= Endpoint layers for reversed diffusion
Reversing the diffusion sign changes which rapid modes remain bounded. In the explicit model
$$
-\varepsilon^2y''+2(x-1)y'=0,\qquad y(0)=A,\quad y(2)=B,
$$
integration gives $y'=C\exp((x-1)^2/\varepsilon^2)$. Thus
$$
y(x)=A+(B-A)\frac{\int_0^x e^{(s-1)^2/\varepsilon^2}\,ds}{\int_0^2 e^{(s-1)^2/\varepsilon^2}\,ds}.
$$
The numerator and denominator are dominated by endpoints. Away from both endpoints, the limit is $(A+B)/2$, since the two endpoint contributions in the denominator are equal. The leading <boundary layer> factors are $e^{-2x/\varepsilon^2}$ and $e^{-2(2-x)/\varepsilon^2}$, of width $\varepsilon^2$.
On the interior scale $z=(x-1)/\varepsilon$, $Y''-2zY'=0$ has $Y'=C e^{z^2}$, so a bounded leading inner solution cannot connect unequal constants at both infinities. For equations with extra reaction or forcing, the selected bulk value and smaller interior corrections depend on those terms and their <solvability condition>; the simple-zero drift alone does not determine them.
Back to article page
