Second order asymptotics and uniqueness for self-similar profiles to a singular diffusion equation with gradient absorption

TL;DR

This paper investigates self-similar solutions with finite time extinction for a singular diffusion equation with gradient absorption, establishing existence, uniqueness in one dimension, and detailed asymptotic behavior in higher dimensions.

Contribution

It provides the first rigorous existence and uniqueness results for these solutions in one dimension and characterizes their asymptotic behavior in higher dimensions.

Findings

Existence and uniqueness in 1D for self-similar solutions.

Existence of radially symmetric solutions in higher dimensions.

Asymptotic behavior of solutions as |x|→∞.

Abstract

Solutions in self-similar form presenting finite time extinction to the singular diffusion equation with gradient absorption $$\partial_t u - \mathrm{div}(|\nabla u|^{p-2}\nabla u) +|\nabla u|^{q}=0 \qquad {\rm in} \ (0,\infty)\times\mathbb{R}^N$$ are studied when $N\geq1$ and the exponents $(p,q)$ satisfy $p_c=\frac{2N}{N+1}$, $p-1<q<\frac{p}{2}$. Existence and uniqueness of such a solution are established in dimension $N=1$. In dimension $N\geq2$, existence of radially symmetric self-similar solutions is proved and a fine description of their behavior as $|x|\to\infty$ is provided.