The spectrum of a family of fourth-order equations near the global attractor

The thin-film and quantum drift diffusion equations belong to a fourth-order family of evolution equations proposed in ref. 16 to be analogous to the (second-order) porous medium family. They are $2$-Wasserstein gradient ($W_2$) flows of the generalized Fisher information $I(u)$ just as the porous medium family was shown to be the $W_2$ gradient flow of the generalized entropy $E(u)$ by Otto. The identity $I(u)=\mathrm{\mid }{\mathrm{\nabla }}_{W_2}E(u){\mathrm{\mid }}^2\mathrm{/}2$ (formally) becomes ${\text{Hess}}_{W_2}I(u_{\ast })={\text{Hess}}_{W_2}^{2}E(u_{\ast })$ when linearizing the equation around its self-similar solution $u_{\ast }$. We couple this relation with the diagonalization of ${\text{Hess}}_{W_2}E(u_{\ast })$ for the porous medium flow computed in ref. 38. This yields information about the leading- and higher-order asymptotics of the equation on ${\mathbb{R}}^N$ which — outside of special cases — was inaccessible previously.