• 대한수학회논문집
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• 제20권4호
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• pp.731-749
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• 2005

In this paper, we study the two dimensional g-Navier­Stokes equations on some unbounded domain ${\Omega}\;{\subset}\;R^2$. We prove the existence of the global attractor for the two dimensional g-Navier­Stokes equations under suitable conditions. Also, we estimate the dimension of the global attractor. For this purpose, we exploit the concept of asymptotic compactness used by Rosa for the usual Navier-Stokes equations.

• East Asian mathematical journal
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• 제32권3호
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• pp.377-385
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• 2016

Global attractor is a basic concept to study the long-time behavior of solutions of the various equations. This paper is investigated with the existence of a global attractor for the beam equation $$u_{tt}+{\Delta}^2u-{\nabla}{\cdot}\{{\sigma}({\mid}{\nabla}u{\mid}^2){\nabla}u\}+f(u)+a(x)g(u_t)=h,$$ using multipliers technique and Nakao's Lemma.

• 대한수학회보
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• 제50권1호
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• pp.143-159
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• 2013

This paper is concerned with the long time behavior for the solution semiflow of the coupled two-compartment Gray-Scott equations with the homogeneous Neumann boundary condition on a bounded domain of space dimension $n{\leq}3$. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the equations possesses a global attractor in $H^k({\Omega})^4$ ($k{\geq}0$) space.

• 대한수학회논문집
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• 제32권3호
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• pp.579-600
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• 2017

In this paper we consider a semilinear pseudoparabolic equation with polynomial nonlinearity and infinite delay. We first prove the existence and uniqueness of weak solutions by using the Galerkin method. Then, we prove the existence of a compact global attractor for the continuous semigroup associated to the equation. The existence and exponential stability of weak stationary solutions are also investigated.

• 충청수학회지
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• 제26권3호
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• pp.631-639
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• 2013

We investigate the existence of a global attractor for the Cauchy problem $$x^{\prime}(t)=Ax(t)+F(x(t)),\;x(0)=x_0{\in}X$$ on a Banach space X according to the remark in You and Yuan's paper.

• 대한수학회지
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• 제49권2호
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• pp.357-378
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• 2012

In this paper, we consider the convective Cahn-Hilliard equation. Based on the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors, we prove that the convective Cahn-Hilliard equation possesses a global attractor in $H^k$($k\geq0$) space, which attracts any bounded subset of $H^k({\Omega})$ in the $H^k$-norm.

• 대한수학회논문집
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• 제37권2호
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• pp.423-443
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• 2022

We study the existence and long-time behavior of weak solutions to a class of strongly degenerate semilinear parabolic equations with exponential nonlinearities on ℝ^N. To overcome some significant difficulty caused by the lack of compactness of the embeddings, the existence of a global attractor is proved by combining the tail estimates method and the asymptotic a priori estimate method.

• 대한수학회지
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• 제58권5호
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• pp.1279-1298
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• 2021

The aim of this paper is to prove the existence of the global attractor of the Cauchy problem for a semilinear degenerate hyperbolic equation involving strongly degenerate elliptic differential operators. The attractor is characterized as the unstable manifold of the set of stationary points, due to the existence of a Lyapunov functional.

• 대한수학회논문집
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• 제36권3호
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• pp.527-548
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• 2021

In this paper we prove the existence of global weak solutions, the exponential stability of a stationary solution and the existence of a global attractor for the three-dimensional convective Brinkman-Forchheimer equations with finite delay and fast growing nonlinearity in bounded domains with homogeneous Dirichlet boundary conditions.

• 대한수학회보
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• 제37권2호
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• pp.273-283
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• 2000

In this paper we consider two delay equations with in-finite delay. We will give two sufficient conditions for the positive and zero equilibriums of these equations to be a global attractor respectively.