On a singularly perturbed system of two second-order equations in the case of intersecting roots of the degenerate equation (Q1033657)

There is studied a boundary value problem with a small parameter \(\varepsilon\) for the unknown functions \(u(x,\varepsilon),\;v(x,\varepsilon)\): \[ \varepsilon^2\,u''=g(u,v,x),\quad v''=f(u,v,x),\quad 0<x<1,\tag{1} \] \[ u'|_{x=0}=u'|_{x=1}=0,\quad v|_{x=0}=v^0,\;v|_{x=1}=v^1\tag{2,} \] where \(v^0\), \(v^1\) are constants. Let the equation \(g(u,v,x)=0\) have two roots with respect to \(u\): \(u=\varphi_1(v,x)\),\ \(u=\varphi_2(v,x)\). By means of these functions the solution \(\hat{v}(x)\), \ \(\hat{u}(x)=\varphi(\hat{v}(x),x)\) of the degenerate problem (1), (2) (with \(\varepsilon=0\)) is constructed. The authors prove that the problem (1), (2) has an asymptotic solution \(u_s(x,\varepsilon)=\hat{u}(x)+O(\varepsilon^{2/3})\),\ \(v_s(x,\varepsilon)=\hat{v}(x)+O(\varepsilon^{2/3})\), \(0\leq x \leq 1\), and the functions \(u_s(x,\varepsilon)\), \(v_s(x,\varepsilon)\) are the stationary solution of the corresponding boundary value problem with suitable initial data.