A singularly perturbed ``reaction-diffusion-transfer'' system which degenerates into a system with a finite equation and a first-order partial differential equation
We consider the homogeneous Neumann boundary value problem for the following singularly perturbed semilinear parabolic system: \[ \left. \begin{aligned} \varepsilon {\partial u\over \partial t} & + \varepsilon b(x) {\partial u\over \partial x}- \varepsilon^3 a_1(x) {\partial^2 u\over \partial x^2}= f(u, v, x, t)\\ {\partial v\over \partial t} & + b(x) {\partial v\over \partial x}- \varepsilon^2 a_2(x) {\partial^2 v\over \partial x^2}= g(u, v, x, t)\end{aligned} \right\}\quad 0< x< 1, \] where \(\varepsilon> 0\) is a small parameter. We construct an asymptotic approximation for small values of \(\varepsilon\) for the solution up to the order \(\varepsilon^2\). The arrangement of the small parameter \(\varepsilon\) corresponds to slow diffusion in both equations, fast reactions in the first equation (the reaction rate is of order \(\varepsilon^{-1}\)) and slow reactions in the second equation.
homogeneous Neumann boundary value problem
scientific article; zbMATH DE number 859640