A singulary perturbed ``reaction-diffusion-transfer'' system which degenerates into a system of two first-order differential equations (Q1903958)

A system of two parabolic semilinear equations with small parameter \(\varepsilon\), \[ \begin{aligned} \varepsilon\partial_t u+ b_1(x) \partial_x u- \varepsilon^2 a_1(x) \partial^2_x u & = f(u, v, x, t),\\ \varepsilon\partial_t v+ b_2(x) \partial_x v- \varepsilon^2 a_2(x) \partial^2_x v & = g(u, v, x, t),\quad 0< x< 1,\quad 0< t< T\end{aligned} \] is considered. In the general case the initial functions \(u= \phi(x)\), \(v= \psi(x)\) as \(t= 0\), \(0< x< 1\) are not compatible to boundary data \(\partial_x u\), \(\partial_x v= 0\) as \(x= 0\) and \(x= 1\). Some modification of the boundary layer method is applied to construct two first terms of the asymptotic expansion of the solution \(u, v(x, t, \varepsilon)\) as \(\varepsilon\to 0\) up to order \(O(\varepsilon^2)\). The estimate of the remainder \(O(\varepsilon^2)\) is obtained.