Asymptotic Behaviour near Finite-Time Extinction for the Fast Diffusion Equation

1997 ◽  
Vol 139 (1) ◽  
pp. 83-98 ◽  
Author(s):  
Victor A. Galaktionov ◽  
Lambertus A. Peletier
Keyword(s):  
Diffusion Equation ◽  
Asymptotic Behaviour ◽  
Finite Time ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Finite Time Extinction ◽  
Time Extinction

scholarly journals Extinction for a Singular Diffusion Equation with Strong Gradient Absorption Revisited

2018 ◽  
Vol 18 (4) ◽  
pp. 785-797
Author(s):  
Razvan Gabriel Iagar ◽  
Philippe Laurençot
Keyword(s):  
Diffusion Equation ◽  
Finite Time ◽  
Extinction Time ◽  
Extinction Rates ◽  
Optimal Decay ◽  
Singular Diffusion ◽  
Finite Time Extinction ◽  
Strong Gradient ◽  
The One ◽  
Time Extinction

AbstractWhen {2N/(N+1)<p<2} and {0<q<p/2}, non-negative solutions to the singular diffusion equation with gradient absorption\partial_{t}u-\Delta_{p}u+|\nabla u|^{q}=0\quad\text{in }(0,\infty)\times% \mathbb{R}^{N}vanish after a finite time. This phenomenon is usually referred to as finite-time extinction and takes place provided the initial condition {u_{0}} decays sufficiently rapidly as {|x|\to\infty}. On the one hand, the optimal decay of {u_{0}} at infinity guaranteeing the occurrence of finite-time extinction is identified. On the other hand, assuming further that {p-1<q<p/2}, optimal extinction rates near the extinction time are derived.

scholarly journals The critical exponent for fast diffusion equation with nonlocal source

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Chunxiao Yang ◽  
Linghua Kong ◽  
Yingxue Wu ◽  
Qing Tian
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Critical Exponent ◽  
Finite Time ◽  
Global Solutions ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Nonlocal Source ◽  
Nonlinear Anal ◽  
The Cauchy Problem

Abstract This paper considers the Cauchy problem for fast diffusion equation with nonlocal source $u_{t}=\Delta u^{m}+ (\int_{\mathbb{R}^{n}}u^{q}(x,t)\,dx )^{\frac{p-1}{q}}u^{r+1}$ u t = Δ u m + ( ∫ R n u q ( x , t ) d x ) p − 1 q u r + 1 , which was raised in [Galaktionov et al. in Nonlinear Anal. 34:1005–1027, 1998]. We give the critical Fujita exponent $p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}$ p c = m + 2 q − n ( 1 − m ) − n q r n ( q − 1 ) , namely, any solution of the problem blows up in finite time whenever $1< p\le p_{c}$ 1 < p ≤ p c , and there are both global and non-global solutions if $p>p_{c}$ p > p c .

Optimal rates of convergence to the singular Barenblatt profile for the fast diffusion equation

2016 ◽  
Vol 146 (2) ◽  
pp. 309-324 ◽  
Author(s):  
Marek Fila ◽  
Michael Winkler
Keyword(s):  
Initial Data ◽  
Diffusion Equation ◽  
Asymptotic Behaviour ◽  
Rates Of Convergence ◽  
Fast Diffusion ◽  
Time And Space ◽  
Fast Diffusion Equation ◽  
Optimal Rates Of Convergence ◽  
The Difference ◽  
Near Extinction

We study the asymptotic behaviour of solutions of the fast diffusion equation near extinction. For a class of initial data, the asymptotic behaviour is described by a singular Barenblatt profile. We complete previous results on rates of convergence to the singular Barenblatt profile by describing a new phenomenon concerning the difference between the rates in time and space.

scholarly journals A continuum of extinction rates for the fast diffusion equation

2011 ◽  
Vol 10 (4) ◽  
pp. 1129-1147 ◽  
Author(s):  
Michael Winkler ◽  
Juan-Luis Vázquez ◽  
Marek Fila
Keyword(s):  
Diffusion Equation ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Extinction Rates
Sign in / Sign up

Export Citation Format

Share Document