On one singularly perturbed system of ordinary differential equations with multiple root of the degenerate equation
Consider the two-dimensional system of singularly perturbed differential equations \[\begin{aligned}\varepsilon^2\frac{d^2u}{dx^2} & =h(x)(u-\varphi(v,x))^2-\varepsilon F_1(u,v,x,\varepsilon), \\ \varepsilon\frac{d^2v}{dx^2} & =f(u,v,x,\varepsilon)\end{aligned} \tag{1}\] on the interval \([0,1]\) with the boundary conditions \[u(0,\varepsilon)=u^0,\ u(1,\varepsilon)=u^1,\ \frac{dv}{dx} (0,\varepsilon)=q^0,\ \frac{dv}{dx}(1,\varepsilon)=q^1.\tag{2}\] It is assumed that the degenerate system \[h(x)(u-\varphi(x,v))^2=0,\ f(u,v,x,0)=0\] has the double root \(u=\varphi(x,v)\) and the simple root \(v=\overline v_0(x)\) on \([0,1]\). The author proves the existence of a solution \((u(x,\varepsilon),v(x,\varepsilon))\) of (1), (2) for sufficiently small \(\varepsilon\) by means of the method of lower and upper solutions. To construct lower and upper solutions, the author makes the ansatz for the solution \((u(x,\varepsilon),v(x,\varepsilon))\) \[u(x,\varepsilon)= \overline u(x,\varepsilon)+\Pi u(\xi,\varepsilon)+\widetilde\Pi u (\widetilde\xi,\varepsilon)+Pu(\zeta,\varepsilon)+\widetilde Pu (\widetilde\zeta,\varepsilon),\] \[v(x,\varepsilon)=\overline v(x,\varepsilon)+\Pi v(\xi,\varepsilon)+\widetilde\Pi v (\widetilde\xi,\varepsilon)+Pv(\zeta,\varepsilon)+\widetilde Pv (\widetilde\zeta,\varepsilon),\] with \[\xi=\frac{x}{\sqrt{\varepsilon}},\ \zeta=\frac{x}{\varepsilon}, \ \widetilde\xi=\frac{1-x} {\sqrt{\varepsilon}}, \ \widetilde\zeta=\frac{1-x}{\varepsilon}.\] \(\Pi,\widetilde\Pi,P,\widetilde P\) are the boundary layer functions. The author presents a procedure to determine these functions which differs essentially from the case of simple roots. It is shown that the boundary layer functions exhibit a multizone structure.
scientific article; zbMATH DE number 7139492