• Communications of the Korean Mathematical Society
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• v.29 no.4
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• pp.505-518
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• 2014

Considered here is the first initial boundary value problem for the two-dimensional g-Navier-Stokes equations in bounded domains. We first study the long-time behavior of strong solutions to the problem in term of the existence of a global attractor and global stability of a unique stationary solution. Then we study the long-time finite dimensional approximation of the strong solutions.

• Journal of the Korean Mathematical Society
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• v.58 no.5
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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.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1059-1068
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• 2015

The paper is devoted to studying a fourth-order degenerate parabolic equation, which arises in fluid, phase transformation and biology. Based on the existence and uniqueness of one semi-discrete problem, two types of approximate solutions are introduced. By establishing some necessary uniform estimates for those approximate solutions, the existence and uniqueness of the corresponding parabolic problem are obtained. Moreover, the long time asymptotic behavior is established by the entropy functional method.

• Communications of the Korean Mathematical Society
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• v.28 no.4
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• pp.751-766
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• 2013

We study the existence and long-time behavior of solutions to the following semilinear degenerate parabolic equation on $\mathbb{R}^N$: $$\frac{{\partial}u}{{\partial}t}-div({\sigma}(x){\nabla}u+{\lambda}u+f(u)=g(x)$$, under a new condition concerning a variable non-negative diffusivity ${\sigma}({\cdot})$. Some essential difficulty caused by the lack of compactness of Sobolev embeddings is overcome here by exploiting the tail-estimates method.

• East Asian mathematical journal
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• v.32 no.3
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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.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.4
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• pp.445-468
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• 2020

A long-time behavior of global solutions for a dispersive-dissipative equation with time-dependent damping terms is investigated under null Dirichlet boundary condition. By virtue of an appropriate new Lyapunov function and the Lojasiewicz-Simon inequality, we show that any global bounded solution converges to a steady state and get the rate of convergence as well, when damping coefficients are integrally positive and positive-negative, respectively. Moreover, under the assumptions on on-off or sign-changing damping, we derive an asymptotic stability of solutions.

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.443-459
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• 2006

The prey-predator system with a single cross-diffusion pressure is known to possess a local solution with the maximal existence time $T\;{\leq}\;{\infty}$. By obtaining the bounds of $W\array_2^1$-norms of the local solution independent of T we establish the global existence of the solution. And the long-time behaviors of the global solution are analyzed when the diffusion rates $d_1\;and\;d_2$ are sufficiently large.

• Structural Engineering and Mechanics
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• v.52 no.6
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• pp.1069-1084
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• 2014

The long-term performance of plates resting on viscoelastic foundations is a major concern in the analysis of soil-structure interaction. As a powerful mathematical tool, fractional calculus may address these plate-on-foundation problems. In this paper, a fractionalized Zener model is proposed to study the time-dependent behavior of a uniformly loaded rectangular thin foundation plate. By use of the viscoelastic-elastic correspondence principle and the Laplace transforms, the analytical solutions were obtained in terms of the Mittag-Leffler function. Through the analysis of a numerical example, the calculated plate deflection, bending moment and foundation reaction were compared to those from ideal elastic and standard viscoelastic models. It is found that the upper and lower bound solutions of the plate response estimated by the proposed model can be determined using the elastic model. Based on a parametric study, the impacts of model parameters on the long-term performance of a foundation plate were systematically investigated. The results show that the two spring stiffnesses govern the upper and lower bound solutions of the plate response. By varying the values of the fractional differential order and the coefficient of viscosity, the time-dependent behavior of a foundation plate can be accurately captured. The fractional differential order seems to be dependent on the mechanical properties of the ground soil. A sandy foundation will have a small fractional differential order while in order to simulate the creeping of clay foundation, a larger fractional differential order value is needed. The fractionalized Zener model is capable of accounting for the primary and secondary consolidation processes of the foundation soil and can be used to predict the plate performance over many decades of time.

• Bulletin of the Korean Chemical Society
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• v.32 no.9
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• pp.3443-3447
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• 2011

The spectroscopic behavior of catechin (5,7,3',4'-tetrahydroxyflavan-3-ol), has been studied in the presence and the absence of air using UV-vis absorption spectrophotometry and fluorescence spectroscopy. The UV-vis absorption spectrum of catechin shows a very sharp and strong absorption maximum peak at 275 nm in deaerated water. New absorption maximum peaks appeared in aerated water, as well as in basic aqueous solution, caused by the oxidation of catechin. The absorbances in the UV-vis absorption spectrum of catechin decreased when the solution was left in the dark for a long time. The fluorescence emission spectrum of catechin after a long time period differs markedly from that in freshly prepared solution; the fluorescence maxia shifted as time passes after adding catechin to the solutions. When the deaerated basic catechin solutions were left in the dark for a long time, their fluorescence quantum yields were found to be nearly zero. This suggests that the oxidized catechin molecules were seen to have slowly undergone successive chemical reactions in basic buffer solution.

• Journal of applied mathematics & informatics
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• v.3 no.2
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• pp.169-182
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• 1996

This paper presents the numerical experiment of a dis-cretized loaded cable with periodic forcing. There appeared to be var-ious type of nonlinear oscillations over a wide range of fequencies and amplitudes for the periodic forcing term. The same forcing term can give rise to large or small oscillation by solving initial value problem and observing the solutions after a long time.