Singularly perturbed boundary value problems in case of exchange of stability (Q1279950)

The authors study the boundary value problem \[ \varepsilon^2\frac{d^2u}{dx^2}=g(u,v,x,\varepsilon),\quad u'(0)=0,\;u'(1)=0, \quad \frac{d^2v}{dx^2}=f(u,v,x,\varepsilon),\quad v(0)=v^0, v(1)=v^1, \] where \(u\) and \(v\) are scalars. Using the method of upper and lower solutions, the authors prove the existence of a solution together with an asymptotic behavior as \(\varepsilon>0\) goes to zero. Some basic assumptions are the following : 1. The degenerate equation \(g(u,v,x,0)=0\) has two solutions \(u=\varphi_1(v,x)\) and \(u=\varphi_2(v,x)\), which intersect along a curve \(v=k(x)\). 2. The degenerate problem has a composite solution, i.e. for some \(x_0\in (0,1)\) the problems \[ \begin{aligned} \frac{d^2v}{dx^2}&=f(\varphi_1(v,x),v,x,0),\quad v(0)=v^0,\;v(x_0)=k(x_0), \\ \frac{d^2v}{dx^2}&=f(\varphi_2(v,x),v,x,0),\quad v(x_0)=k(x_0),\;v(1)=v^1,\end{aligned} \] have solutions \(v_1\) and \(v_2\) such that \(v'_1(x_0)=v'_2(x_0)\). 3. The two families of equilibria, \(u=\varphi_1(v_j(x),x)\) and \(u=\varphi_2(v_j(x),x)\), to \[ \frac{d^2u}{d\xi^2}=g(u,v_j(x),x,0), \] intersect transversally, i.e. they exchange the stability. This is characterized by sign conditions on the functions \(g_u(\varphi_i(v_j(x),x),v_j(x),x,0)\). Some examples modeling biomolecular reactions with fast reaction rate are given.