A singularly perturbed problem of chemical kinetics (Q798834)

The authors study the initial-boundary value problems \(\partial u^ I/\partial t=\mu^ 2_ 1\partial^ 2u^ I/\partial x^ 2_ 1\), \(u^ I(x_ 1,0)=0,\quad\partial u^ I/\partial x_ 1(0,t)=0;\quad\partial u^{II}/\partial t=\mu^ 2_ 2\partial^ 2u^{II}/\partial x^ 2_ 2,\quad u^{II}(x_ 2,0)=1,\quad\partial u^{II}/\partial x_ 2(0,t)=0,\) which are coupled through the interface conditions \(u^ I(\ell,t)=u^{II}(\ell,t)\) and \([a_ 1\partial u^ I/\partial x_ 1+a_ 2\partial u^{II}/\partial x_ 2+a_ 3u^ I+a_ 4u^{II}+a_ 5](\ell,t)=0,\) in the case when \(\mu_ 1,\mu_ 2\) and \(\mu_ 1/\mu_ 2\) are small. Using asymptotic analysis they are able to prove the existence of a solution \(\{u^ I,u^{II}\}\) and to construct a uniformly valid expansion for \(\{u^ I,u^{II}\}\) that features layer behavior in a neighborhood of \(x=\ell\).