Asymptotics of solutions to the porous medium equation near conical singularities

TL;DR

This work analyzes the porous medium equation on manifolds with conical singularities and shows that short-time asymptotics near cone tips are dictated by the Laplacian spectrum on the cross-section. By reformulating via $v=u^m$ and establishing maximal $L^q$-regularity through a bounded $H^\infty$-calculus for the closed extension $\underline\Delta$, the authors obtain a rigorous framework to track asymptotics. The key finding is that the asymptotics involve powers and logarithmic terms $x^{-q_j^-}$ and $x^{-q_j^-}\ln x$ where $q_j^\pm$ derive from the cross-section spectrum, yielding a spectral-invariant description of early-time behavior. The results connect geometric features of conical singularities to precise nonlinear parabolic dynamics, with a robust functional-analytic backbone via Clément-Li theory and spectral calculus.

Abstract

We show that, on a manifold with conical singularities, the asymptotics of the solutions to the porous medium equation near the conical points are determined by the spectrum of the Laplacian on the cross-section of the cone. The key to this result is a precise description of the maximal domain of the cone Laplacian.

Theorems & Definitions (31)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 1.5
• Corollary 1.6
• Definition 1.7
• Theorem 1.8
• Theorem 2.1
• Theorem 2.2