Solutions with snaking singularities for the fast diffusion equation

TL;DR

This paper constructs solutions to the fast diffusion equation that are globally defined in time and exhibit singularities along moving curves, with detailed analysis of their behavior near these singularities.

Contribution

It introduces a method to construct solutions with prescribed singularities along moving curves for the fast diffusion equation, expanding understanding of singular solution structures.

Findings

Solutions exist for all real times with singularities on specified curves.

Detailed asymptotic behavior of solutions near the singular set.

Framework for constructing solutions with dynamic singularities.

Abstract

We construct solutions of the fast diffusion equation, which exist for all $t\in\mathbb{R}$ and are singular on the set $\Gamma(t):= \{ \xi(s) ; -\infty <s \leq ct \}$, $c>0$, where $\xi\in C^3(\mathbb{R};\mathbb{R}^n)$, $n\geq 2$. We also give a precise description of the behavior of the solutions near $\Gamma(t)$.