• Title, Summary, Keyword: parabolic equation

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EXISTENCE AND LARGE TIME BEHAVIOR OF SOLUTIONS TO A FOURTH-ORDER DEGENERATE PARABOLIC EQUATION

  • LIANG, BO;WANG, MEISHAN;WANG, YING
    • 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.

Model Reference Adaptive Control of a Time-Varying Parabolic System

  • Hong, Keum-Shik;Yang, Kyung-Jinn;Kang, Dong-Hunn
    • Journal of Mechanical Science and Technology
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    • v.14 no.2
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    • pp.168-176
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    • 2000
  • Related to the error dynamics of an adaptive system, averaging theorems are developed for coupled differential equations which consist of ordinary differential equations and a parabolic partial differential equation. The results are then applied to the convergence analysis of the parameter estimate errors in the model reference adaptive control of a nonautonomous parabolic partial differential equation with lowly time-varying parameters.

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A Parabolic Model to the Modified Mild Slope Equation (수정 완경사 파랑식에 대한 포물형 근사식 모형)

  • Seo, Seung-Nam;Lee, Jong-Chan
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.18 no.4
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    • pp.360-371
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    • 2006
  • In order to calculate waves propagating into the shallow water region, a generalized parabolic approximate model is presented. The model is derived from the modified mild slope equation and includes all the existing parabolic models presented in the paper. Numerical results are presented in comparison to laboratory data of Berkhoff et al.(1982). The existing parabolic model shows almost same accuracy against the modified parabolic model and both results of models stand in closer agreement to the laboratory data. Therefore the existing parabolic model based on mild slope equation is a useful tool to compute shallow water waves which turns out to be more fast and stable in computational aspect.

UNIQUENESS OF SOLUTIONS FOR A DEGENERATE PARABOLIC EQUATION WITH ABSORPTION

  • Lee, Jin Ho;Jang, Seong Hee
    • Korean Journal of Mathematics
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    • v.5 no.2
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    • pp.151-167
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    • 1997
  • We estimate the interior Lipschitz norm and maximum of the solution for degenerate parabolic equations with absorption. Also obtain the growth rate of the solution $u$ in terms of time $t$. From this we show the uniqueness of solution with respect to the initial trace.

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MULTIPLICITY OF SOLUTIONS AND SOURCE TERMS IN A NONLINEAR PARABOLIC EQUATION UNDER DIRICHLET BOUNDARY CONDITION

  • Choi, Q-Heung;Jin, Zheng-Guo
    • Bulletin of the Korean Mathematical Society
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    • v.37 no.4
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    • pp.697-710
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    • 2000
  • We investigate the existence of solutions of the nonlinear heat equation under Dirichlet boundary conditions on $\Omega$ and periodic condition on the variable t, $Lu-D_tu$+g(u)=f(x, t). We also investigate a relation between multiplicity of solutions and the source terms of the equation.

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Application of the Polar Parabolic Equation Method for Sound Propagation over a Smooth Sea Mountain in the Ocean (해저구릉 위로의 음의 전파를 설명하기 위한 Polar PE의 적용)

  • You, Chul-Soo;Yoon, Suk-Wang
    • The Journal of the Acoustical Society of Korea
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    • v.14 no.2E
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    • pp.37-42
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    • 1995
  • The polar parabolic equation method (Polar PE) which introduces a series of "cascaded" boundary fitting coordinates into the parabolic equation method has been verified as a good numerical method for atmospheric sound propagation over a curved surface and hills. Polar PE is applied here to underwater sound propagation over a sea mountain assuming locally reacting boundary sea bottom and pressure release water surface for the boundary conditions. Calculations are presented for underwater propagation over a 450 m high sea mountain. Feasibility of Polar PE application for underwater sound propagation over a smooth mountain is discussed.

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A Comparative Study of Efficient Transient Analysis Algorithm for Parabolic Equations (Parabolic 방정식의 효율적인 시간해석 알고리즘에 대한 비교연구)

  • 최창근;이은진;유원진
    • Proceedings of the Computational Structural Engineering Institute Conference
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    • pp.68-74
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    • 1998
  • A finite element analysis for physical phenomenon which are governed by parabolic equation, has some inefficiencies caused by much computational time and large storage space. In this paper, a comparative study is performed to suggest the best efficient transient analysis algorithms for parabolic equations. First, the general finite element analysis techniques are summarized in views of formulation procedures, treatments of convection terms. and time stepping methods. Results of several combinations applied to one dimensional convection-diffusion equation and Burger equation are represented and compared using some criteria such as accuracy, stability, and computational time. Through the results, some guidelines to select a algorithm for solving parabolic equations are proposed for diffusion dominant and convection dominant cases. Finally applicability of two dimensional extension of the result is also discussed.

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TWO NEW BLOW-UP CONDITIONS FOR A PSEUDO-PARABOLIC EQUATION WITH LOGARITHMIC NONLINEARITY

  • Ding, Hang;Zhou, Jun
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1285-1296
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    • 2019
  • This paper deals with the blow-up phenomenon of solutions to a pseudo-parabolic equation with logarithmic nonlinearity, which was studied extensively in recent years. The previous result depends on the mountain-pass level d (see (1.6) for its definition). In this paper, we obtain two blow-up conditions which do not depend on d. Moreover, the upper bound of the blow-up time is obtained.

NEGATIVELY BOUNDED SOLUTIONS FOR A PARABOLIC PARTIAL DIFFERENTIAL EQUATION

  • FANG ZHONG BO;KWAK, MIN-KYU
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.4
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    • pp.829-836
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    • 2005
  • In this note, we introduce a new proof of the unique-ness and existence of a negatively bounded solution for a parabolic partial differential equation. The uniqueness in particular implies the finiteness of the Fourier spanning dimension of the global attractor and the existence allows a construction of an inertial manifold.