• 대한수학회보
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• 제52권4호
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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.

• 대한수학회지
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• 제57권6호
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• pp.1347-1372
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• 2020

We study bifurcation for the following fractional Schrödinger equation $$\{\left.\begin{eqnarray}(-{\Delta})^su+V(x)u&=&{\lambda}f(u)&&{\text{in}}\;{\Omega}\\u&>&0&&{\text{in}}\;{\Omega}\\u&=&0&&{\hspace{32}}{\text{in}}\;{\mathbb{R}}^n{\backslash}{\Omega}\end{eqnarray}\right$$ where 0 < s < 1, n > 2s, Ω is a bounded smooth domain of ℝⁿ, (-∆)^s is the fractional Laplacian of order s, V is the potential energy satisfying suitable assumptions and λ is a positive real parameter. The nonlinear term f is a positive nondecreasing convex function, asymptotically linear that is $\lim_{t{\rightarrow}+{\infty}}\;{\frac{f(t)}{t}}=a{\in}(0,+{\infty})$. We discuss the existence, uniqueness and stability of a positive solution and we also prove the existence of critical value and the uniqueness of extremal solutions. We take into account the types of Bifurcation problem for a class of quasilinear fractional Schrödinger equations, we also establish the asymptotic behavior of the solution around the bifurcation point.

• 전자공학회논문지S
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• 제34S권9호
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• pp.109-117
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• 1997

A new relaxation algorithm based on distribution of matched errors and possibility is proposed to solve efficiently correspondence problem. This algorithm can be applied to various method, such as BMA, feature-, and region-based matching methods, by modifying its smoothness function. It consists of two stages which are transformation and iteration process. In transformation stage, the errors obtained by any matching algorithm are transformed to possibility values according to these statistical distribution. Each grade of possility is updated by some constraints which are defined as smoothness, uniqueness, and discontinuity factor in iteration stage. The discontinuity factor is used to reserve discontinuity of disparity. In conventional methods, it is difficult to find proper weights and stop condition, because only two factors, smoothness and uniqueness, have been used. However, in the proposed mthod, the more smoothing is not ocurred because of discontinuity factor. And it is efective to the various image, even if the image has a severe noise and repeating patterns. In addition, it is shown that the convergence rate and the quality of output are improved.

• 대한수학회보
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• 제56권1호
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• pp.245-251
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• 2019

Assume that ${\Omega}$ is a bounded domain in ${\mathbb{R}}^n$ with $n{\geq}2$. We study positive solutions to the problem, ${\Delta}u=u^p$ in ${\Omega}$, $u(x){\rightarrow}{\infty}$ as $x{\rightarrow}{\partial}{\Omega}$, where p > 1. Such solutions are called boundary blow-up solutions of ${\Delta}u=u^p$. We show that a boundary blow-up solution exists in any bounded domain if 1 < p < ${\frac{n}{n-2}}$. In particular, when n = 2, there exists a boundary blow-up solution to ${\Delta}u=u^p$ for all $p{\in}(1,{\infty})$. We also prove the uniqueness under the additional assumption that the domain satisfies the condition ${\partial}{\Omega}={\partial}{\bar{\Omega}}$.

• 대한수학회보
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• 제35권2호
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• pp.325-338
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• 1998

Some nonlocal boundary value problems which arise from a feedback control problem are considered. We give a precise statement of the mathematical problems and then prove the existence and uniqueness of the solutions. We consider the Dirichlet type boundary value problem and the Neumann type boundary value problem with nonlinear boundary conditions. We also provide a regularity results for the solutions.

• Kyungpook Mathematical Journal
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• 제62권4호
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• pp.615-640
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• 2022

In this paper, we investigate a time-dependent double obstacle problem associated with the model of European call option pricing with transaction costs. We prove the existence and uniqueness of a W^2,1_p,loc solution to the problem. We then characterize the behavior of the free boundaries in terms of continuity and values of limit points.

• Kyungpook Mathematical Journal
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• 제23권2호
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• pp.193-201
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• 1983

• 대한수학회지
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• 제31권2호
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• pp.245-254
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• 1994

• Kyungpook Mathematical Journal
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• 제27권1호
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• pp.43-46
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• 1987

In this paper we consider the semilinear hyperbolic symmetric system of the first-order. The existence and uniqueness of the solution are proved, under certain conditions, some properties of the solution are investigated.

• Korean Journal of Mathematics
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• 제19권2호
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• pp.111-127
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• 2011

We prove the uniqueness solvability of the Tricomi problem for the elliptic - hyperbolical equation of the second type by using a new representation of the solution in the generalized class R.