On the cauchy problem for a very fast diffusion equation

1996 ◽  
Vol 21 (9-10) ◽  
pp. 1349-1365 ◽  
Author(s):  
Jong Sheng Guo
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
The Cauchy Problem

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 .

Cauchy problem for fast diffusion equation with localized reaction

2011 ◽  
Vol 74 (7) ◽  
pp. 2508-2514 ◽  
Author(s):  
Xueli Bai ◽  
Shuangshuang Zhou ◽  
Sining Zheng
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Localized Reaction

On the equation ut = ∆uα + uβ

1993 ◽  
Vol 123 (2) ◽  
pp. 373-390 ◽  
Author(s):  
Yuan-Wei Qi
Keyword(s):  
Dynamical System ◽  
Blow Up ◽  
Uniqueness Result ◽  
Fast Diffusion ◽  
Negative Solution ◽  
Fast Diffusion Equation ◽  
Infinite Dimensional ◽  
The Cauchy Problem ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System

SynopsisThe Cauchy problem of ut, = ∆uα + uβ, where 0 < α < l and α>1, is studied. It is proved that if 1< β<α + 2/n then every nontrivial non-negative solution is not global in time. But if β>α+ 2/n there exist both blow-up solutions and global positive solutions which decay to zero as t–1/(β–1) when t →∞. Thus the famous Fujita result on ut = ∆u + up is generalised to the present fast diffusion equation. Furthermore, regarding the equation as an infinite dimensional dynamical system on Sobolev space W1,s (W2.s) with S > 1, a non-uniqueness result is established which shows that there exists a positive solution u(x, t) with u(., t) → 0 in W1.s (W2.s) as t → 0.

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
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