## 2 Materials and Methods  
We consider the following problem:\[\epsilon^{4}\Delta u-u_{t}=f(u,v,\mathbf{x},\epsilon),\ \ \epsilon^{2}\Delta v-v_{t}=g(u,v, \mathbf{x},\epsilon),\ \ \mathbf{x}\in D,\ t\in\mathbb{R}^{+}; \tag{1}\] \[\frac{\partial u}{\partial n_{\Gamma}}(\mathbf{x}_{\Gamma},t)= \frac{\partial v}{\partial n_{\Gamma}}(\mathbf{x}_{\Gamma},t)=0,\ \ \mathbf{x}_{\Gamma}\in\Gamma,\ t\in\mathbb{R}^{+};\] \[u(\mathbf{x},0)=u^{0}(\mathbf{x}),\ \ v(\mathbf{x},0)=v^{0}( \mathbf{x}),\ \mathbf{x}\in\overline{D}.\]  
where \(D\subset\mathbb{R}^{2}\) is a connected domain in the coordinate space \(\mathbf{x}=(x_{1},x_{2})\) with a closed smooth boundary \(\Gamma\); \(\epsilon\in(0,\epsilon_{0}]\) is a small parameter; \(\frac{\partial}{\partial n_{\Gamma}}\) is the derivative with respect to the internal normal to the curve \(\Gamma\); and the functions \(u^{0}(\mathbf{x})\), \(v^{0}(\mathbf{x})\) are smooth in \(\overline{D}\) and satisfy the matching conditions  
\[\frac{\partial u^{0}}{\partial n_{\Gamma}}(\mathbf{x}_{\Gamma})=\frac{ \partial v^{0}}{\partial n_{\Gamma}}(\mathbf{x}_{\Gamma})=0,\ \ \mathbf{x}_{\Gamma}\in\Gamma.\]  
It is assumed that the functions \(f(u,v,\mathbf{x},\epsilon)\) and \(g(u,v,\mathbf{x},\epsilon)\) have jumps on some surface \(I_{u}\times I_{v},\times\gamma\), where \(\gamma\) is a simple, smooth closed curve lying entirely in the domain \(D\):  
\[f(u,v,\mathbf{x},\epsilon) = \begin{cases}f^{(-)}(u,v,\mathbf{x},\epsilon),&u\in I_{u},\,v\in I _{v},\,\mathbf{x}\in\overline{D}^{(-)}\,,\\ f^{(+)}(u,v,\mathbf{x},\epsilon),&u\in I_{u},\,v\in I_{v},\,\mathbf{x}\in D^{ (+)}\,;\end{cases}  \tag{2}\] \[g(u,v,\mathbf{x},\epsilon) = \begin{cases}g^{(-)}(u,v,\mathbf{x},\epsilon),&u\in I_{u},\,v\in I _{v},\,\mathbf{x}\in\overline{D}^{(-)}\,,\\ g^{(+)}(u,v,\mathbf{x},\epsilon),&u\in I_{u},\,v\in I_{v},\,\mathbf{x}\in D^{ (+)}\,,\end{cases} \]  
where \(\overline{D}^{(-)}\cup\overline{D}^{(+)}=\overline{D}\) and \(\overline{D}^{(-)}\cap\overline{D}^{(+)}=\gamma\). For definiteness, we assume that \(D^{(-)}\) is the domain enclosed between the internal curve \(\gamma\) and the boundary \(\Gamma\). In turn, \(D^{(+)}\) is the domain bounded by \(\gamma\). The construction of an asymptotic approximation to the solution is a generating part of the algorithm. A requirement for the functions \(f^{(\mp)}(u,v,\mathbf{x},\epsilon)\) and \(g^{(\mp)}(u,v,\mathbf{x},\epsilon)\) to be sufficient to construct the \(n\)th-order asymptotic is that these functions are of a class \(C^{n+2}(I_{u}\times I_{v}\times\overline{D}^{(\mp)}\times[0,\epsilon_{0}])\), where \(I_{u}\) and \(I_{v}\) are some permissible intervals for changing variables \(u\) and \(v\), respectively. In order to appreciate the behavior of the solution, it is enough to construct its asymptotic approximation of the zero or first order. When proving the existence and stability theorems, we use the possibility of constructing an asymptotic approximation of the order \(n=3\). We define the solution to Problem (1) as follows:  
Definition 1: A couple of functions \((u_{\epsilon}(\mathbf{x},t),v_{\epsilon}(\mathbf{x},t))\) of the class \(C\big{(}\overline{D}\times\mathbb{R}_{0}^{+}\big{)}\cap C^{1,0}\big{(} \overline{D}\times\mathbb{R}^{+}\big{)}\cap C^{2,1}\big{(}(D\backslash\gamma) \times\mathbb{R}^{+}\big{)}\) are called the solution to Problem (1) if they satisfy the equations (1) when \((\mathbf{x},t)\in(D\backslash\gamma)\times\mathbb{R}^{+}\) and the boundary and initial conditions of this problem.