The Spectrum of a Family of Fourth-Order Nonlinear Diffusions Near the Global Attractor

Abstract

The thin film and quantum drift diffusion equations belong to a fourth-order family of evolution equations proposed in [21] to be analogous to the (second-order) porous medium family. They are 2-Wasserstein (=d 2) gradient flows of the generalized Fisher information I(v) just as the porous medium family was shown to be the d 2 gradient flow of the generalized entropy E(v) by Otto [41]. The identity aI(v) = bE(v) + |∇ d 2 E(v)|2/2 implies a Hess d 2 I(v *) = Hess d 2 E(v *)(b + Hess d 2 E(v *)) formally, when the equation is rescaled and linearized around the resulting self-similar critical profile v *. We couple this relation with the diagonalization of Hess d 2 E(v *) for the porous medium flow computed in [46]. This yields information about the leading- and higher-order asymptotics of the equation on R N which—outside of special cases—was inaccessible previously.