scholarly journals Non-Self-Similar Dead-Core Rate for the Fast Diffusion Equation with Dependent Coefficient

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Liping Zhu ◽  
Zhengce Zhang
Keyword(s):  
Diffusion Equation ◽  
Single Point ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Dead Core ◽  
Core Profile ◽  
Self Similar ◽  
Similar Solutions ◽  
Compactness Arguments ◽  
Core Problem

We consider the dead-core problem for the fast diffusion equation with spatially dependent coefficient and show that the temporal dead-core rate is non-self-similar. The proof is based on the standard compactness arguments with the uniqueness of the self-similar solutions and the precise estimates on the single-point final dead-core profile.

scholarly journals Spatial Profile of the Dead Core for the Fast Diffusion Equation with Dependent Coefficient

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Zhengce Zhang ◽  
Biao Wang
Keyword(s):  
Maximum Principle ◽  
Diffusion Equation ◽  
Single Point ◽  
Fast Diffusion ◽  
Spatial Profile ◽  
Fast Diffusion Equation ◽  
Dead Core ◽  
Core Profile ◽  
The Dead ◽  
Core Problem

We consider the dead-core problem for the fast diffusion equation with spatially dependent coefficient and obtain precise estimates on the single-point final dead-core profile. The proofs rely on maximum principle and require much delicate computation.

Non-self-similar dead-core rate for the fast diffusion equation with strong absorption

Nonlinearity ◽  
2010 ◽  
Vol 23 (3) ◽  
pp. 657-673 ◽  
Author(s):  
Jong-Shenq Guo ◽  
Chia-Tung Ling ◽  
Philippe Souplet
Keyword(s):  
Diffusion Equation ◽  
Strong Absorption ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Dead Core ◽  
Self Similar

scholarly journals Similarity solutions for a quasilinear parabolic equation

1995 ◽  
Vol 37 (2) ◽  
pp. 253-266 ◽  
Author(s):  
Jong-Sheng Guo
Keyword(s):  
Diffusion Equation ◽  
Quasilinear Parabolic Equation ◽  
Similarity Solutions ◽  
Fast Diffusion ◽  
Equation Approach ◽  
Fast Diffusion Equation ◽  
Self Similar ◽  
Similar Solutions ◽  
Quasilinear Parabolic ◽  
Slow Diffusion Equation

AbstractIn this paper, we use an ordinary differential equation approach to study the existence of similarity solutions for the equation u1 = Δ(uα) + θu–β in Rn × (0, ∞) where β > 0, θ ∈ [0, 1}, and n ≥ 1. This includes the slow diffusion equation when α > = 1, and the standard heat equation when α = 1, and the fast diffusion equation when 0 < α < 1. We prove that there are forward self-similar solutions for this equation with initial data of the form c|x|p, where p = 2/(α + β) if θ = 1; p ≥ 0 and 2 + (1 – α)p > 0 if θ = 0, for some positive constant c.

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.

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 Fast rate of dead core for fast diffusion equation with strong absorption

2010 ◽  
Vol 9 (2) ◽  
pp. 397-411
Author(s):  
Chunlai Mu ◽  
◽  
Jun Zhou ◽  
Yuhuan Li ◽  
◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Fast Rate ◽  
Strong Absorption ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Dead Core
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