Differential and Integral Equations Volume 15, Number 8, August 2002, Pages 1009–1023 ASYMPTOTIC DESCRIPTION OF VANISHING IN A FAST-DIFFUSION EQUATION WITH ABSORPTION ∗ Manuel del Pino, and Mariel S´ aez Departamento de Ingenier´ ıa Matem´atica, Centro de Modelamiento Matem´ atico Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile (Submitted by: L.A. Peletier) Abstract. In this paper we study the Cauchy problem ut =Δu m − u m in R N × (0, ∞), u(x, 0) = u0(x), with N-2 N <m< 1 and N ≥ 2. If u0 ≡ 0 is a non- negative, compactly supported function such that and u m 0 ∈ H 1 (R N ) ∩ L ∞ (R N ), then the solution u vanishes identically after a least finite time T> 0. We prove the asymptotic formula u(x, t) ∼ (1 − m) 1 1-m (T − t) 1 1-m w m * (|x − ¯ x|) as t ↑ T , for certain uniquely determined ¯ x ∈ R N . Here w* is the unique positive radial solution of Δw − w + w p =0 in R N , w(x) → 0 as |x|→∞, where p =1/m. 1. Introduction This paper deals with analysis of the finite-time extinction phenomenon in solutions of the Cauchy problem, u t =Δu m − u m , in R N × (0, ∞) (1.1) u(x, 0) = u 0 (x), in R N , (1.2) where 0 <m< 1. u 0 ≡ 0 is a non-negative, compactly supported function which we additionally assume to be such that u m 0 ∈ H 1 (R N ) ∩ L ∞ (R N ). Asymptotic behavior of solutions for the more general problem u t =Δu m − u α , in R N × (0, ∞) (1.3) Accepted for publication: September 2001. AMS Subject Classifications: 35K15, 35K55. * The authors have been supported by FONDAP and by FONDECYT Lineas Comple- mentarias grant 8000010. 1009