Singularly perturbed partially dissipative systems of equations (Q2036373)

In this paper, the author considers the stationary partially dissipative system consisting of the two equations \[ \varepsilon^2 \left(\frac{d^2u}{dx^2}-w(x)\frac{du}{dx}\right)=F(u,v,x,\varepsilon), \] \[ \varepsilon^2 \frac{dv}{dx}=f(u,v,x,\varepsilon), \ x\in(0;1) \] and with the boundary conditions \[ u(0,\varepsilon)=u^0,\ v(0,\varepsilon)=v^0,\ u(1,\varepsilon)=u^1, \] where \(\varepsilon>0\) is a small parameter and \(w, F,\) and \(f\) are given sufficiently smooth functions, with \(F(u,v,x,0)=0\) and \(f(u,v,x,0)=0.\) Other key assumptions are as follows: \begin{itemize} \item The function \(f\) has the form \[ f(u,v,x,\varepsilon) = -(v - \varphi(u,x))^3 + \varepsilon f_1(u,v,x,\varepsilon). \] \item The equation \[ F(u,\varphi(u,x),x,0) =: g(u,x) = 0 \] has a root \(u=\bar u_0(x),\) \(x\in[0;1],\) and \[ \frac{\partial g}{\partial u}(\bar u_0(x),x) > 0, \ x\in[0;1]. \] \end{itemize} The purpose of the paper is to obtain conditions under which the boundary value problem has a multizonal boundary layer solution \(u(x,\varepsilon),\) \(v(x,\varepsilon)\) for sufficiently small \(\varepsilon\), that is, a solution that tends to the solution of degenerate system (\(\varepsilon=0\)) as \(\varepsilon\to 0^+\) on the interval \(0 < x < 1\) and to construct an asymptotic approximation of this solution in the parameter \(\varepsilon\) on the whole interval \(0 \leq x \leq 1\) including the boundary layers, that is, small neighborhoods of the boundary points \(x = 0\) and \(x = 1,\) where the solution \(u(x,\varepsilon),\) \(v(x,\varepsilon)\) significantly differs from the solution of the degenerate system.