Existence and uniqueness of very singular solutions for a fast diffusion equation with gradient absorption

TL;DR

This paper proves the existence and uniqueness of radially symmetric self-similar very singular solutions for a nonlinear diffusion equation with gradient absorption, advancing understanding of such solutions in singular diffusion models.

Contribution

It establishes the first rigorous proof of existence and uniqueness of these solutions for the specified equation, including the conditions under which they occur.

Findings

Existence of radially symmetric self-similar solutions

Uniqueness of these solutions under certain conditions

Characterization of solutions in the singular diffusion context

Abstract

Existence and uniqueness of radially symmetric self-similar very singular solutions are proved for the singular diffusion equation with gradient absorption {equation*} \partial_t u -\Delta_{p}u+|\nabla u|^q=0, \ \hbox{in} \ (0,\infty)\times\real^N, {equation*} where $2N/(N+1)