• Title, Summary, Keyword: weak solutions

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GLOBAL WEAK SOLUTIONS FOR THE RELATIVISTIC VLASOV-KLEIN-GORDON SYSTEM IN TWO DIMENSIONS

  • Xiao, Meixia;Zhang, Xianwen
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.2
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    • pp.591-598
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    • 2018
  • This paper is concerned with global existence of weak solutions to the relativistic Vlasov-Klein-Gordon system. The energy of this system is conserved, but the interaction term ${\int}_{{\mathbb{R}}^n}\;{\rho}{\varphi}dx$ in it need not be positive. So far existence of global weak solutions has been established only for small initial data [9, 14]. In two dimensions, this paper shows that the interaction term can be estimated by the kinetic energy to the power of ${\frac{4q-4}{3q-2}}$ for 1 < q < 2. As a consequence, global existence of weak solutions for general initial data is obtained.

UNIQUENESS OF POSITIVE STEADY STATES FOR WEAK COMPETITION MODELS WITH SELF-CROSS DIFFUSIONS

  • Ko, Won-Lyul;Ahn, In-Kyung
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.2
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    • pp.371-385
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    • 2004
  • In this paper, we investigate the uniqueness of positive solutions to weak competition models with self-cross diffusion rates under homogeneous Dirichlet boundary conditions. The methods employed are upper-lower solution technique and the variational characterization of eigenvalues.

WEAK SOLUTIONS OF THE EQUATION OF MOTION OF MEMBRANE WITH STRONG VISCOSITY

  • Hwang, Jin-Soo;Nakagiri, Shin-Ichi
    • Journal of the Korean Mathematical Society
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    • v.44 no.2
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    • pp.443-453
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    • 2007
  • We study the equation of a membrane with strong viscosity. Based on the variational formulation corresponding to the suitable function space setting, we have proved the fundamental results on existence, uniqueness and continuous dependence on data of weak solutions.

LOCAL REGULARITY OF THE STEADY STATE NAVIER-STOKES EQUATIONS NEAR BOUNDARY IN FIVE DIMENSIONS

  • Kim, Jaewoo;Kim, Myeonghyeon
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.3
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    • pp.557-569
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    • 2009
  • We present a new regularity criterion for suitable weak solutions of the steady-state Navier-Stokes equations near boundary in dimension five. We show that suitable weak solutions are regular up to the boundary if the scaled $L^{\frac{5}{2}}$-norm of the solution is small near the boundary. Our result is also valid in the interior.

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SHALLOW ARCHES WITH WEAK AND STRONG DAMPING

  • Gutman, Semion;Ha, Junhong
    • Journal of the Korean Mathematical Society
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    • v.54 no.3
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    • pp.945-966
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    • 2017
  • The paper develops a rigorous mathematical framework for the behavior of arch and membrane like structures. Our main goal is to incorporate moving point loads. Both the weak and the strong damping cases are considered. First, we prove the existence and the uniqueness of the solutions. Then it is shown that the solution in the weak damping case is the limit of the strong damping solutions, as the strong damping vanishes. The theory is applied to a car moving on a bridge.

GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR A KELLER-SEGEL-FLUID MODEL WITH NONLINEAR DIFFUSION

  • Chung, Yun-Sung;Kang, Kyungkeun;Kim, Jaewoo
    • Journal of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.635-654
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    • 2014
  • We consider the Cauchy problem for a Keller-Segel-fluid model with degenerate diffusion for cell density, which is mathematically formulated as a porus medium type of Keller-Segel equations coupled to viscous incompressible fluid equations. We establish the global-in-time existence of weak solutions and bounded weak solutions depending on some conditions of parameters such as chemotactic sensitivity and consumption rate of oxygen for certain range of diffusive exponents of cell density in two and three dimensions.

GROUND STATE SOLUTIONS OF NON-RESONANT COOPERATIVE ELLIPTIC SYSTEMS WITH SUPERLINEAR TERMS

  • Chen, Guanwei
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.789-801
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    • 2014
  • In this paper, we study the existence of ground state solutions for a class of non-resonant cooperative elliptic systems by a variant weak linking theorem. Here the classical Ambrosetti-Rabinowitz superquadratic condition is replaced by a general super quadratic condition.

EXISTENCE AND MULTIPLICITY OF WEAK SOLUTIONS FOR SOME p(x)-LAPLACIAN-LIKE PROBLEMS VIA VARIATIONAL METHODS

  • AFROUZI, G.A.;SHOKOOH, S.;CHUNG, N.T.
    • Journal of applied mathematics & informatics
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    • v.35 no.1_2
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    • pp.11-24
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    • 2017
  • Using variational methods, we study the existence and multiplicity of weak solutions for some p(x)-Laplacian-like problems. First, without assuming any asymptotic condition neither at zero nor at infinity, we prove the existence of a non-zero solution for our problem. Next, we obtain the existence of two solutions, assuming only the classical Ambrosetti-Rabinowitz condition. Finally, we present a three solutions existence result under appropriate condition on the potential F.