Forward-backward threshold for gradient-weighted exponential diffusion ($F^{\prime }(p)=|p|e^{-p^2/(2{\lambda }^2)}(2-p^2/{\lambda }^2)$)

For one-dimensional flux $F(p)=p|p|e^{-p^2/(2{\lambda }^2)}$, the nonlinear diffusion equation is $u_t=F^{\prime }(u_x)u_{xx}$. The sole one-dimensional member of the principal diffusion coefficients is positive for $0<|u_x|<\sqrt{2}\lambda$, negative above $\sqrt{2}\lambda$, 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.