Anisotropic and isotropic persistent singularities of solutions of the fast diffusion equation

TL;DR

This paper investigates various types of persistent singularities in solutions to the fast diffusion equation, including anisotropic and isotropic cases, constructing new solutions and clarifying their distributional properties.

Contribution

It introduces new solutions with anisotropic singularities and clarifies the distributional and integrability properties of existing solutions, including those with snaking and isotropic singularities.

Findings

Constructed solutions with anisotropic singularities.

Established that some solutions solve the equation in the distributional sense.

Discussed existence of solutions with anisotropic singularities in critical cases.

Abstract

The aim of this paper is to study a class of positive solutions of the fast diffusion equation with specific persistent singular behavior. First, we construct new types of solutions with anisotropic singularities. Depending on parameters, either these solutions solve the original equation in the distributional sense, or they are not locally integrable in space-time. We show that the latter also holds for solutions with snaking singularities, whose existence has been proved recently by M. Fila, J.R. King, J. Takahashi, and E. Yanagida. Moreover, we establish that in the distributional sense, isotropic solutions whose existence was proved by M. Fila, J. Takahashi, and E. Yanagida in 2019, actually solve the corresponding problem with a moving Dirac source term. Last, we discuss the existence of solutions with anisotropic singularities in a critical case.