Large time behavior for the fast diffusion equation with critical absorption

Original · EN

We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption ∂ₜu-Δuᵐ+uq=0 in (0,∞)×ⁿ with mc:=(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.