• Honam Mathematical Journal
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• v.15 no.1
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• pp.105-120
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• 1993

The auxiliary principle technique is used to prove the uniqueness and the existence of solutions for a class of nonlinear variational inequalities and suggest an innovative iterative algorithm for computing the approximate solution of variational inequalities. Error estimates for the finite element approximation of the solution of variational inequalities are derived, which refine the previous known results. An example is given to illustrate the applications of the results obtained. Several special cases are considered and studied.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.143-154
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• 2003

Analytical solutions for the viscous Cahn-Hilliard equation are considered. Existence and uniqueness of the solution are shown. The exponential decay of the solution in H$^2$-norm, which is an improvement of the result in Elliott and Zheng[5]. We also compare the early stages of evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation, which has been given as an open question in Novick-Cohen[8].

• Journal of the Korean Mathematical Society
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• v.41 no.4
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• pp.747-754
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• 2004

In this paper we investigate the existence and uniqueness of almost periodic and pseudo almost periodic solution for nonlinear differential equation with reflection of argument. For the case of almost periodic forced term, we consider the frequency modules of the solutions.

• Bulletin of the Korean Mathematical Society
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• v.47 no.1
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• pp.81-87
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• 2010

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem(BVP) $-D_0^{\alpha}+u(t)=\lambda[f(t, u(t))+q(t)]$, 0 < t < 1 u(0) = u(1) = 0, where $\lambda$ > 0 is a parameter, 1 < $\alpha$ $\leq$ 2, $D_{0+}^{\alpha}$ is the standard Riemann-Liouville differentiation, f : [0, 1] ${\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is continuous, and q(t) : (0, 1) $\rightarrow$ [0, $+\infty$] is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when $\lambda$ in some interval. Our approach is based on Leray-Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity which was essential for the technique used in almost all existed literature on f.

• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1365-1388
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• 2019

In this paper we consider a class of nonlinear nonlocal parabolic equations involving p-Laplacian operator where the nonlocal quantity is present in the diffusion coefficient which depends on $L^p$-norm of the gradient and the nonlinear term is of polynomial type. We first prove the existence and uniqueness of weak solutions by combining the compactness method and the monotonicity method. Then we study the existence of global attractors in various spaces for the continuous semigroup generated by the problem. Finally, we investigate the existence and exponential stability of weak stationary solutions to the problem.

• Communications of the Korean Mathematical Society
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• v.36 no.3
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• pp.447-463
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• 2021

In this paper we study the existence and long-time behavior of weak solutions to a class of quasilinear degenerate parabolic equations involving weighted p-Laplacian operators with a new class of nonlinearities. First, we prove the existence and uniqueness of weak solutions by combining the compactness and monotone methods and the weak convergence techniques in Orlicz spaces. Then, we prove the existence of global attractors by using the asymptotic a priori estimates method.

• Korean Journal of Mathematics
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• v.30 no.4
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• pp.571-592
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• 2022

By established a Banach space with the Hausdorff distance, we introduce the alternative fixed-point theorem to explore the existence and uniqueness of a fixed subset of Y and investigate the stability of set-valued Euler-Lagrange functional equations in this space. Some properties of the Hausdorff distance are furthermore explored by a short and simple way.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.1003-1031
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• 2020

We consider the following Dirichlet initial boundary value problem with a gradient absorption and a nonlocal source $$\frac{{\partial}u}{{\partial}t}-div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)={\lambda}u^k{\displaystyle\smashmargin{2}{\int\nolimits_{\Omega}}}u^sdx-{\mu}u^l{\mid}{\nabla}u{\mid}^q$$ in a bounded domain Ω ⊂ ℝ^N, where p > 1, the parameters k, s, l, q, λ > 0 and µ ≥ 0. Firstly, we establish local existence for weak solutions; the aim of this part is to prove a crucial priori estimate on |∇u|. Then, we give appropriate conditions in order to have existence and uniqueness or nonexistence of a global solution in time. Finally, depending on the choices of the initial data, ranges of the coefficients and exponents and measure of the domain, we show that the non-negative global weak solution, when it exists, must extinct after a finite time.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.945-966
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• 2017

The paper develops a rigorous mathematical framework for the behavior of arch and membrane like structures. Our main goal is to incorporate moving point loads. Both the weak and the strong damping cases are considered. First, we prove the existence and the uniqueness of the solutions. Then it is shown that the solution in the weak damping case is the limit of the strong damping solutions, as the strong damping vanishes. The theory is applied to a car moving on a bridge.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1411-1425
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• 2016

We provide a partial differential equation for European options on a stock whose price process follows a general geometric Riemannian Brownian motion. The existence and the uniqueness of solutions to the partial differential equation are investigated, and then an expression of the value for European options is obtained using the fundamental solution technique. Proper Riemannian metrics on the real number field can make the distribution of return rates of the stock induced by our model have the character of leptokurtosis and fat-tail; in addition, they can also explain option pricing bias and implied volatility smile (skew).