Large time behavior for the fast diffusion equation with critical absorption

2014/09/07 by Benachour, Said, Iagar, Razvan Gabriel, Laurencot, Philippe
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1409.2154

Abstract

We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption ∂_tu-Δu^m+u^q=0 \hboxin (0,∞)×\real^N with m_c:=(N-2)₊/N < m < 1 and q=m+2/N. Given an initial condition u₀ 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 u₀. 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.

Related

• Large time behavior for a quasilinear diffusion equation with critical gradient absorption
• Large Time Behavior of a Nonlocal Diffusion Equation with Absorption and Bounded Initial Data
• Asymptotic large time behavior of singular solutions of the fast diffusion equation
• Asymptotic behavior for the fast diffusion equation with absorption and singularity
• Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data: the subcritical case