Long-time asymptotics for the porous medium equation: The spectrum of the linearized operator

TL;DR

This paper analyzes the complete spectrum of the linearized operator around the Barenblatt profile for the porous medium equation, providing insights into solution convergence and asymptotics.

Contribution

It explicitly computes the spectrum of the displacement Hessian operator, revealing its discrete nature and eigenfunctions, which improve understanding of solution behavior near the attractor.

Findings

Spectrum of the displacement Hessian is purely discrete

Eigenfunctions reveal symmetries and convergence rates

Provides detailed asymptotic behavior of solutions

Abstract

We compute the complete spectrum of the displacement Hessian operator, which is obtained from the confined porous medium equation by linearization around its stationary attractor, the Barenblatt profile. On a formal level, the operator is conjugate to the Hessian of the entropy via similarity transformation. We show that the displacement Hessian can be understood as a self-adjoint operator and find that its spectrum is purely discrete. The knowledge of the complete spectrum and the explicit information about the corresponding eigenfunctions give new insights on the convergence and higher order asymptotics of solutions to the porous medium equation towards its attractor. More precisely, the inspection of the eigenfunctions allows to identify symmetries in $\mathbbm{R}^N$ with flows whose rates of convergence are faster than the uniform, translation-governed bound. The present work complements the analogous study of Denzler & McCann for the fast-diffusion equation.