Asymptotics for slowly converging evolution equations

TL;DR

This paper analyzes the conditions under which solutions to certain non-linear elliptic or parabolic equations converge slowly, revealing a necessary condition and characterizing their convergence behavior, with implications for geometric variational problems.

Contribution

It introduces the Adams-Simon non-negativity condition as a necessary criterion for slowly converging solutions, extending understanding beyond previous positivity assumptions.

Findings

Identifies the Adams-Simon non-negativity condition as necessary for slow convergence.

Characterizes the rate and direction of convergence of solutions.

Partially confirms Thom's gradient conjecture in infinite-dimensional settings.

Abstract

We investigate slowly converging solutions for non-linear evolution equations of elliptic or parabolic type. These equations arise from the study of isolated singularities in geometric variational problems. Slowly converging solutions have previously been constructed assuming the Adams-Simon positivity condition. In this study, we identify a necessary condition for slowly converging solutions to exist, which we refer to as the Adams-Simon non-negativity condition. Additionally, we characterize the rate and direction of convergence for these solutions. Our result partially confirms Thom's gradient conjecture in the context of infinite-dimensional problems.