!=\!\left. \frac{du^{(k)}}{dx} \right|_{x=1}\!=\!
\left.\frac{dv^{(k)}}{dx} \right|_{x=0} \!=\!\left. \frac{dv^{(k)}}{dx} \right|_{x=1}\!=0,

\end{equation}
where $c>0$ is a sufficiently big constant and

\begin{equation*}
\mathcal{F}_1^{(k-1)}(u^{(k-1)},v^{(k-1)},x):=-f(u^{(k-1)},v^{(k-1)},x,\varepsilon)+cu^{(k-1)},\;\;
\mathcal{F}_2^{(k-1)}(u^{(k-1)},v^{(k-1)},x):=-g(u^{(k-1)},v^{(k-1)},x,\varepsilon)+cv^{(k-1)}.

\end{equation*}
The solutions to \eqref{ProblemPao} can be expressed explicitly  as

\begin{equation}
\label{ell_sol_pre_expl} 
\hat{u}^{(k)}(x) = \int\limits_{0}^{1}\!\!G_1(x,s)\mathcal{F}_1^{(k-1)}(u^{(k-1)},v^{(k-1)},s)ds, \;\;
\hat{v}^{(k)}(x) = \int\limits_{0}^{1}\!\!G_2(x,s)\mathcal{F}_2^{(k-1)}(u^{(k-1)},v^{(k-1)},s)ds

\end{equation}
and are of $C^{1}[0,1]\cap C^{2}((0,1)\backslash x_0)$ for $u^{(0)},v^{(0)} \in C([0,1])$ class \hyperref[csl:8]{(Stakgold \& Holst, 2011)}.

Following Pao \hyperref[csl:5]{(Pao, 1992)} we consider monotone sequences

\begin{equation*}
\tilde{U} \leq \tilde{u}^{(k-1)} \leq \tilde{u}^{(k)} \leq \bar{u}^{(k)} \leq \bar{u}^{(k-1)} \leq \bar{U}, \quad
\tilde{V} \leq \tilde{v}^{(k-1)} \leq \tilde{v}^{(k)} \leq \bar{v}^{(k)} \leq \bar{v}^{(k-1)} \leq \bar{V}.

\end{equation*}
It is worth noting that in \hyperref[csl:5]{(Pao, 1992)} monotonicity is proven for $C^2$ functions on the basis of maximum principle, in our case we use

\noindent
\textbf{Lemma 2} .
Assume $w(x) \in C^{1}[0,1]\cap C^{2}((0,1)\backslash x_0)$ for some continuous $c(x)>0$ satisfies

\begin{equation}
\label{lemm} 
-w''(x)+c(x)w(x) \geq 0, \; x \in (0,1)\backslash x_0,\;\;
w'(x_0-0)\geq w'(x_0+0),\;\;
w'(0)\leq 0 \leq w'(1),

\end{equation}
then  $w(x) \geq 0,\; x\in [0,1]$.

From now on we consider \eqref{ProblemPao} for the supersequence (for the subsequence reasoning is analogous). In \eqref{ell_sol_pre_expl} there exist limits of the left-hand sides therefore  there exist limits of the left-hand sides.

Utilizing Levi's Theorem from explicit form  \eqref{ell_sol_pre_expl}

\begin{eqnarray}
&\bar{u}_{\varepsilon}(x) = \int\limits_{0}^{1}\!\!G_1(x,s)\bar{\mathcal{F}}_1(s)ds, \quad
\bar{v}_{\varepsilon}(x) = \int\limits_{0}^{1}\!\!G_2(x,s)\bar{\mathcal{F}}_2(s)ds, \;\;\label{ell_sol_expl} \\
&\bar{\mathcal{F}}_1(s)=\lim_{k\rightarrow\infty}\mathcal{F}_1^{(k)}(u^{(k)},v^{(k)},s)\;
\left(=
\mathcal{F}_1^{(k)}(\bar{u}_\varepsilon,\bar{v}_\varepsilon,s)
\right),\quad
\bar{\mathcal{F}}_2(s)=\lim_{k\rightarrow\infty}\mathcal{F}_2^{(k)}(u^{(k)},v^{(k)},s)\;
\left(=
\mathcal{F}_2^{(k)}(\bar{u}_\varepsilon,\bar{v}_\varepsilon,s)
\right)\notag

\end{eqnarray}
we conclude continuity of the limits which coupled with expressions in parentheses allows then to conclude from \eqref{ell_sol_pre_expl} that  $\bar{u}_{\varepsilon}(x),\, \bar{v}_{\varepsilon}(x) \in C^{1}[0,1]\cap C^{2}((0,1)\backslash x_0)$ and are indeed solutions of the stationary problem \eqref{ell_syst} in the sense of Definition 1.

\section*{The locally uniqueness and stability of the stationary solution}\label{sec4}

\begin{theorem}
\label{theorem_2} 
Suppose Proposition\ref{H1}-\ref{H5} hold. Then for sufficiently small $\varepsilon > 0$ there exists locally unique and asymptotically stable in the sense of Lyapunov solution $(u_\varepsilon(x), v_\varepsilon(x))$ to the problem \eqref{par_syst} having the internal transition layer in the vicinity of the point $x_0$  with the domain of attraction not less than
$[\tilde{U}(x,\varepsilon),\bar{U}(x,\varepsilon)]\times[\tilde{V}(x,\varepsilon),\bar{V}(x,\varepsilon)]$.

\end{theorem}
Proof of this theorem is based on sub-supersolutions method.
Introduce in $\overline{D_T}$ functions:

\begin{eqnarray}
\label{ParabolicSupSub1} 
&\tilde{U}_T(x,t,\varepsilon)=
u_{\varepsilon}(x) + \left(\tilde{U}-u_{\varepsilon}(x) \right)e^{-\varepsilon \lambda  t},
\tilde{V}_T(x,t,\varepsilon)=
u_{\varepsilon}(x) + \left(\tilde{V}-v_{\varepsilon}(x) \right)e^{-\varepsilon \lambda  t},\\
&\bar{U}_T(x,t,\varepsilon)=
u_{\varepsilon}(x) + \left(\bar{U}-u_{\varepsilon}(x) \right)e^{-\varepsilon \lambda  t},
\bar{V}_T(x,t,\varepsilon)=
u_{\varepsilon}(x) + \left(\bar{V}-v_{\varepsilon}(x) \right)e^{-\varepsilon \lambda  t},

\end{eqnarray}
where $\left(u_{\varepsilon}(x),v_{\varepsilon}(x)\right)$ - any solution of \eqref{ell_syst}.
Also for the initial functions in \eqref{par_syst} we demand for all $x\in [0,1]$

\begin{equation*}
\tilde{U}_T(x,0,\varepsilon)=\tilde{U}(x,\varepsilon) \leq u^0(x) \leq  \bar{U}(x,\varepsilon) = \bar{U}_T(x,0,\varepsilon),\;\;
\tilde{V}_T(x,0,\varepsilon)=\tilde{V}(x,\varepsilon) \leq v^0(x) \leq  \bar{V}(x,\varepsilon) = \bar{V}_T(x,0,\varepsilon).

\end{equation*}
These functions \eqref{ParabolicSupSub1} are indeed the lower and the upper solutions as defined and proven in \hyperref[csl:9]{(Mel'nikova, 2019)} .

We define the iterative process for the initial boundary value problem as

\begin{eqnarray}
\label{ProblemPao_parabolic} 
&\dfrac{\partial y^{(k)}}{\partial t}-\varepsilon^{4}\dfrac{\partial^{2}y^{(k)}}{\partial x^2}+cy^{(k)}=
\mathcal{F}_1^{(k-1)}(y^{(k-1)},z^{(k-1)},x,t),
\;\; y^{(k)}_x(0,t)=y^{(k)}_x(1,t)=0,
\;\; y^{(k)}(x,0)=u^{0}(x),
\\
&\dfrac{\partial z^{(k)}}{\partial t}-\varepsilon^{2}\dfrac{\partial^{2}z^{(k)}}{\partial x^2}+cz^{(k)}=
\mathcal{F}_2^{(k-1)}(y^{(k-1)},z^{(k-1)},x,t),
\;\; z^{(k)}_x(0,t)=z^{(k)}_x(1,t)=0,
\;\; z^{(k)}(x,0)=v^{0}(x).\notag

\end{eqnarray}
where

\begin{equation*}
\mathcal{F}_1^{(k)}(y^{(k)},z^{(k)},x,t) := -f(y^{(k)},z^{(k)},x,\varepsilon)+cy^{(k)}, \;\;
\mathcal{F}_2^{(k)}(y^{(k)},z^{(k)},x,t) := -g(y^{(k)},z^{(k)},x,\varepsilon)+cz^{(k)}.

\end{equation*}

Following Pao $\hyperref[csl:5]{(Pao, 1992)}$ and using a proposition analogous to Lemma 1 for parabolic systems \hyperref[csl:10]{(Levashova et al., 2018)} we obtain monotone sequences

\begin{equation*}
\tilde{U}_T \leq \tilde{y}^{(k-1)} \leq \tilde{y}^{(k)} \leq \bar{y}^{(k)} \leq \bar{y}^{(k-1)}
\leq \bar{U}_T, \quad
\tilde{V}_T \leq \tilde{z}^{(k-1)} \leq \tilde{z}^{(k)} \leq \bar{z}^{(k)} \leq \bar{z}^{(k-1)}
\leq \bar{V}_T.

\end{equation*}

Denote

\begin{eqnarray*}
& \varkappa^{(k)}_{1}(t) :=
f^{(-)}(y^{(k)}(x_0,t),z^{(k)}(x_0,t),x_0,\varepsilon)-f^{(+)}(y^{(k)}(x_0,t),z^{(k)}(x_0,t),x_0,\varepsilon),
\\
& f^{(k)}_0(x,t) :=
f(y^{(k)}(x,t),z^{(k)}(x,t),x,\varepsilon) +  \Theta(x-x_0)\varkappa^{(k)}_{1,sub}(t),
\\
& \varkappa^{(k)}_{2}(t) :=
g^{(-)}(y^{(k)}(x_0,t),z^{(k)}(x_0,t),x_0,\varepsilon)-g^{(+)}(y^{(k)}(x_0,t),z^{(k)}(x_0,t),x_0,\varepsilon),
\\
& g^{(k)}_0(x,t) :=
g(y^{(k)}(x,t),z^{(k)}(x,t),x,\varepsilon) +  \Theta(x-x_0)\varkappa^{(k)}_{2}(t),

\end{eqnarray*}
therefore

\begin{eqnarray*}
f(y^{(k)}(x,t),z^{(k)}(x,t),x,\varepsilon) \equiv  f^{(k)}_0(x,t) + \Theta(x-x_0)\varkappa^{(k)}_{1}(t),
\;\;\;&
g(y^{(k)}(x,t),z^{(k)}(x,t),x,\varepsilon) \equiv  g^{(k)}_0(x,t) + \Theta(x-x_0)\varkappa^{(k)}_{2}(t),\\
\mathcal{F}_1^{(k-1)}(y^{(k-1)},z^{(k-1)},x,t) \equiv \mathcal{F}_{1,0}^{(k-1)}(x,t)+  \Theta(x-x_0)\varkappa^{(k)}_{1}(t),
\;\;\;&
\mathcal{F}_2^{(k-1)}(y^{(k-1)},z^{(k-1)},x,t) \equiv \mathcal{F}_{2,0}^{(k-1)}(x,t)+  \Theta(x-x_0)\varkappa^{(k)}_{2}(t).

\end{eqnarray*}
Here we introduce

\begin{equation*}
Y^{(k)}(x,t) :=
\int\limits_{0}^{1}\!\!G_1(x,s,t)u^{0}(s)ds +
\int\limits_{0}^{t}\!\!d\tau\!\!\int\limits_{0}^{1}\!\!G_1(x,s,t-\tau)\mathcal{F}_{1,0}^{(k-1)}(s,\tau)ds,\;\;
U^{(k)}(x,t) :=
\int\limits_{0}^{t}\!\!d\tau\!\!\int\limits_{x_0}^{1}\!\!G_1(x,s,t-\tau)\varkappa^{(k)}_{1}(\tau)ds

\end{equation*}
The function $Y^{(k)}(x,t)$ is obviously a classical solution to the problem

\begin{equation}
\label{par_part_lin_prob_y} 
\frac{\partial Y^{(k)}}{\partial t}-\varepsilon^{4}\frac{\partial^{2}Y^{(k)}}{\partial x^2}+c\mkern 1mu Y^{(k)}=
\mathcal{F}_{1,0}^{(k-1)}(x,t),
\;\; Y^{(k)}_x(0,t)=Y^{(k)}_x(1,t)=0,
\;\; Y^{(k)}(x,0)=u^{0}(x).

\end{equation}
Considering $U^{(k)}(x,t)$ as a function defined on $\overline{D^{a}_T}$,  utilizing the reasoning of Friedman \hyperref[csl:11]{(Friedman, 1983)} and the estimates \hyperref[csl:12]{(Sobolevsky, 1961)} it can be proven that   $U^{(k)}(x,t)\in C\left(\overline{D^{a}_T} \right)$ and

\begin{equation*}
\tilde{U}^{(k)}_x(x,t) :=
\int\limits_{0}^{t}\!\!d\tau\!\!\int\limits_{x_0}^{1}\!\!\frac{\partial}{\partial x}G_1(x,s,t-\tau)\varkappa^{(k)}_{1}(\tau)ds
\in C\left(\overline{D^{a}_T}\right),

\end{equation*}
moreover, $U^{(k)}_x(x,t) \equiv \tilde{U}^{(k)}_x(x,t)$ for $x\in D^{a}_T$.
Lagrange's Theorem provides that  $U^{(k)}_x(x_0,t) = \tilde{U}^{(k)}_x(x_0,t)$, $U^{(k)}_x(1,t) = \tilde{U}^{(k)}_x(1,t)$ for $t\in(0,T]$ and due to \hyperref[csl:12]{(Sobolevsky, 1961)} or \hyperref[csl:13]{(Ladyzenskaja et al., 1968)}
$U^{(k)}_x(x,t)\equiv \tilde{U}^{(k)}_x(x,t)\rightrightarrows{x\in [-a,1+a]}{t\rightarrow +0}0 \equiv  U^{(k)}_x(x,0) $,
hence $U^{(k)}_x(x,t) \in C\left(\overline{D_T}\right) \subset C\left(\overline{D^{a}_T}\right)$.
Furthermore, with regard to \hyperref[csl:11]{(Friedman, 1983}; \hyperref[csl:12]{Sobolevsky, 1961)}
the equation  $\tilde{U}^{(k)}_t - \varepsilon^4\tilde{U}^{(k)}_{xx}+ c\tilde{U}^{(k)}=\Theta(x-x_0)\varkappa^{(k)}_{1}(t),\; (x,t)\in D^{(-)}_{T}\cup D^{(+)}_{T}$ is being satisfied in the classical sense, therefore it concludes in $y^{(k)}(x,t)=Y^{(k)}(x,t)+U^{(k)}(x,t)$ being the solution to \eqref{par_part_lin_prob_y} in the sense of Definition~\ref{par_sol_def}  and having an explicit form

\begin{equation*}
y^{(k)}(x,t) =
\int\limits_{0}^{1}\!\!G_1(x,s,t)u^{0}(s)ds +
\int\limits_{0}^{t}\!\!d\tau\!\!\int\limits_{0}^{1}\!\!G_1(x,s,t-\tau)\mathcal{F}_{1}^{(k-1)}(y^{(k-1)},z^{(k-1)},s,\tau)ds,.

\end{equation*}
Same for $z^{(k)}(x,t)$.

Now we consider, for example, supersequence ~--- using the same steps as in the previous paragraph it can be proven firstly that
$\left(\bar{y}^{(k)}(x,t),\bar{z}^{(k)}(x,t)\right)\rightrightarrows{\overline{D_T}}{k\rightarrow \infty}
\left(\bar{y}_\varepsilon(x,t),\bar{z}_\varepsilon(x,t)\right)\in C\left(\overline{D_T} \right)$, then that  the pair

\begin{eqnarray*}
&\bar{y}_\varepsilon(x,t) =
\int\limits_{0}^{1}\!\!G_1(x,s,t)u^{0}(s)ds +
\int\limits_{0}^{t}\!\!d\tau\!\!\int\limits_{0}^{1}\!\!G_1(x,s,t-\tau)\bar{\mathcal{F}}_{1}(s,\tau)ds,\\
&\bar{z}_\varepsilon(x,t) =
\int\limits_{0}^{1}\!\!G_2(x,s,t)v^{0}(s)ds +
\int\limits_{0}^{t}\!\!d\tau\!\!\int\limits_{0}^{1}\!\!G_2(x,s,t-\tau)\bar{\mathcal{F}}_{2}(s,\tau)ds,\\
&\bar{\mathcal{F}}_{1}(s,\tau) = \lim_{k\rightarrow\infty}\mathcal{F}_1^{(k)}(y^{(k)},z^{(k)},s,\tau)\;
\left(
=\mathcal{F}_1(\bar{y}_\varepsilon,\bar{z}_\varepsilon,s,\tau)
\right),\quad
\bar{\mathcal{F}}_{2}(s,\tau) = \lim_{k\rightarrow\infty}\mathcal{F}_2^{(k)}(y^{(k)},z^{(k)},s,\tau)\;
\left(
=\mathcal{F}_2(\bar{y}_\varepsilon,\bar{z}_\varepsilon,s,\tau)
\right)

\end{eqnarray*}
is indeed the solution to \eqref{par_syst} in the sense of Definition \eqref{par_sol_def}.

Finally, due to uniqueness of the solution \hyperref[csl:5]{(Pao, 1992)} to the initial boundary value problem we have
$y_\varepsilon(x,t):=\bar{y}_\varepsilon(x,t)=\tilde{y}_\varepsilon(x,t)$,
$z_\varepsilon(x,t):=\bar{z}_\varepsilon(x,t)=\tilde{z}_\varepsilon(x,t)$.
and from \eqref{ParabolicSupSub1} follows that

\begin{equation*}
\lim_{t\rightarrow+\infty}\|y_\varepsilon(x,t) - u_\varepsilon(x)\|_{C[0,1]} = 0,\quad
\lim_{t\rightarrow+\infty}\|z_\varepsilon(x,t) - v_\varepsilon(x)\|_{C[0,1]} = 0.

\end{equation*}

\section*{Conclusion}
Though only the one-dimensional problems are considered in this paper it is already sufficient for development and justification of various models in physics especially when numerical experiments are preferable or simply unavoidable. Moreover, as a natural step forward our approach with several slight adjustments can be extended to the 2D problems which are proven to be extremely prolific for modelling.

\appendix

\section*{The Proof of Lemma 1}.

\begin{proof}
The existence of the fundamental exponentially bounded solution immediately follows from the fact that $\tilde{\nu}^{(\mp)}(\tilde{v}^{(\mp)}(\tau))$ are bounded continuous functions on $\mathbb{R}_{\mp}$ and the book\hyperref[csl:14]{(Coppel, 1978)}. The inequalities for derivatives and  positivity are the result of linearity of the equation in question and functions

\begin{equation*}
\overline{W}^{(\mp)}(\tau) := \exp{\left(-\mkern-10mu
\int\limits_{q^*}^{\tilde{v}^{(\mp)}(\tau)} \mkern-10mu \frac{ds_1}{\left(\Phi^{(\mp)}(s_1)\right)^2}
\int\limits_{s_1}^{\psi^{(\mp)}(x_0)} \mkern-10mu \tilde{\nu}^{(\mp)}(s_2)ds_2
\right)}, \quad
\underline{W}^{(\mp)}(\tau) = \frac{\Phi^{(\mp)}(\tilde{v}^{(\mp)}(\tau))}{\Phi^{(\mp)}(q^*)}

\end{equation*}
being respectively super- and subsolutions to problems

\begin{equation*}
W^{(\mp)}_{\tau \tau}(\tau)-\tilde{\nu}^{(\mp)}(\tilde{v}^{(\mp)}(\tau))W^{(\mp)}(\tau)=0,\;\;\tau \in (0,T^{(\mp)}),\;\;
W^{(\mp)}(0)=1,\;\;
W^{(\mp)}(T^{(\mp)})=\delta^{(\mp)}

\end{equation*}
for any $T^{(+)}>0$, $T^{(-)}<0$ and $\delta^{(\mp)}$ such that $\underline{W}^{(\mp)}(T^{(\mp)})\leq\delta^{(\mp)}\leq\overline{W}^{(\mp)}(T^{(\mp)})$.

\end{proof}

\selectlanguage{english}
\FloatBarrier
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