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 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.

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 → ∞.

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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