The long-time behavior of the solution to a non-linear diffusion problem in population genetics and combustion

1983 ◽  
Vol 104 (4) ◽  
pp. 473-484 ◽  
Author(s):  
William I. Newman
Keyword(s):  
Population Genetics ◽  
Diffusion Problem ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Linear Diffusion ◽  
Non Linear ◽  
Long Time

scholarly journals Long-time behavior of solutions for a fractional diffusion problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ailing Qi ◽  
Die Hu ◽  
Mingqi Xiang
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Attractors ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Initial Boundary Value

AbstractThis paper deals with the asymptotic behavior of solutions to the initial-boundary value problem of the following fractional p-Kirchhoff equation: $$ u_{t}+M\bigl([u]_{s,p}^{p}\bigr) (-\Delta )_{p}^{s}u+f(x,u)=g(x)\quad \text{in } \Omega \times (0, \infty ), $$ u t + M ( [ u ] s , p p ) ( − Δ ) p s u + f ( x , u ) = g ( x ) in  Ω × ( 0 , ∞ ) , where $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N is a bounded domain with Lipschitz boundary, $N>ps$ N > p s , $0< s<1<p$ 0 < s < 1 < p , $M:[0,\infty )\rightarrow [0,\infty )$ M : [ 0 , ∞ ) → [ 0 , ∞ ) is a nondecreasing continuous function, $[u]_{s,p}$ [ u ] s , p is the Gagliardo seminorm of u, $f:\Omega \times \mathbb{R}\rightarrow \mathbb{R}$ f : Ω × R → R and $g\in L^{2}(\Omega )$ g ∈ L 2 ( Ω ) . With general assumptions on f and g, we prove the existence of global attractors in proper spaces. Then, we show that the fractal dimensional of global attractors is infinite provided some conditions are satisfied.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

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