Square-root reaction problem with nested boundary layers ($\epsilon y^{\prime \prime }-\sqrt{y}+\epsilon k=0$)

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.

After $y={\epsilon }^{2}Y$ and $x=\epsilon \xi$ or a translated version, the layer equation is $Y^{\prime \prime }-\sqrt{Y}+k=0$. Multiplying by $Y^{\prime }$ and matching $Y=k^2$, $Y^{\prime }=0$ fixes the energy constant to $k^3/3$ and gives the displayed factorization. The increasing branch below $k^2$ approaches it at positive infinity; the increasing branch above it leaves it at negative infinity and grows like ${\xi }^4/144$. This supplies both interfaces of the square-root reaction problem with nested boundary layers.