Asymptotics of the solution of a singularly perturbed system of equations with a multizone internal layer (Q2027294)

The authors consider the asymptotic behavior of the solutions for a singularly perturbed boundary value problem \[ \varepsilon^2 \frac{\mathrm{d}^2u}{\mathrm{d}x^2}=F(u,v,x,\varepsilon), \ \varepsilon\frac{\mathrm{d}^2v}{\mathrm{d}x^2}=f(u,v,x,\varepsilon),\ 0 < x < 1, \] \[ \frac{\mathrm{d}u}{\mathrm{d}x}(0,\varepsilon)=\frac{\mathrm{d}u}{\mathrm{d}x}(1,\varepsilon)=0, \ \frac{\mathrm{d}v}{\mathrm{d}x}(0,\varepsilon)=\frac{\mathrm{d}v}{\mathrm{d}x}(1,\varepsilon)=0, \] where \(\varepsilon > 0\) is a small parameter and \(u(x,\varepsilon)\) and \(v(x,\varepsilon)\) are scalar functions. The function \(F\) has the form \[ F(u,v,x,\varepsilon)=\left(u-\varphi_1(v,x)\right)^2\left(u-\varphi_2(v,x)\right)\left(u-\varphi_3(v,x)\right)-\varepsilon F_1(u,v,x,\varepsilon) \] with \[ \varphi_1(v,x) < \varphi_2(v,x) < \varphi_3(v,x), \ (v,x)\in I_v\times [0,1]. \] The regular part and internal layer part of the asymptotics are constructed in the form of series in integer powers of \(\sqrt{\varepsilon}\) in the Subsection 3.1 and Subsection 2.3, respectively. By combining the asymptotic expansion techniques and the method of lower and upper solutions, the authors prove that for sufficiently small parameter values the problem has a solution with a rapid transition from the double root of the degenerate equation to a simple root in a neighborhood of some interior point of the interval.