Large time behavior for the fast diffusion equation with critical absorption

TL;DR

This paper investigates the long-term behavior of solutions to a fast diffusion equation with critical absorption, revealing that solutions asymptotically resemble a scaled Barenblatt profile with logarithmic correction, regardless of initial decay rate.

Contribution

It extends previous results by showing asymptotic behavior without requiring specific lower bounds on initial data, and derives sharp gradient estimates and universal lower bounds.

Findings

Solutions asymptotically match a Barenblatt profile with logarithmic scaling.

The analysis provides sharp gradient estimates and a universal lower bound.

Results hold for general exponents q > 1.

Abstract

We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption $$ \partial_{t}u-\Delta u^m+u^q=0 \quad \quad \hbox{in} \ (0,\infty)\times\real^N\, $$ with $m_c:=(N-2)_{+}/N < m < 1$ and $q=m+2/N$. Given an initial condition $u_0$ decaying arbitrarily fast at infinity, we show that the asymptotic behavior of the corresponding solution $u$ is given by a Barenblatt profile with a logarithmic scaling, thereby extending a previous result requiring a specific algebraic lower bound on $u_0$. A by-product of our analysis is the derivation of sharp gradient estimates and a universal lower bound, which have their own interest and hold true for general exponents $q > 1$.