Dynamics near extinction time of a singular diffusion equation

2002 ◽  
Vol 323 (2) ◽  
pp. 281-318 ◽  
Author(s):  
Shu-Yu Hsu
Keyword(s):  
Diffusion Equation ◽  
Extinction Time ◽  
Singular Diffusion ◽  
Near 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.

On the Risk of Extinction for a Population Subject to a Random Markov Evolution with Jumps

2004 ◽  
Vol 11 (02) ◽  
pp. 105-121 ◽  
Author(s):  
Mario Abundo
Keyword(s):  
Diffusion Coefficient ◽  
Diffusion Equation ◽  
Extinction Risk ◽  
Jump Diffusion ◽  
Extinction Time ◽  
Simple Diffusion ◽  
One Dimensional ◽  
Risk Of Extinction ◽  
Lower Population ◽  
Markov Evolution

We consider a one-dimensional population whose evolution is described by a jump-diffusion equation and we study the effects of changing the coefficients of the equation on the extinction time, that is the instant at which the population becomes arbitrarily small. It is shown that, under the same diffusion coefficient, if one reduces the drift and the size of jumps, the speed of extinction increases; moreover, the probability of reaching a higher population state than the present one before reaching a lower population size decreases. If the diffusion coefficient is state-independent, the speed of extinction increases with it. Furthermore, if no jumps are allowed (i.e. for a simple-diffusion equation), then under certain conditions on the coefficients of the equation both large and small values of the diffusion coefficient result in a higher extinction risk.

scholarly journals On a singular diffusion equation

1995 ◽  
Vol 3 (3) ◽  
pp. 523-542 ◽  
Author(s):  
Panagiota Daskalopoulos ◽  
Manuel A. del Pino
Keyword(s):  
Diffusion Equation ◽  
Singular Diffusion
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