scholarly journals Global dynamics of solutions for a sixth-order parabolic equation describing continuum evolution of film-free surface

2022 ◽  
Vol 27 (1) ◽  
pp. 19-37
Author(s):  
Ning Duan ◽  
Xiaopeng Zhao
Keyword(s):  
Free Surface ◽  
Parabolic Equation ◽  
Diffusion Equation ◽  
Existence Theorem ◽  
Global Attractor ◽  
Global Attractors ◽  
Sixth Order ◽  
Global Dynamics ◽  
Regularity Estimates ◽  
Iteration Technique

This paper is concerned with a sixth-order diffusion equation, which describes continuum evolution of film-free surface. By using the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors we verified the existence of global attractor for this surface diffusion equation in the spaces H3(Ω) and fractional-order spaces Hk(Ω), where 0 ≤ k < ∞.

scholarly journals Regularity of Global Attractor for the Reaction-Diffusion Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Hong Luo
Keyword(s):  
Sobolev Space ◽  
Diffusion Equation ◽  
Existence Theorem ◽  
Global Attractor ◽  
Reaction Diffusion ◽  
Iteration Procedure ◽  
Reaction Diffusion Equation ◽  
Bounded Subset ◽  
Regularity Estimates ◽  
Regularity Of Global Attractor

By using an iteration procedure, regularity estimates for the linear semigroups, and a classical existence theorem of global attractor, we prove that the reaction-diffusion equation possesses a global attractor in Sobolev spaceHkfor allk>0, which attracts any bounded subset ofHk(Ω) in theHk-norm.

Global Attractor of the Liquid Helium-4 System in Hk Space

2013 ◽  
Vol 444-445 ◽  
pp. 731-737
Author(s):  
Zhi Bo Hou ◽  
Li Mei Li
Keyword(s):  
Existence Theorem ◽  
Liquid Helium ◽  
Global Attractor ◽  
Iteration Procedure ◽  
Helium 4 ◽  
Bounded Set ◽  
Regularity Estimates

In this paper, by using an iteration procedure, regularity estimates of the linear semi-groups and a generalized existence theorem of global attractor, we prove that the liquid helium-4 system possesses a global attractor in space for all , which attracts any bounded set of in the-norm.

scholarly journals Global dynamics of a fourth-order parabolic equation describing crystal surface growth

2019 ◽  
Vol 24 (2) ◽  
pp. 159-175 ◽  
Author(s):  
Ning Duan ◽  
Xianyun Xu
Keyword(s):  
Parabolic Equation ◽  
Global Attractor ◽  
Fourth Order ◽  
Crystal Surface ◽  
Surface Growth ◽  
Global Dynamics ◽  
Initial Value ◽  
Order Parabolic Equation ◽  
Fourth Order Parabolic Equation

In this paper, we study the global dynamics for the solution semiflow of a fourth-order parabolic equation describing crystal surface growth. We show that the equation has a global attractor in H4per(Ω) when the initial value belongs to H1per(Ω).

scholarly journals ASYMPTOTIC DYNAMICS OF A NEW MECHANOCHEMICAL MODEL IN BIOLOGICAL PATTERNS

2017 ◽  
Vol 22 (2) ◽  
pp. 252-269 ◽  
Author(s):  
Aibo Liu ◽  
Changchun Liu
Keyword(s):  
Boundary Conditions ◽  
Existence Theorem ◽  
Bounded Domain ◽  
Space Dimension ◽  
Neumann Boundary ◽  
Global Attractors ◽  
Regularity Estimates ◽  
Asymptotic Dynamics ◽  
Mechanochemical Model ◽  
Biological Patterns

In this paper, we prove the existence of attractor for a new mechanochemical model with Neumann boundary conditions on a bounded domain of space dimension n ≤ 3. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the mechanochemical model possesses a global attractor and Hk attractor.

scholarly journals Attractors for Nonautonomous Parabolic Equations without Uniqueness

2010 ◽  
Vol 2010 ◽  
pp. 1-17
Author(s):  
Cung The Anh ◽  
Nguyen Dinh Binh
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Global Existence ◽  
Global Attractor ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equation ◽  
Global Attractors ◽  
Degenerate Parabolic ◽  
Uniform Global Attractor

Using the theory of uniform global attractors of multivalued semiprocesses, we prove the existence of a uniform global attractor for a nonautonomous semilinear degenerate parabolic equation in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness. The Kneser property of solutions is also studied, and as a result we obtain the connectedness of the uniform global attractor.

scholarly journals Global Attractors for the Higher-Order Evolution Equation

2020 ◽  
Vol 5 (1) ◽  
pp. 195-210
Author(s):  
Erhan Pişkin ◽  
Hazal Yüksekkaya
Keyword(s):  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Global Solution ◽  
Global Attractor ◽  
Type Equation ◽  
Global Attractors ◽  
Higher Order ◽  
Evolution Type

AbstractIn this paper, we obtain the existence of a global attractor for the higher-order evolution type equation. Moreover, we discuss the asymptotic behavior of global solution.

Regularity Estimates for Nondivergence Parabolic Equations on Generalized Orlicz Spaces

2020 ◽  
Author(s):  
The Anh Bui
Keyword(s):  
Parabolic Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Harmonic Analysis ◽  
Parabolic Equations ◽  
Orlicz Spaces ◽  
Representation Theorems ◽  
New Approach ◽  
Analysis Techniques ◽  
Regularity Estimates

Abstract In this paper, by using a new approach, we prove regularity estimates for the solution to the nondivergence parabolic equation on generalized Orlicz spaces. Our approach can be viewed as a combination of representation theorems in partial differential equations and harmonic analysis techniques.

Global Dynamics of a Parasite-Host Model with Nonlinear Incidence Rate

2015 ◽  
Vol 25 (08) ◽  
pp. 1550102 ◽  
Author(s):  
Yilei Tang
Keyword(s):  
Incidence Rate ◽  
Global Attractor ◽  
Exponential Function ◽  
Limit Cycles ◽  
Hopf Bifurcations ◽  
Global Dynamics ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Dynamical Behaviors ◽  
Fractional Function

The paper is concerned with the effect of a nonlinear incidence rate Sp Iq on dynamical behaviors of a parasite-host model. It is shown that the global attractor of the parasite-host model is an equilibrium if q = 1, which is similar to that of the parasite-host model with a nonlinear incidence rate of the fractional function [Formula: see text]. However, when q is greater than one, more positive equilibria appear and limit cycles arise from Hopf bifurcations at the positive equilibria for the model with the incidence rate Sp Iq. It reveals that the nonlinear incidence rate of the exponential function Sp Iq for generic p and q can lead to more complicated and richer dynamics than the bilinear incidence rate or the fractional incidence rate for this model.

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