## Differential and Integral Equations

### Asymptotic description of vanishing in a fast-diffusion equation with absorption

#### Abstract

In this paper we study the Cauchy problem $$u_t = \Delta u^m-u^m \ \hbox{ in } \mathbb R^N\times (0,\infty) ,$$ $u(x,0) = u_0(x),$ with $\frac{N-2}{N} < m < 1$ and $N\ge 2$. If $u_0 \not\equiv 0$ is a non-negative, compactly supported function such that and $u_0^m \in H^1(\mathbb R^N)\cap L^\infty(\mathbb R^N)$, then the solution $u$ vanishes identically after a least finite time $T>0$. We prove the asymptotic formula $$u(x,t) \sim (1-m)^{1\over 1-m} (T-t)^{1 \over 1-m}w_*^m (|x-\bar x| )$$ as $t\uparrow T$, for certain uniquely determined $\bar x\in \mathbb R^N$. Here $w_*$ is the unique positive radial solution of $$\Delta w- w + w^p =0 \ \hbox{ in } \mathbb R^N ,$$ $$w(x)\to 0 \hbox{ as } |x|\to \infty ,$$ where $p= 1/m$.

#### Article information

Source
Differential Integral Equations, Volume 15, Number 8 (2002), 1009-1023.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060782

Mathematical Reviews number (MathSciNet)
MR1895576

Zentralblatt MATH identifier
1011.35084

#### Citation

del Pino, Manuel; Sáez, Mariel. Asymptotic description of vanishing in a fast-diffusion equation with absorption. Differential Integral Equations 15 (2002), no. 8, 1009--1023. https://projecteuclid.org/euclid.die/1356060782