 Square-root reaction problem with nested boundary layers

For fixed $k>0$ and endpoint values $y(0)=0$, $y(1)=1$, the outer solution is $y\sim {\epsilon }^2{k}^2$. The left boundary layer has width $\epsilon$. At the right, a layer of width $\sqrt{\epsilon }$ carries the order-one rise, and a translated layer of width $\epsilon$ connects its inner edge to the small plateau. The dominant balance depends on both amplitude and derivative scale; using the same scale at both ends loses a necessary match.