• Communications of the Korean Mathematical Society
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• v.12 no.4
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• pp.869-880
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• 1997

In this paper, we give sufficient conditions of certain elliptic systems involving competing iteractions with nonlinear diffusion rates. The existence of positive solution depends on the sign of the first eigenvalue of operators of Schr$\ddot{o}$dinger type. More precisely, if the sign of such operators are either both positive or both negative, then system has a positive solution. The main tool employed is the fixed point index of compact operator on positive cones.

• Bulletin of the Korean Chemical Society
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• v.8 no.3
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• pp.196-200
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• 1987

A nonlinear theory presented previously is applied to the Oregonator, which is a model for the Belousov-Zhabotinskii reaction, to study instability near the critical point driven by diffusions. The result shows that the theory may be applied to an actual system.

• Bulletin of the Korean Mathematical Society
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• v.32 no.1
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• pp.123-132
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• 1995

Our interest is to study the existence of positive solutions to the following elliptic system involving competing interaction $$ (1) { -\partial(x,u,\upsilon)\Delta u = uf(x,u,v) { - \psi(x,u,\upsilon)\Delta \upsilon = \upsilon g(x,u,\upsilon) { \frac{\partial n}{\partial u} + ku = 0 on \partial\Omega { \frac{\partial n}{\partial\upsilon} + \sigma\upsilon = 0 $$ in a bounded region $\Omega$ in $R^n$ with a smooth boundary, where the diffusion terms $\varphi, \psi$ are strictly positive nondecreasing function, and k, $\sigma$ are positive constants. Also we assume that the growth rates f, g are $C^1$ monotone functions. The variables u, $\upsilon$ may represent the population densities of the interacting species in problems from ecology, microbiology, immunology, etc.

• Ocean Systems Engineering
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• v.12 no.3
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• pp.359-384
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• 2022

We develop in this work a new well-balanced preserving-positivity path-conservative central-upwind scheme for Saint-Venant-Exner (SVE) model. The SVE system (SVEs) under some considerations, is a nonconservative hyperbolic system of nonlinear partial differential equations. This model is widely used in coastal engineering to simulate the interaction of fluid flow with sediment beds. It is well known that SVEs requires a robust treatment of nonconservative terms. Some efficient numerical schemes have been proposed to overcome the difficulties related to these terms. However, the main drawbacks of these schemes are what follows: (i) Lack of robustness, (ii) Generation of non-physical diffusions, (iii) Presence of instabilities within numerical solutions. This collection of drawbacks weakens the efficiency of most numerical methods proposed in the literature. To overcome these drawbacks a reformulation of the central-upwind scheme for SVEs (CU-SVEs for short) in a path-conservative version is presented in this work. We first develop a finite-volume method of the first order and then extend it to the second order via the averaging essentially non oscillatory (AENO) framework. Our numerical approach is shown to be well-balanced positivity-preserving and shock-capturing. The resulting scheme could be seen as a predictor-corrector method. The accuracy and robustness of the proposed scheme are assessed through a carefully selected suite of tests.