• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.11 no.1
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• pp.13-30
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• 2007

In this paper we study the numerical solutions of the stochastic functional differential equations of the following form $$du(x,\;t)\;=\;f(x,\;t,\;u_t)dt\;+\;g(x,\;t,\;u_t)dB(t),\;t\;>\;0$$ with initial data $u(x,\;0)\;=\;u_0(x)\;=\;{\xi}\;{\in}\;L^p_{F_0}\;([-{\tau},0];\;R^n)$. Here $x\;{\in}\;R^n$, ($R^n$ is the ${\nu}\;-\;dimenional$ Euclidean space), $f\;:\;C([-{\tau},\;0];\;R^n)\;{\times}\;R^{{\nu}+1}\;{\rightarrow}\;R^n,\;g\;:\;C([-{\tau},\;0];\;R^n)\;{\times}\;R^{{\nu}+1}\;{\rightarrow}\;R^{n{\times}m},\;u(x,\;t)\;{\in}\;R^n$ for each $t,\;u_t\;=\;u(x,\;t\;+\;{\theta})\;:\;-{\tau}\;{\leq}\;{\theta}\;{\leq}\;0\;{\in}\;C([-{\tau},\;0];\;R^n)$, and B(t) is an m-dimensional Brownian motion.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.137-161
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• 2022

In this article, we study the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation (1) u⁽⁴⁾(x) + ωu''(x) + a(x)u(x) = f(x, u(x)), ∀x ∈ ℝ where a(x) is not required to be either positive or coercive, and F(x, u) = ∫^u₀ f(x, v)dv is of subquadratic or superquadratic growth as |u| → ∞, or satisfies only local conditions near the origin (i.e., it can be subquadratic, superquadratic or asymptotically quadratic as |u| → ∞). To the best of our knowledge, there is no result published concerning the existence and multiplicity of homoclinic solutions for (1) with our conditions. The proof is based on variational methods and critical point theory.

• Communications of the Korean Mathematical Society
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• v.38 no.1
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• pp.137-151
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• 2023

In this paper we prove the existence of nontrivial weak solutions to the boundary value problem -G₁u = u³ + f(x, y, u) in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, where Ω is a bounded domain with smooth boundary in ℝ³, G₁ is a Grushin type operator, and f(x, y, u) is a lower order perturbation of u³ with f(x, y, 0) = 0. The nonlinearity involved is of critical exponent, which differs from the existing results in [11, 12].

• Korean Journal of Mathematics
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• v.21 no.2
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• pp.203-212
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• 2013

We investigate the existence of solutions of the one-dimensional wave system $$u_{tt}-u_{xx}+{\mu}g(u+v)=f(u+v)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,\\v_{tt}-v_{xx}+{\nu}g(u+v)=f(u+v)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,$$ with Dirichlet boundary condition. We find them by applying linking inequlaities.

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.695-706
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• 1996

In the control system $ \dot{x} = f(t,x(t)) + g(t,x(t))\dot{u}, x(0) = \bar{x}, t \in [0,T], $ this paper shows that the map from u with $L^1(m)$-topology to $x_u$ with $L^1(\mu)$-topology is Lipschitz continuous where f is $C^1$, $\mu$ is the Stieltjes measure derived from the function g which is not smooth in the variable t and $x_u$ is the solution of the above system corresponding to u under the assumption that $\dot{u}$ is bounded.

• Korean Journal of Mathematics
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• v.23 no.1
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• pp.153-161
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• 2015

In this work, we consider an elliptic system $$\left{\array {-{\Delta}u=au+bv+{\delta}_1u+-{\delta}_2u^-+f_1(x,u,v) && in\;{\Omega},\\-{\Delta}v=bu+cv+{\eta}_1v^+-{\eta}_2v^-+f_2(x,u,v) && in\;{\Omega},\\{\hfill{70}}u=v=0{\hfill{90}}on\;{\partial}{\Omega},}$$, where ${\Omega}{\subset}R^N$ be a bounded domain with smooth boundary. We prove that the system has at least two nontrivial solutions by applying linking theorem.

• Communications for Statistical Applications and Methods
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• v.12 no.3
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• pp.545-556
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• 2005

Hjort(1990) obtained the nonparametric Bayes estimator $\^{F}_{c,a}$ of $F_0$ with respect to beta processes in the random censorship model. Let $X_1,{\cdots},X_n$ be i.i.d. $F_0$ and let $C_1,{\cdot},\;C_n$ be i.i.d. G. Assume that $F_0$ and G are continuous. This paper shows that {$\^{F}_{c,a}$(u){\|}0 < u < T} converges weakly to a Gaussian process whenever T < $\infty$ and $\~{F}_0({\tau})\;<\;1$.

• Proceedings of the Korean Society of Computational and Applied Mathematics Conference
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• 2003.09a
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• pp.6-6
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• 2003

In this work we consider the mathematical formulation and numerical resolution of the linear feedback control problem for Boussinesq equations. The controlled Boussinesq equations is given by $$\frac{{\partial}u}{{\partial}t}-{\nu}{\Delta}u+(u{\cdot}{\nabla}u+{\nabla}p={\beta}{\theta}g+f+F\;\;in\;(0,\;T){\times}\;{\Omega}$$, $${\nabla}{\cdot}u=0\;\;in\;(0,\;T){\times}{\Omega}$$, $$u|_{{\partial}{\Omega}=0,\;u(0,x)=\;u_0(x)$$ $$\frac{{\partial}{\theta}}{{\partial}t}-k{\Delta}{\theta}+(u{\cdot}){\theta}={\tau}+T,\;\;in(0,\;T){\times}{\Omega}$$ $${\theta}|_{{\partial}{\Omega}=0,\;\;{\theta}(0,X)={\theta}_0(X)$$, where $\Omega$ is a bounded open set in $R^{n}$, n=2 or 3 with a $C^{\infty}$ boundary ${\partial}{\Omega}$. The control is achieved by means of a linear feedback law relating the body forces to the velocity and temperature field, i.e., $$f=-{\gamma}_1(u-U),\;\;{\tau}=-{\gamma}_2({\theta}-{\Theta}}$$ where (U,$\Theta$) are target velocity and temperature. We show that the unsteady solutions to Boussinesq equations are stabilizable by internal controllers with exponential decaying property. In order to compute (approximations to) solution, semi discrete-in-time and full space-time discrete approximations are also studied. We prove that the difference between the solution of the discrete problem and the target solution decay to zero exponentially for sufficiently small time step.

• Korean Journal of Environmental Agriculture
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• v.16 no.3
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• pp.212-215
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• 1997

Parent isolates and $first-generation(F_1)$ progenies from single-sporangium of Phytophthora infestans were investigated for the acqusition of metalaxyl and streptomycin sulfate(SM) resistance. The parent isolates, KM10, U6, CDB6 showed resistance to metalaxyl, and F817, DNC303, DN107 to SM. Especially, the average mycelial growth of $F_1$ progenies of F817, CDB6, JD1 was more than 51% than parent isolates on 25ppm metalaxyl added V-8 juice agar, whereas those of KM10, U6, JD1 were extended more than 51% similar on 200ppm SM treatment. They acquired resistance on the basis of the standard in this experiment. There were positive correlations between parent isolates and $F_1$colonies in resistance. The $F_1$ colonies obtained from parent isolates showing above interinediate resistance on metalaxyl amended agar acquired much resistance.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.185-202
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• 2019

In this paper, we investigate many types of stability, like (uniform stability, exponential stability and h-stability) of the first order dynamic equations of the form $$\{u^{\Delta}(t)=Au(t)+f(t),\;\;t{\in}{\mathbb{T}},\;t>t_0\\u(t_0)=x{\in}D(A),$$ and $$\{u^{\Delta}(t)=Au(t)+f(t,u),\;\;t{\in}{\mathbb{T}},\;t>t_0\\u(t_0)=x{\in}D(A),$$ in terms of the stability of the homogeneous equation $$\{u^{\Delta}(t)=Au(t),\;\;t{\in}{\mathbb{T}},\;t>t_0\\u(t_0)=x{\in}D(A),$$ where f is rd-continuous in $t{\in}{\mathbb{T}}$ and with values in a Banach space X, with f(t, 0) = 0, and A is the generator of a $C_0$-semigroup $\{T(t):t{\in}{\mathbb{T}}\}{\subset}L(X)$, the space of all bounded linear operators from X into itself. Here D(A) is the domain of A and ${\mathbb{T}}{\subseteq}{\mathbb{R}}^{{\geq}0}$ is a time scale which is an additive semigroup with property that $a-b{\in}{\mathbb{T}}$ for any $a,b{\in}{\mathbb{T}}$ such that a > b. Finally, we give illustrative examples.