scholarly journals On the Well Posedness and Refined Estimates for the Global Attractor of the TYC Model

2010 ◽  
Vol 2010 (1) ◽  
pp. 405816 ◽  
Author(s):  
RanaD Parshad ◽  
JuanB Gutierrez
Keyword(s):  
Global Attractor ◽  
Well Posedness

scholarly journals Existence and approximation of attractors for nonlinear coupled lattice wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lianbing She ◽  
Mirelson M. Freitas ◽  
Mauricio S. Vinhote ◽  
Renhai Wang
Keyword(s):  
Asymptotic Behavior ◽  
Global Attractor ◽  
Wave Equations ◽  
Upper Semicontinuity ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Asymptotic Compactness ◽  
Finite Dimensional ◽  
Lattice Wave ◽  
Infinite Lattices

<p style='text-indent:20px;'>This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in <inline-formula><tex-math id="M1">\begin{document}$ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $\end{document}</tex-math></inline-formula>. We then prove that the solution semigroup has a unique global attractor in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula>. We finally prove that this attractor can be approximated in terms of upper semicontinuity of <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula> by a finite-dimensional global attractor of a <inline-formula><tex-math id="M4">\begin{document}$ 2(2n+1) $\end{document}</tex-math></inline-formula>-dimensional truncation system as <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.</p>

ASYMPTOTIC DYNAMICS OF 2D FRACTIONAL COMPLEX GINZBURG–LANDAU EQUATION

2013 ◽  
Vol 23 (12) ◽  
pp. 1350202 ◽  
Author(s):  
HONG LU ◽  
SHUJUAN LÜ ◽  
ZHAOSHENG FENG
Keyword(s):  
Global Attractor ◽  
Landau Equation ◽  
Fractal Dimensions ◽  
Well Posedness ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Spatial Dimensions ◽  
Asymptotic Dynamics ◽  
Hausdorff And Fractal Dimensions ◽  
The Galerkin Method

In this paper, we consider the well-posedness and asymptotic behaviors of solutions of the fractional complex Ginzburg–Landau equation with the initial and periodic boundary conditions in two spatial dimensions. We explore the existence and uniqueness of global smooth solution by means of the Galerkin method and establish the existence of the global attractor. The estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractor are presented.

scholarly journals Global well-posedness and attractors for the hyperbolic Cahn–Hilliard–Oono equation in the whole space

2016 ◽  
Vol 26 (07) ◽  
pp. 1357-1384 ◽  
Author(s):  
Anton Savostianov ◽  
Sergey Zelik
Keyword(s):  
Growth Rate ◽  
Schrödinger Equation ◽  
Global Attractor ◽  
Schrodinger Equation ◽  
Energy Space ◽  
Strichartz Estimates ◽  
Well Posedness ◽  
Whole Space

We prove the global well-posedness of the so-called hyperbolic relaxation of the Cahn–Hilliard–Oono equation in the whole space [Formula: see text] with the nonlinearity of the sub-quintic growth rate. Moreover, the dissipativity and the existence of a smooth global attractor in the naturally defined energy space is also verified. The result is crucially based on the Strichartz estimates for the linear Schrödinger equation in [Formula: see text].

scholarly journals Allen–Cahn equation with strong irreversibility

2018 ◽  
Vol 30 (04) ◽  
pp. 707-755
Author(s):  
GORO AKAGI ◽  
MESSOUD EFENDIEV
Keyword(s):  
Energy Dissipation ◽  
Global Attractor ◽  
Comparison Principle ◽  
Obstacle Problem ◽  
Well Posedness ◽  
Smoothing Effect ◽  
Cahn Equation ◽  
Non Linear ◽  
Long Time

This paper is concerned with a fully non-linear variant of the Allen–Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. The main purposes of this paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviours of solutions. More precisely, by derivingpartialenergy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solutionu(x,t) converges to a solution of an elliptic obstacle problem ast→ +∞.

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