• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1573-1594
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• 2017

This paper deals with the existence and energy estimates of solutions for a class of degenerate nonlocal problems involving sub-linear nonlinearities, while the nonlinear part of the problem admits some hypotheses on the behavior at origin or perturbation property. In particular, for a precise localization of the parameter, the existence of a non-zero solution is established requiring the sublinearity of nonlinear part at origin and infinity. We also consider the existence of solutions for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz. In what follows, by combining two algebraic conditions on the nonlinear term which guarantees the existence of two solutions as well as applying the mountain pass theorem given by Pucci and Serrin, we establish the existence of the third solution for our problem. Moreover, concrete examples of applications are provided.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.811-821
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• 2013

We study the initial value problem of the exponential wave equation in $\math{R}^{n+1}$ for small initial data. We shows, in the case of $n=1$, the global existence of solution by applying the formulation of first order quasilinear hyperbolic system which is weakly linearly degenerate. When $n{\geq}2$, a vector field method is applied to show the stability of a trivial solution ${\phi}=0$.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.265-291
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• 2012

In this paper, we consider the regularity theory and the existence of smooth solution of a degenerate fully nonlinear equation describing the evolution of the rolling stones with nonconvex sides: $\{M(h)=h_t-F(t,z,z^{\alpha}h_{zz})\;in\;\{0<z{\leq}1\}{\times}[0,T] \\ h_t(z,t)=H(h_z(z,t),h)\;{on}\;\{z=0\}$. We establish the Schauder theory for $C^{2,{\alpha}}$-regularity of h.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1697-1705
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• 2016

In this article, we consider the problem on finding non-degenerate nary m-ic forms having an $n{\times}n$ matrix A as a linear isomorphism. We show that it is equivalent to solve a linear diophantine equation. In particular, we find all integral ternary cubic forms having A as a linear isomorphism, for any $A{\in}GL_3({\mathbb{Z}})$. We also give a family of non-degenerate cubic forms F such that F(x) = N always has infinitely many integer solutions if exists.

• The Transactions of the Korea Information Processing Society
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• v.1 no.3
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• pp.409-417
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• 1994

In extracting contours from elevation matrix, the most important problem is to solve the degenerate case where four cross point arise in a grid. This paper describes a technique which performs the checking of degenerate case and determination of next point simultaneously. It requires minimum number of array indexing. Also this paper proposes a technique which reduces the number of array indexing by using LOWER/HIGHER information of a cross point designated according to height difference of grid vertices. In addition, we describe a data structure which is proper for representing cross points and tracing them.

• 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.

• Journal of the Korean Institute of Navigation
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• v.24 no.5
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• pp.361-371
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• 2000

This paper treats a genetic algorithm for ship scheduling problem in set packing formulation. We newly devised a partition based representation of solution and compose initial population using a domain knowledge of problem which results in saving calculation cost. We established replacement strategy which makes each individual not to degenerate during evolutionary process and applied adaptive mutate operator to improve feasibility of individual. If offspring is feasible then an improve operator is applied to increase objective value without loss of feasibility. A computational experiment was carried out with real data and showed a useful result for a large size real world problem.

• Kyungpook Mathematical Journal
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• v.60 no.1
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• pp.145-161
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• 2020

In this article, we study the solvability of the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations with nonstandard growth and nonlocal terms. We prove the existence of weak solutions of the considered problem under more general conditions. In addition, we investigate the behavior of the solution when the problem is homogeneous.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.671-676
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• 2018

In this paper we prove that the gradient ideal of a Morse polynomial is radical. This gives a generic class of polynomials whose gradient ideals are radical. As a consequence we reclaim a previous result that the unconstrained polynomial optimization problem for Morse polynomials has a finite convergence.

• Korean Journal of Mathematics
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• v.12 no.1
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• pp.91-106
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• 2004

We prove the regularity of a moving interface of the solutions of the initial value problem of equation (1.1) is $C^{\infty}$.