scholarly journals Uniqueness and time oscillating behaviour of finite points blow-up solutions of the fast diffusion equation

2019 ◽  
Vol 150 (6) ◽  
pp. 2849-2870
Author(s):  
Kin Ming Hui
Keyword(s):  
Initial Data ◽  
Diffusion Equation ◽  
Bounded Domain ◽  
Positive Constant ◽  
Blow Up ◽  
Fast Diffusion ◽  
Bounded Domains ◽  
Fast Diffusion Equation

AbstractLet n ⩾ 3 and 0 < m < (n − 2)/n. We extend the results of Vazquez and Winkler (2011, J. Evol. Equ. 11, no. 3, 725–742) and prove the uniqueness of finite points blow-up solutions of the fast diffusion equation ut = Δum in both bounded domains and ℝn × (0, ∞). We also construct initial data such that the corresponding solution of the fast diffusion equation in bounded domain oscillates between infinity and some positive constant as t → ∞.

On the quenching set for a fast diffusion equation: regional quenching

2005 ◽  
Vol 135 (3) ◽  
pp. 585-602
Author(s):  
R. Ferreira ◽  
A. de Pablo ◽  
F. Quirós ◽  
J. D. Rossi
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Blow Up ◽  
Single Point ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Flux Boundary Condition ◽  
Bounded Interval ◽  
Quenching Set ◽  
Type Boundary

We study positive solutions of a very fast diffusion equation, ut = (um−1ux)x, m < 0, in a bounded interval, 0 < x < L, with a quenching-type boundary condition at one end, u (0, t) = (T − t)1/(1 − m) and a zero-flux boundary condition at the other, (um −1ux)(L, t) = 0. We prove that for m ≥ −1 regional quenching is not possible: the quenching set is either a single point or the whole interval. Conversely, if m < −1 single-point quenching is impossible, and quenching is either regional or global. For some lengths the above facts depend on the initial data. The results are obtained by studying the corresponding blow-up problem for the variable v = um −1.

scholarly journals Extinction and Nonextinction for the Fast Diffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chunlai Mu ◽  
Li Yan ◽  
Yi-bin Xiao
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Initial Data ◽  
Diffusion Equation ◽  
Dirichlet Boundary ◽  
Fast Diffusion ◽  
First Eigenvalue ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Fast Diffusion Equation ◽  
The First Eigenvalue

This paper deals with the extinction and nonextinction properties of the fast diffusion equation of homogeneous Dirichlet boundary condition in a bounded domain ofRNwithN>2. For0<m<1, under appropriate hypotheses, we show thatm=pis the critical exponent of extinction for the weak solution. Furthermore, we prove that the solution either extinct or nonextinct in finite time depends strongly on the initial data and the first eigenvalue of-Δwith homogeneous Dirichlet boundary.

On the quenching set for a fast diffusion equation: regional quenching

2005 ◽  
Vol 135 (3) ◽  
pp. 585-602
Author(s):  
R. Ferreira ◽  
A. de Pablo ◽  
F. Quirós ◽  
J. D. Rossi
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Blow Up ◽  
Single Point ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Flux Boundary Condition ◽  
Bounded Interval ◽  
Quenching Set ◽  
Type Boundary

We study positive solutions of a very fast diffusion equation, ut = (um−1ux)x, m < 0, in a bounded interval, 0 < x < L, with a quenching-type boundary condition at one end, u (0, t) = (T − t)1/(1 − m) and a zero-flux boundary condition at the other, (um −1ux)(L, t) = 0. We prove that for m ≥ −1 regional quenching is not possible: the quenching set is either a single point or the whole interval. Conversely, if m < −1 single-point quenching is impossible, and quenching is either regional or global. For some lengths the above facts depend on the initial data. The results are obtained by studying the corresponding blow-up problem for the variable v = um −1.

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.

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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