scholarly journals Fractal Dimension of a Random Invariant Set and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Gang Wang ◽  
Yanbin Tang
Keyword(s):  
Fractal Dimension ◽  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Invariant Set ◽  
Invariant Sets ◽  
Abstract Result ◽  
Random Attractors ◽  
Finite Fractal Dimension ◽  
Degenerate Parabolic ◽  
Random Invariant Set

We prove an abstract result on random invariant sets of finite fractal dimension. Then this result is applied to a stochastic semilinear degenerate parabolic equation and an upper bound is obtained for the random attractors of fractal dimension.

scholarly journals REGULARITY AND FRACTAL DIMENSION OF PULLBACK ATTRACTORS FOR A NON-AUTONOMOUS SEMILINEAR DEGENERATE PARABOLIC EQUATION

2013 ◽  
Vol 55 (2) ◽  
pp. 431-448 ◽  
Author(s):  
CUNG THE ANH ◽  
TANG QUOC BAO ◽  
LE THI THUY
Keyword(s):  
Fractal Dimension ◽  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Pullback Attractor ◽  
Formula Group ◽  
Degenerate Parabolic ◽  
Image Position ◽  
Initial Boundary Value

AbstractConsidered here is the pullback attractor of the process associated with the first initial boundary value problem for the non-autonomous semilinear degenerate parabolic equation \begin{linenomath} u_t-\text{div}(\sigma(x)\nabla u)+f(u)=g(x,t) \end{linenomath} in a bounded domain Ω in ℝN (N≥2). We prove the regularity in the space L2p−2(Ω)∩ $D_0^2(\Omega,\sigma)$, and estimate the fractal dimension of the pullback attractor in L2(Ω).

scholarly journals On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Parabolic Equation ◽  
Newtonian Fluid ◽  
Weak Solutions ◽  
Mixed Type ◽  
Degenerate Parabolic Equation ◽  
Type Equation ◽  
Existence Of Weak Solutions ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Increasing Function

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

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