• Journal of the Korean Mathematical Society
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• v.37 no.5
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• pp.793-803
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• 2000

An impulsive semilinear parabolic equation subject to Robin boundary condition is considered. We prove that for certain classes of impulsive sources and continuous initial data, the solutions of the problem under consideration blow up in the desired time interval.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.3
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• pp.501-516
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• 2013

The well-known Vlasov-Poisson equation describes plasma physics as nonlinear first-order partial differential equations. Because of the nonlinear condition from the self consistency of the Vlasov-Poisson equation, many problems occur: the existence, the numerical solution, the convergence of the numerical solution, and so on. To solve the problems, a viscosity term (a second-order partial differential equation) is added. In a viscosity term, the Vlasov-Poisson equation changes into a parabolic equation like the Fokker-Planck equation. Therefore, the Schauder fixed point theorem and the classical results on parabolic equations can be used for analyzing the Vlasov-Poisson equation. The sequence and the convergence results are obtained from linearizing the Vlasove-Poisson equation by using a fixed point theorem and Gronwall's inequality. In numerical experiments, an implicit first-order scheme is used. The numerical results are tested using the changed viscosity terms.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.3
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• pp.287-308
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• 2018

This paper deals with blow-up phenomena for an initial boundary value problem of a quasilinear parabolic equation with time-dependent coefficient in a bounded star-shaped region under nonlinear boundary flux. Using the auxiliary function method and differential inequality technique, we establish some conditions on time-dependent coefficient and nonlinear functions for which the solution u(x, t) exists globally or blows up at some finite time $t^*$. Moreover, some upper and lower bounds for $t^*$ are derived in higher dimensional spaces. Some examples are presented to illustrate applications of our results.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1149-1159
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• 2015

This paper is concerned with a generalized 2D parabolic equation with a nonautonomous perturbation $$-{\Delta}u_t+{\alpha}^2{\Delta}^2u_t+{\mu}{\Delta}^2u+{\bigtriangledown}{\cdot}{\vec{F}}(u)+B(u,u)={\epsilon}g(x,t)$$. Under some proper assumptions on the external force term g, the upper semicontinuity of pullback attractors is proved. More precisely, it is shown that the pullback attractor $\{A_{\epsilon}(t)\}_{t{\epsilon}{\mathbb{R}}}$ of the equation with ${\epsilon}>0$ converges to the global attractor A of the equation with ${\epsilon}=0$.

• Kyungpook Mathematical Journal
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• v.48 no.4
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• pp.593-611
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• 2008

Global attractivity and oscillatory behavior of the following nonlinear impulsive parabolic differential equation which is a general form of many population models $$\array{\{{{\frac {{\partial}u(t,x)}{{\partial}t}=\Delta}u(t,x)-{\delta}u(t,x)+f(u(t-\tau,x)),\;t{\neq}t_k,\\u(t^+_k,x)-u(t_k,x)=g_k(u(t_k,x)),\;k{\in}I_\infty,}\;\;\;\;\;\;\;\;(*)$$ are considered. Some new sufficient conditions for global attractivity and oscillation of the solutions of (*) with Neumann boundary condition are established. These results no only are true but also improve and complement existing results for (*) without diffusion or impulses. Moreover, when these results are applied to the Nicholson's blowflies model and the model of Hematopoiesis, some new results are obtained.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.2 no.3
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• pp.134-142
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• 1990

A wide-angle approximation in the parabolic equation method is presented to calculate wave transformation in the shallow water. The parabolic approximation to the mild-slope equation is obtain-ed by the use of a splitting matrix, which leads to a generalized equation in form. A numerical model based on a finite difference scheme is presented and computational results are provided to test the model against the laboratory measurements of circular and elliptical shoals. The numerical results are in good agreement with most of experimental data. Therefore it can be concluded that the model shows greater capability to reproduce the characteristics of waves in the refractive focus.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1597-1612
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• 2016

There are fruitful results on degenerate parabolic-hyperbolic equations recently following the idea of $Kru{\check{z}}kov^{\prime}s$ doubling variables device. This paper is devoted to the well-posedness of nonhomogeneous boundary problem for degenerate parabolic-hyperbolic equations with spatially dependent second order operator, which has not caused much attention. The novelty is that we use the boundary flux triple instead of boundary layer to treat this problem.

• Journal of the Korean Mathematical Society
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• v.58 no.4
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• pp.1001-1017
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• 2021

In this paper, we first apply parabolic inequalities and a maximum principle to give a new proof for symmetry and monotonicity of solutions to fractional elliptic equations with gradient term by the method of moving planes. Under the condition of suitable initial value, by maximum principles for the fractional parabolic equations, we obtain symmetry and monotonicity of positive solutions for each finite time to nonlinear fractional parabolic equations in a bounded domain and the whole space. More generally, if bounded domain is a ball, then we show that the solution is radially symmetric and monotone decreasing about the origin for each finite time. We firmly believe that parabolic inequalities and a maximum principle introduced here can be conveniently applied to study a variety of nonlocal elliptic and parabolic problems with more general operators and more general nonlinearities.

• Journal of Ocean Engineering and Technology
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• v.15 no.2
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• pp.6-10
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• 2001

Water waves propagate over irregular bottom bathymetry are transformed by refraction, diffraction, shoaling, reflection etc. Principal factor of wave transform is bottom bathymetry, but in case of current field, current is another important factor which effect wave transformation. The governing equation of this study is develope as wave-current equation type to investigate the effect of wave-current interaction. It starts from Berkhoff's(1972) mild slope equation and is transformed to time-dependent hyperbolic type equation by using variational principal. Finally the governing equation is shown as a parabolic type equation by splitting method. This wave-current model was applied to the kwangan beach which is located at Pusan. The numerical simulation results of this model show the characteristics of wave transformation and flow pattern around the Kwangan beach fairly well.

• The Journal of the Acoustical Society of Korea
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• v.36 no.1
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• pp.1-11
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• 2017

In this study, novel approximate form of the square-root operator of three dimensional acoustic Parabolic Equation (3D PE) is proposed using a rational approximant for two variables. This form has two advantages in comparison with existing approximation studies of the square-root operator. One is the wide-angle capability. The proposed form has wider angle accuracy to the inclination angle of ${\pm}62^{\circ}$ from the range axis of 3D PE at the bearing angle of $45^{\circ}$, which is approximately three times the angle limit of the existing 3D PE algorithm. Another is that the denominator of our approximate form can be expressed into the product of one-dimensional operators for depth and cross-range. Such a splitting form is very preferable in the numerical analysis in that the 3D PE can be easily transformed into the tridiagonal matrix equation. To confirm the capability of the proposed approximate form, comparative study of other approximation methods is conducted based on the phase error analysis, and the proposed method shows best performance.