= Forward-backward threshold for gradient-weighted exponential diffusion
{title2=$F'(p)=|p|e^{-p^2/(2\lambda^2)}(2-p^2/\lambda^2)$}
For one-dimensional flux $F(p)=p|p|e^{-p^2/(2\lambda^2)}$, the <nonlinear diffusion equation> is $u_t=F'(u_x)u_{xx}$. The sole one-dimensional member of the <principal diffusion coefficients> is positive for $0<|u_x|<\sqrt2\lambda$, negative above $\sqrt2\lambda$, and zero at the two endpoints. Negative diffusion gives formal sharpening with <ill-posedness> through high-frequency growth. The extra factor $|p|$ matters: the unweighted exponential conductance has threshold $\lambda$ instead.
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