The spectrum of a family of fourth-order nonlinear diffusions near the global attractor∗

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 2Wasserstein (= d2) gradient flows of the generalized Fisher information I(v) just as the porous medium family was shown to be the d2 gradient flow of the generalized entropy E(v) by Otto [40]. The identity aI(v) = bE(v) + |∇d2E(v)|2/2 implies aHessd2 I(v∗) = Hessd2 E(v∗)(b + Hessd2 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 Hessd2 E(v∗) for the porous medium flow computed in [45]. This yields information about the leadingand higher-order asymptotics of the equation on R which — outside of special cases — was inaccessible previously.