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Kim, Daewook The Youngnam Mathematical Society 2016 East Asian mathematical journal Vol.32 No.3
In this paper, we study the viscoelastic Kirchhoff type equation with the following nonlinear source and time-varying delay $$u_{tt}-M(x,t,{\parallel}{\nabla}u(t){\parallel}^2){\Delta}u+{\int_{0}^{t}}h(t-{\tau})div[a(x){\nabla}u({\tau})]d{\tau}\\+{\parallel}u{\parallel}^{\gamma}u+{\mu}_1u_t(x,t)+{\mu}_2u_t(x,t-s(t))=0.$$ Under the smallness condition with respect to Kirchhoff coefficient and the relaxation function and other assumptions, we prove the uniform decay rate of the Kirchhoff type energy.
Kim, Daewook The Youngnam Mathematical Society 2016 East Asian mathematical journal Vol.32 No.5
In this paper, we study the generalized Kirchhoff type equation in the presence of past and finite history $$\large u_{tt}-M(x,t,{\tau},\;{\parallel}{\nabla}u(t){\parallel}^2){\Delta}u+{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_0}^t}\;h(t-{\tau})div[a(x){\nabla}u({\tau})]d{\tau}\\\hspace{25}-{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{-{\infty}}}^t}\;k(t-{\tau}){\Delta}u(x,t)d{\tau}+{\mid}u{\mid}^{\gamma}u+{\mu}_1u_t(x,t)+{\mu}_2u_t(x,t-s(t))=0.$$ Under the smallness condition with respect to Kirchhoff coefficient and the relaxation function and other assumptions, we prove the expoential decay rate of the Kirchhoff type energy.
STABILIZATION FOR THE VISCOELASTIC KIRCHHOFF TYPE EQUATION WITH A NONLINEAR SOURCE
Kim, Daewook The Youngnam Mathematical Society 2016 East Asian mathematical journal Vol.32 No.1
In this paper, we study the viscoelastic Kirchhoff type equation with a nonlinear source $$u^{{\prime}{\prime}}-M(x,t,{\parallel}{\bigtriangledown}u(t){\parallel}^2){\bigtriangleup}u+{\int}_0^th(t-{\tau})div[a(x){\bigtriangledown}u({\tau})]d{\tau}+{\mid}u{\mid}^{\gamma}u=0$$. Under the smallness condition with respect to Kirchhoff coefficient and the relaxation function and other assumptions, we prove the uniform decay rate of the Kirchhoff type energy.
Some Boundary Feedback Control of a Nonlinear String Equation of Kirchhoff-Carrier Type
Daewook Kim,Il Hyo Jung 제어로봇시스템학회 2011 제어로봇시스템학회 국제학술대회 논문집 Vol.2011 No.10
In this paper we study the boundary control for Nonlinear problem concerning the hyperbolic equation suggested in the study of the viscoelastic string model of Kirchhoff-Carrier type. We also give some numerical simulation for each suggested boundary conditions by feedback control.
Kim, Daewook The Youngnam Mathematical Society 2018 East Asian mathematical journal Vol.34 No.1
In this paper, we study the viscoelastic Kirchhoff type equation with a nonlinear source for each independent kernels h and g with respect to Volterra terms. Under the smallness condition with respect to Kirchhoff coefficient and the relaxation function and other assumptions, we prove the uniform decay rate of the Kirchhoff type energy.
Kim, Daewook,Kim, Dojin,Hong, Keum-Shik,Jung, Il Hyo Hindawi Publishing Corporation 2014 The Scientific World Journal Vol.2014 No.-
<P>The first objective of this paper is to prove the existence and uniqueness of global solutions for a Kirchhoff-type wave equation with nonlinear dissipation of the form <I>Ku</I>′′ + <I>M</I>(|<I>A</I><SUP>1/2</SUP><I>u</I>|<SUP>2</SUP>)<I>Au</I> + <I>g</I>(<I>u</I>′) = 0 under suitable assumptions on <I>K</I>, <I>A</I>, <I>M</I>(·), and <I>g</I>(·). Next, we derive decay estimates of the energy under some growth conditions on the nonlinear dissipation <I>g</I>. Lastly, numerical simulations in order to verify the analytical results are given.</P>
A model for predicting transport velocity in gas fluidized-beds
Kim, Daewook,Won, Yoo Sube,Khurram, Muhammad Shahzad,Joo, Ji Bong,Choi, Jeong-Hoo,Ryu, Ho-Jung Elsevier 2018 Advanced powder technology Vol.29 No.12
<P><B>Abstract</B></P> <P>This study improved the model that used the correlation of the particle entrainment rate to determine the transport velocity. It proposed the new absolute value of the local slope as the criterion of the model for locating the transport velocity in the relationship between dimensionless velocity (U/U<SUB>t</SUB>) and the reciprocal of the entrainment rate (1/K<SUB>i</SUB> <SUP>∗+</SUP>). It indicated that the criterion depended on properties of fluidized particles and increased with Archemedes number. The Archemedes number was modified by substituting the critical diameter, the maximum particle diameter at which the sum of the interparticle adhesion forces gave a dominant influence to the particle entrainment rate, for the diameter of particles in case that the mean diameter of particles was smaller than the critical diameter. A correlation was suggested to calculate the absolute value of the local slope at the transport velocity. The new model was successful covering the effect of particle properties in predicting the transport velocity.</P> <P><B>Highlights</B></P> <P> <UL> <LI> This model determined the transport velocity from the correlation of entrainment rate. </LI> <LI> The criterion for locating the transport velocity was proposed. </LI> <LI> The criterion increased with Archimedes number. </LI> <LI> A correlation for the criterion was successful in predicting the transport velocity. </LI> </UL> </P> <P><B>Graphical abstract</B></P> <P>Archimedes number versus absolute slope at transport velocity.</P> <P>[DISPLAY OMISSION]</P>
MATHEMATICAL MODELLING FOR THE AXIALLY MOVING PLATE WITH INTERNAL TIME DELAY
Kim, Daewook The Youngnam Mathematical Society 2021 East Asian mathematical journal Vol.37 No.5
In [1, 2], we studied the string-like system with time-varying delay. Unlike the string system, the plate system must consider both longitudinal and transverse strains. First, we consider the physical phenomenon of an axially moving plate concerning kinetic energy, potential energy, and work dones. By the energy conservation law in physics, we have a nonlinear plate-like system with internal time delay.
Energy decay rate for a von Karman system with a boundary nonlinear delay term
Kim, Daewook,Park, Jong Yeoul,Kang, Yong Han Elsevier 2018 COMPUTERS & MATHEMATICS WITH APPLICATIONS - Vol.75 No.9
<P><B>Abstract</B></P> <P>In this paper, we show the energy decay rate for a von Karman system with a boundary nonlinear delay term. This work is devoted to investigate the influence of kernel function g and the effect of the boundary nonlinear term <SUB> μ 1 </SUB> <SUP> | <SUB> u t </SUB> ( t ) | m − 1 </SUP> <SUB> u t </SUB> ( t ) , a boundary nonlinear time delay term <SUB> μ 2 </SUB> <SUP> | <SUB> u t </SUB> ( t − τ ) | m − 1 </SUP> <SUB> u t </SUB> ( t − τ ) and prove energy decay rates of solutions when g do not necessarily decay exponentially and the boundary condition has a time delay.</P>
Daewook Kim,Jin-Mun Jeong 충청수학회 2021 충청수학회지 Vol.34 No.3
In this paper, we consider the approximate controllability for a class of semilinear integro-differential functional control equations in which nonlinear terms of given equations satisfy quasi-homogeneous properties. The main method used is to make use of the surjective theorems that is similar to Fredholm alternative in the nonlinear case under restrictive assumptions. The sufficient conditions for the approximate controllability is obtain which is different from previous results on the system operator, controller and nonlinear terms. Finally, a simple example to which our main result can be applied is given.