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A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions

Jan Eisner, Milan Kučera, Martin Väth (2016)

Applications of Mathematics

Given a reaction-diffusion system which exhibits Turing's diffusion-driven instability, the influence of unilateral obstacles of opposite sign (source and sink) on bifurcation and critical points is studied. In particular, in some cases it is shown that spatially nonhomogeneous stationary solutions (spatial patterns) bifurcate from a basic spatially homogeneous steady state for an arbitrarily small ratio of diffusions of inhibitor and activator, while a sufficiently large ratio is necessary in the...

A well-posedness result for a mass conserved Allen-Cahn equation with nonlinear diffusion

Kettani, Perla El, Hilhorst, Danielle, Lee, Kai (2017)

Proceedings of Equadiff 14

In this paper, we prove the existence and uniqueness of the solution of the initial boundary value problem for a stochastic mass conserved Allen-Cahn equation with nonlinear diffusion together with a homogeneous Neumann boundary condition in an open bounded domain of ℝ n with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion.

An IMEX scheme for reaction-diffusion equations: application for a PEM fuel cell model

István Faragó, Ferenc Izsák, Tamás Szabó, Ákos Kriston (2013)

Open Mathematics

An implicit-explicit (IMEX) method is developed for the numerical solution of reaction-diffusion equations with pure Neumann boundary conditions. The corresponding method of lines scheme with finite differences is analyzed: explicit conditions are given for its convergence in the ‖·‖∞ norm. The results are applied to a model for determining the overpotential in a proton exchange membrane (PEM) fuel cell.

An Optimal Control Problem for a Predator-Prey Reaction-Diffusion System

N. C. Apreutesei (2010)

Mathematical Modelling of Natural Phenomena

An optimal control problem is studied for a predator-prey system of PDE, with a logistic growth rate of the prey and a general functional response of the predator. The control function has two components. The purpose is to maximize a mean density of the two species in their habitat. The existence of the optimal solution is analyzed and some necessary optimality conditions are established. The form of the optimal control is found in some particular...

Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics.

Gonzalo Galiano, María Luisa Garzón, Ansgar Jüngel (2001)

RACSAM

En este trabajo se estudia de modo analítico y numérico un problema en ecuaciones diferenciales en derivadas parciales que modela la dinámica de dos poblaciones afectadas por la presión poblacional inter e intraespecíficas y por un potencial medioambiental. Debido a los términos de difusión cruzada, el problema es fuertemente no lineal por lo que el principio del máximo y los métodos relacionados con el mismo no pueden ser aplicados. En primer lugar demostramos la existencia de soluciones débiles...

Analysis of pattern formation using numerical continuation

Vladimír Janovský (2022)

Applications of Mathematics

The paper deals with the issue of self-organization in applied sciences. It is particularly related to the emergence of Turing patterns. The goal is to analyze the domain size driven instability: We introduce the parameter L , which scales the size of the domain. We investigate a particular reaction-diffusion model in 1-D for two species. We consider and analyze the steady-state solution. We want to compute the solution branches by numerical continuation. The model in question has certain symmetries....

Analysis of total variation flow and its finite element approximations

Xiaobing Feng, Andreas Prohl (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε , and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our...

Analysis of total variation flow and its finite element approximations

Xiaobing Feng, Andreas Prohl (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since...

Application of a center manifold theory to a reaction-diffusion system of collective motion of camphor disks and boats

Shin-Ichiro Ei, Kota Ikeda, Masaharu Nagayama, Akiyasu Tomoeda (2014)

Mathematica Bohemica

Unidirectional motion along an annular water channel can be observed in an experiment even with only one camphor disk or boat. Moreover, the collective motion of camphor disks or boats in the water channel exhibits a homogeneous and an inhomogeneous state, depending on the number of disks or boats, which looks like a kind of bifurcation phenomena. In a theoretical research, the unidirectional motion is represented by a traveling wave solution in a model. Hence it suffices to investigate a linearized...

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