• East Asian mathematical journal
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• 제39권1호
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• pp.87-92
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• 2023

In this paper, we study the nonlinear hyperbolic system with epsilon perturbation s_tt - div[c(x)∇s] + (ε - 1)s_t = |s|^γs. Under the conditions of c, γ and other assumptions, we prove the exponential behavior rates of the modified energy perturbation.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제9권1호
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• pp.19-30
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• 2002

We consider one class of bursting oscillation models, that is square-wave burster. One of the interesting features of these models is that periodic bursting solution need not to be unique or stable for arbitrarily small values of a singular perturbation parameter $\epsilon$. Recent results show that the bursting solution is uniquely determined and stable for most of the ranges of the small parameter $\epsilon$. In this paper, we present a condition of uniqueness and stability of periodic bursting solutions for all sufficiently small values of $\epsilon$ > 0.

• 대한수학회지
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• 제52권6호
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• pp.1149-1159
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• 2015

This paper is concerned with a generalized 2D parabolic equation with a nonautonomous perturbation $$-{\Delta}u_t+{\alpha}^2{\Delta}^2u_t+{\mu}{\Delta}^2u+{\bigtriangledown}{\cdot}{\vec{F}}(u)+B(u,u)={\epsilon}g(x,t)$$. Under some proper assumptions on the external force term g, the upper semicontinuity of pullback attractors is proved. More precisely, it is shown that the pullback attractor $\{A_{\epsilon}(t)\}_{t{\epsilon}{\mathbb{R}}}$ of the equation with ${\epsilon}>0$ converges to the global attractor A of the equation with ${\epsilon}=0$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제17권4호
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• pp.267-278
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• 2013

We consider zero range processes with density dependent jump rates g given by $g=g(n,k)=g_1(n)g_2(k/n)$ with $g_1(x)=x^{-\alpha}$ and $$g_2(x)=\{^{x^{-\alpha}\;if\;a&lt;x}_{Mx^{-\alpha}\;if\;x{\leq}a}$$. (0.1) In this case, with 1/2 < a < 1 and ${\alpha}$ > 0, we show that non-complete condensation occurs with maximum cluster size an. More precisely, for any ${\epsilon}$ > 0, there exists $M^*$ > 0 such that, for any 0 < M ${\leq}M^*$, the maximum cluster size is between (a - ${\epsilon}$)n and (a + ${\epsilon}$)n for large n. This provides a simple example of non-complete condensation under perturbation of rates which are deep in the range of perfect condensation (e.g. ${\alpha}$ >> 1) and supports the instability of the condensation transition.

• 대한기계학회논문집
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• 제17권12호
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• pp.3208-3216
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• 1993

Heat transfer characteristics of the laminar oscillating flow in a circular pipe have been studied under the condition that the wall temperature of the pipe is distributed sinusoidally with the axial direction. The axial velocity was assumed to be uniform in radial direction and the temperature field was analyzed by means of the perturbation method. The results show that the difference between wall and section-time-averaged fluid temperature increases as the oscillating frequency increases and eventually converges to a constant value which is determined by the ratio of swept distance to the characteristic length of wall temperature distribution. Also it is shown that the dominant variable in the heat transfer process when swept distance ratio is greater than 1 is not thermal Womersley number(F) but thermal Womersley number multiplied by the square root of swept distance ratio. The variation of the time-averaged Nusselt number is obtained as a function of F. The results indicate that Nusselt number is proportional to $F_{\epsilon}^{1/2}$ when both of F and .epsilon. are much greater than 1.