• East Asian mathematical journal
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• 제39권1호
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• pp.51-63
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• 2023

We establish the interior L^q regularity estimates for spatial gradient of weak solutions to nonlinear parabolic equations with the inhomogeneity which is given by the divergence and nondivergence data.

• 충청수학회지
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• 제20권3호
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• pp.327-332
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• 2007

A variational inequality defined by the second-order cone is considered, and it is shown that a necessary and sufficient condition for a solution of the variational inequality holds under some regularity condition.

• 한국지진공학회:학술대회논문집
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• 한국지진공학회 2003년도 춘계 학술발표회논문집
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• pp.85-92
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• 2003

Hysteretic behaviors of a seismic isolator are identified by using the regularized output error estimator (OEE) based on the secant stiffness model. A proper regularity condition of tangent stiffness for the current OEE is proposed considering the regularity condition of Duhem hysteretic operator. The proposed regularity condition is defined by 12-norm of the tangent stiffness with respect to time. The secant stiffness model for the OEE is obtained by approximating the tangent stiffness under the proposed regularity condition by the secant stiffness at each time step. A least square method is employed to minimize the difference between the calculated response and measured response for the OEE. The regularity condition of the secant stiffness is utilized to alleviate ill-posedness of the OEE and to yield numerically stable solutions through the regularization technique. An optimal regularization factor determined by geometric mean scheme (GMS) is used to yield appropriate regularization effects on the OEE.

• 대한수학회지
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• 제45권2호
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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 $X_1(\mathbb{R}^d)$.

• Kyungpook Mathematical Journal
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• 제49권2호
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• pp.195-202
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• 2009

This paper deals with very weak solutions of the A-harmonic equation $divA(x,{\nabla}u)$ = 0 (*) with the operator $A:{\Omega}{\times}R^n{\rightarrow}R^n$ satisfies some coercivity and controllable growth conditions with Muckenhoupt weight. By using the Hodge decomposition with weight, a regularity property is proved: There exists an integrable exponent $r_1=r_1({\lambda},n,p)$ < p, such that every very weak solution $u{\in}W_{loc}^{1,r}({\Omega},{\omega})$ with $r_1$ < r < p belongs to $W_{loc}^{1,p}({\Omega},{\omega})$. That is, u is a weak solution to (*) in the usual sense.

• 대한수학회지
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• 제55권5호
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• pp.1269-1283
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• 2018

We are concerned with elliptic equations in ${\mathbb{R}}^N$, driven by a non-local integro-differential operator, which involves the fractional Laplacian. The main aim of this paper is to prove the existence of small solutions for our problem with negative energy in the sense that the sequence of solutions converges to 0 in the $L^{\infty}$-norm by employing the regularity type result on the $L^{\infty}$-boundedness of solutions and the modified functional method.

• Journal of applied mathematics & informatics
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• 제9권1호
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• pp.213-230
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• 2002

We deal with the approximate controllability for semilinear systems with time delay in a Hilbert space. First, we show the existence and uniqueness of solutions of the given systems with the mere general Lipschitz continuity of nonlinear operator f from $R\;\times\;V$ to H. Thereafter, it is shown that the equivalence between the reachable set of the semilinear system and that of its corresponding linear system. Finally, we make a practical application of the conditions to the system with only discrete delay.

• 대한수학회지
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• 제38권4호
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• pp.843-852
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• 2001

In this lecture I shall discuss some recent progress in the development of methods for attacking the central questions of the formation and structure of singularities and of global regularity for solutions of the Cauchy problem for nonlinear systems of partial differential equations of hyperbolic type. Applications to the Einstein equations of general relativity and to the equations of compressible fluid flow shall be particularly emphasized and detailed.

• 대한수학회보
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• 제46권3호
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• pp.463-475
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• 2009

This paper deals with the regularity properties for a class of semilinear integrodifferential functional differential equations. It is shown the relation between the reachable set of the semilinear system and that of its corresponding linear system. We also show that the Lipschitz continuity and the uniform boundedness of the nonlinear term can be considerably weakened. Finally, a simple example to which our main result can be applied is given.

• Kyungpook Mathematical Journal
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• 제59권2호
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• pp.241-258
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• 2019

In this paper, we deal with the null controllability of semilinear functional integrodifferential control systems under the Lipschitz continuity of nonlinear terms. Moreover, we establish the regularity and a variation of constant formula for solutions of the given control systems in Hilbert spaces.