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

Abstract

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 $I(u)=|{\nabla }_{W_2}E(u){|}^2/2$ (formally) becomes $\Hess_{W_2} I(u_*) =\Hess_{W_2}^2 E(u_*)$ when linearizing the equation around its self-similar solution $u_{\ast }$. We couple this relation with the diagonalization of $\Hess_{W_2} E(u_*)$ 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.