• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.537-558
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• 2008

Consider a weak solution u of the Navier-Stokes equations in the class $L^2((0,T);X_1(\mathbb{R}^d)^d)$. We establish a new approach to treat the regularity problem for the Navier-Stokes equation in term of the multiplier space $X_1(\mathbb{R}^d)$.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.6 no.1
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• pp.70-76
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• 2006

Some Stampacchia type of generalized weak vector quasivariational-like inequalities for fuzzy mappings was introduced and the existence of solutions to them under non-compact assumption was considered using the particular form of the generalized Ky Fan's section theorem due to Park [15]. As a corollary, Stampacchia type of generalized vector quasivariational-like inequalities for fuzzy mappings was studied under compact assumption using Ky Fan's section theorem [7].

• Journal of applied mathematics & informatics
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• v.32 no.3_4
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• pp.547-554
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• 2014

In this paper, we consider the existence and uniqueness of global weak solution for a sixth-order classical surface-diffusion equation in one spatial dimension. Moreover, the regularity and blow-up of solutions are also studied.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1235-1250
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• 2010

In this paper we establish the existence of at least three weak solutions for Neumann doubly eigenvalue elliptic systems driven by a ($p_1,\ldots,p_n$)-Laplacian. Our main tool is a recent three critical points theorem of B. Ricceri.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1817-1826
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• 2013

The existence of at least three weak solutions is established for a class of quasilinear elliptic equations involving the p(x)-biharmonic operators with Navier boundary value conditions. The technical approach is mainly based on a three critical points theorem due to Ricceri [11].

• Journal of the Chungcheong Mathematical Society
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• v.27 no.4
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• pp.763-769
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• 2014

In this paper, we illustrate a counter example for the converse of a certain conjecture proposed by De Giorgi. De Giorgi suggested a series of conjectures, in which a certain integral condition for singularity or degeneracy of an elliptic operator is satisfied, the solutions are continuous. We construct some singular elliptic operators and solutions such that the integral condition does not hold, but the solutions are continuous.

• Restorative Dentistry and Endodontics
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• v.15 no.1
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• pp.187-200
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• 1990

The purpose of this study was to investigate antimicrobial effect of several irrigation solutions on 7 anaerobes and 2 aerobes, which are found frequently in infected root canals. The antimicrobial effects of normal saline, 3% $H_2O_2$, 0.5% & 3.5% NaOCl, 10% & 50% citric acid and mixed solutions of 3% $H_2O_2$ plus 3.5% NaOCl were compared. No. 80 paper points dipped in bacterial broth were soaked in each irrigation solutions and moved into thioglycolate broth, subcultured in agar plate for bacterial growth. The results were as follows: 1. Normal saline had no antimicrobial effect. 2. Mixed solutions of 3% $H_2O_2$ plus 3.5% NaOCl, 10% citric acid had relatively weak antimicrobial effect. 3.3% $H_2O_2$, 50% citric acid, 0.5% NaOCl showed relatively strong antimicrobial effect. 4. 3.5% NaOCl had the strongest antimicrobial effect among used 7 irrigation solutions.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.1003-1031
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• 2020

We consider the following Dirichlet initial boundary value problem with a gradient absorption and a nonlocal source $$\frac{{\partial}u}{{\partial}t}-div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)={\lambda}u^k{\displaystyle\smashmargin{2}{\int\nolimits_{\Omega}}}u^sdx-{\mu}u^l{\mid}{\nabla}u{\mid}^q$$ in a bounded domain Ω ⊂ ℝ^N, where p > 1, the parameters k, s, l, q, λ > 0 and µ ≥ 0. Firstly, we establish local existence for weak solutions; the aim of this part is to prove a crucial priori estimate on |∇u|. Then, we give appropriate conditions in order to have existence and uniqueness or nonexistence of a global solution in time. Finally, depending on the choices of the initial data, ranges of the coefficients and exponents and measure of the domain, we show that the non-negative global weak solution, when it exists, must extinct after a finite time.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.737-748
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• 2012

The goal of this paper is to study the existence of non-trivial non-negative weak solution for the nonlinear elliptic equation: $$-div(h(x){\nabla}u)=f(x,u)\;in\;{\Omega}$$ with Dirichlet boundary condition in a bounded domain ${\Omega}{\subset}\mathbb{R}^N$, $N{\geq}3$, where $h(x){\in}L^1_{loc}({\Omega})$, $f(x,s)$ has asymptotically linear behavior. The solutions will be obtained in a subspace of the space $H^1_0({\Omega})$ and the proofs rely essentially on a variation of the mountain pass theorem in [12].

• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.1031-1050
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• 2003

In this paper, we assume a density with integrability on the space $L^{\infty}$(0, T; $L^{q_{0}}$) for some $q_{0}$ and T > 0. Under the assumption on the density, we obtain a regularity result for the weak solutions to the compressible Navier-Stokes equations. That is, the supremum of the density is finite and the infimum of the density is positive in the domain $T^3$ ${\times}$ (0, T). Moreover, Moser type iteration scheme is developed for $L^{\infty}$ norm estimate for the velocity.