• Title/Summary/Keyword: Dirichlet Condition

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DOUBLY NONLINEAR PARABOLIC EQUATIONS INVOLVING p-LAPLACIAN OPERATORS VIA TIME-DISCRETIZATION METHOD

  • Shin, Kiyeon;Kang, Sujin
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.6
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    • pp.1179-1192
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    • 2012
  • In this paper, we consider a doubly nonlinear parabolic partial differential equation $\frac{{\partial}{\beta}(u)}{{\partial}t}-{\Delta}_pu+f(x,t,u)=0$ in ${\Omega}{\times}[0,T]$, with Dirichlet boundary condition and initial data given. We prove the existence of a discrete approximate solution by means of the Rothe discretization in time method under some conditions on ${\beta}$, $f$ and $p$.

SIGN CHANGING PERIODIC SOLUTIONS OF A NONLINEAR WAVE EQUATION

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.16 no.2
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    • pp.243-257
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    • 2008
  • We seek the sign changing periodic solutions of the nonlinear wave equation $u_{tt}-u_{xx}=a(x,t)g(u)$ under Dirichlet boundary and periodic conditions. We show that the problem has at least one solution or two solutions whether $\frac{1}{2}g(u)u-G(u)$ is bounded or not.

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Remark for Certain Elliptic PDE with Exponential Nonlinearity in a Bounded Domain

  • Kim, Namkwon
    • Journal of Integrative Natural Science
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    • v.6 no.3
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    • pp.181-182
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    • 2013
  • In this note, we are concerned with a class of semi-linear elliptic pdes with exponential nonlinearity in a bounded domain. Here, the nonlinearity is more or less growing exponentially with power p. We consider the problem under two types of Dirichlet boundary condition. We give existence and non-existence of solutions for those problems and some asymptotics.

TWO JUMPING NONLINEAR TERMS AND A NONLINEAR WAVE EQUATION

  • Jung, Tacksun;Choi, Q-Heung
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.4
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    • pp.675-687
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    • 2009
  • We find the multiple nontrivial solutions of the equation of the form $u_{tt}-u_{xx}=b_1[(u+1)^{+}-1]+b_2[(u+2)^{+}-2]$ with Dirichlet boundary condition. Here we reduce this problem into a two-dimensional problem by using variational reduction method and apply the Mountain Pass theorem to find the nontrivial solutions.

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FINITE ELEMENT ANALYSIS FOR A MIXED LAGRANGIAN FORMULATION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

  • Kim, Hong-Chul
    • Journal of the Korean Mathematical Society
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    • v.34 no.1
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    • pp.87-118
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    • 1997
  • This paper is concerned with a mixed Lagrangian formulation of the wiscous, stationary, incompressible Navier-Stokes equations $$ (1.1) -\nu\Delta u + (u \cdot \nabla)u + \nabla_p = f in \Omega $$ and $$ (1.2) \nubla \cdot u = 0 in \Omega $$ along with inhomogeneous Dirichlet boundary conditions on a portion of the boundary $$ (1.3) u = ^{0 on \Gamma_0 _{g on \Gamma_g, $$ where $\Omega$ is a bounded open domain in $R^d, d = 2 or 3$, or with a boundary $\Gamma = \partial\Omega$, which is composed of two disjoint parts $\Gamma_0$ and $\Gamma_g$.

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PRECONDITIONING $C^1$-QUADRATIC SPLINE COLLOCATION METHOD OF ELLIPTIC EQUATIONS BY FINITE DIFFERENCE METHOD

  • Woo, Gyung-Soo;Kim, Seok-Chan
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.17-27
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    • 2001
  • We discuss a finite difference preconditioner for the$C^1$ Lagrance quadratic spline collocation method for a uniformly elliptic operator with homogeneous Dirichlet boundary conditions. Using the generalized field of values argument, we analyzed eigenvalues of the matrix preconditioned by the matrix corresponding to a finite difference operator with zero boundary condition.

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CRITICAL POINTS AND MULTIPLE SOLUTIONS OF A NONLINEAR ELLIPTIC BOUNDARY VALUE PROBLEM

  • Choi, Kyeongpyo
    • Korean Journal of Mathematics
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    • v.14 no.2
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    • pp.259-271
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    • 2006
  • We consider a semilinear elliptic boundary value problem with Dirichlet boundary condition $Au+bu^+-au^-=t_{1{\phi}1}+t_{2{\phi}2}$ in ${\Omega}$ and ${\phi}_n$ is the eigenfuction corresponding to ${\lambda}_n(n=1,2,{\cdots})$. We have a concern with the multiplicity of solutions of the equation when ${\lambda}_1$ < a < ${\lambda}_2$ < b < ${\lambda}_3$.

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BLOW UP OF SOLUTIONS WITH POSITIVE INITIAL ENERGY FOR THE NONLOCAL SEMILINEAR HEAT EQUATION

  • Fang, Zhong Bo;Sun, Lu
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.16 no.4
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    • pp.235-242
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    • 2012
  • In this paper, we investigate a nonlocal semilinear heat equation with homogeneous Dirichlet boundary condition in a bounded domain, and prove that there exist solutions with positive initial energy that blow up in finite time.

THE EXISTENCE OF THE SOLUTION OF ELLIPTIC SYSTEM APPLYING TWO CRITICAL POINT THEOREM

  • Nam, Hyewon
    • Journal of the Chungcheong Mathematical Society
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    • v.31 no.1
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    • pp.53-64
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    • 2018
  • This paper deals with the study of solutions for the elliptic system with jumping nonlineartity and growth nonlinearity and Dirichlet boundary condition. We apply the two critical point theorem when proving the existence of nontrivial solutions for the elliptic system. We define the energy functional associated to the elliptic system and prove that the functional has two critical values.

EXTINCTION AND POSITIVITY OF SOLUTIONS FOR A CLASS OF SEMILINEAR PARABOLIC EQUATIONS WITH GRADIENT SOURCE TERMS

  • Yi, Su-Cheol
    • Journal of the Chungcheong Mathematical Society
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    • v.30 no.4
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    • pp.397-409
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    • 2017
  • In this paper, we investigated the extinction, positivity, and decay estimates of the solutions to the initial-boundary value problem of the semilinear parabolic equation with nonlinear gradient source and interior absorption terms by using the integral norm estimate method. We found that the decay estimates depend on the choices of initial data, coefficients and domain, and the first eigenvalue of the Laplacean operator with homogeneous Dirichlet boundary condition plays an important role in the proofs of main results.