Large time behavior for the fast diffusion equation with critical absorption

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:260

Article

Benachour, Said¹  Gabriel Iagar, Razvan^2,3  Laurencot, Philippe⁴ 

[1] Univ Lorraine, Inst Elie Cartan, UMR 7502, F-54506 Vandoeuvre Les Nancy, France

[2] Inst Ciencias Matemat ICMAT, Campus Cantoblanco,Nicolas Cabrera 13-15, E-28049 Madrid, Spain

[3] Romanian Acad, Inst Math, POB 1-764, RO-014700 Bucharest, Romania

[4] Univ Toulouse, CNRS, Inst Math Toulouse, UMR 5219, F-31062 Toulouse 9, France

关键词: Large time behavior;    Fast diffusion;    Critical absorption;    Gradient estimates;    Lower bound;   

DOI  :  10.1016/j.jde.2016.02.008

来源: Elsevier

【 摘 要 】

We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption partial derivative(t)u - Delta u(m) + u(q) = 0 in (0, infinity) x R-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. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free