= Endpoint layers for reversed diffusion
For $-\varepsilon^2y''+2(x-x_*)y'+\cdots=f$ on an interval straddling $x_*$, the bounded leading interior solution cannot connect unequal constants. A common bulk value is instead selected by an inner <solvability condition>, and width-$\varepsilon^2$ endpoint layers enforce the two boundary values. At endpoints $x=0,2$ with $x_*=1$, their leading decaying factors are $e^{-2x/\varepsilon^2}$ and $e^{-2(2-x)/\varepsilon^2}$. The interior layer still corrects derivative mismatches at smaller amplitude.
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