• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.143-159
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• 2013

This paper is concerned with the long time behavior for the solution semiflow of the coupled two-compartment Gray-Scott equations with the homogeneous Neumann boundary condition on a bounded domain of space dimension $n{\leq}3$. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the equations possesses a global attractor in $H^k({\Omega})^4$ ($k{\geq}0$) space.

• Journal of the Korean Mathematical Society
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• v.41 no.5
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• pp.875-882
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• 2004

Let (M,g) be a compact m dimensional Einstein manifold with smooth boundary. Let $\Delta$$_{p}$,B be the realization of the p form valued Laplacian with a suitable boundary condition B. Let Spec($\Delta$$_{p}$,B) be the spectrum where each eigenvalue is repeated according to multiplicity. We show that certain geometric properties of the boundary may be spectrally characterized in terms of this data where we fix the Einstein constant.ant.

• Journal of the Korean Mathematical Society
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• v.43 no.5
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• pp.931-950
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• 2006

We study the identification problems of constant parameters appearing in the perturbed sine-Gordon equation with the Neumann boundary condition. The existence of optimal parameters is proved, and necessary conditions are established for several types of observations by utilizing quadratic optimal control theory due to Lions [13].

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.251-261
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• 2010

This paper deals with a predator-prey model with Holling type III response function and cross-diffusion subject to the homogeneous Neumann boundary condition. We first give a priori estimates (positive upper and lower bounds) of positive steady states. Then the non-existence and existence results of non-constant positive steady states are given as the cross-diffusion coefficient is varied, which means that stationary patterns arise from cross-diffusion.

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.383-389
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• 2017

This paper devotes to the study of a diffusive model for unstirred membrane reactors with maintenance energy subject to a homogeneous Neumann boundary condition. It shows that the unique constant steady state is globally asymptotically stable when it exists. This result further implies the non-existence of the non-uniform steady state solution.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.75-87
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• 2018

In this paper, we consider ratio-dependent predator-prey models with disease in the prey under Neumann boundary condition. We investigate sufficient conditions for the existence and non-existence of non-constant positive steady-state solutions by the effects of the induced diffusion rates.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.571-579
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• 2007

Finite difference schemes are considered for a generalization of the Cahn-Hilliard equation with Neumann boundary conditions and the Kuramoto-Sivashinsky equation with a periodic boundary condition, which is of the type $ut+\frac{{\partial}^2} {{\partial}x^2}\;g\;(u,\;u_x,\;u_{xx})=f(u,\;u_x,\;u_{xx})$. Stability and $L^{\infty}$ error estimates of approximate solutions for the corresponding schemes are obtained using the extended Lax-Richtmyer equivalence theorem.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1133-1148
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• 2016

In this paper, we first discuss a representation of solutions to the initial value problem and the initial-boundary value problem for discrete evolution equations $${\sum\limits^l_{n=0}}c_n{\partial}^n_tu(x,t)-{\rho}(x){\Delta}_{\omega}u(x,t)=H(x,t)$$, defined on networks, i.e. on weighted graphs. Secondly, we show that the weight of each link of networks can be uniquely identified by using their Dirichlet data and Neumann data on the boundary, under a monotonicity condition on their weights.

• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.35-64
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• 2021

In this paper, we investigate an initial boundary value problem for a nonlinear pseudoparabolic equation. At first, by applying the Faedo-Galerkin, we prove local existence and uniqueness results. Next, by constructing Lyapunov functional, we establish a sufficient condition to obtain the global existence and exponential decay of weak solutions.

• Journal of the Chungcheong Mathematical Society
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• v.36 no.1
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• pp.39-53
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• 2023

In this paper, we consider a delayed ratio-dependent predator-prey reaction-diffusion system with homogenous Neumann boundary conditions. We study the existence of nonnegative solutions and the stability of the nonnegative equilibria to the system. In particular, we provide a sufficient condition for the positive equilibrium to be globally asymptotically stable.