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A free boundary problem for a predator-prey model with nonlinear prey-taxis

Mohsen Yousefnezhad, Seyyed Abbas Mohammadi, Farid Bozorgnia (2018)

Applications of Mathematics

This paper deals with a reaction-diffusion system modeling a free boundary problem of the predator-prey type with prey-taxis over a one-dimensional habitat. The free boundary represents the spreading front of the predator species. The global existence and uniqueness of classical solutions to this system are established by the contraction mapping principle. With an eye on the biological interpretations, numerical simulations are provided which give a real insight into the behavior of the free boundary...

A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems

Hideki Murakawa (2009)

Kybernetika

This paper deals with nonlinear diffusion problems involving degenerate parabolic problems, such as the Stefan problem and the porous medium equation, and cross-diffusion systems in population ecology. The degeneracy of the diffusion and the effect of cross-diffusion, that is, nonlinearities of the diffusion, complicate its analysis. In order to avoid the nonlinearities, we propose a reaction-diffusion system with solutions that approximate those of the nonlinear diffusion problems. The reaction-diffusion...

Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance

Temur A. Jangveladze, Zurab V. Kiguradze (2010)

Applications of Mathematics

The nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is considered. The asymptotic behavior as t → ∞ of solutions for two initial-boundary value problems are studied. The problem with non-zero conditions on one side of the lateral boundary is discussed. The problem with homogeneous boundary conditions is studied too. The rates of convergence are given. Results presented show the difference between stabilization characters of solutions of these...

Controllability of a parabolic system with a diffusive interface

Jérôme Le Rousseau, Matthieu Léautaud, Luc Robbiano (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

We consider a linear parabolic transmission problem across an interface of codimension one in a bounded domain or on a Riemannian manifold, where the transmission conditions involve an additional parabolic operator on the interface. This system is an idealization of a three-layer model in which the central layer has a small thickness δ . We prove a Carleman estimate in the neighborhood of the interface for an associated elliptic operator by means of partial estimates in several microlocal regions....

Cross-Diffusion Systems with Entropy Structure

Jüngel, Ansgar (2017)

Proceedings of Equadiff 14

Some results on cross-diffusion systems with entropy structure are reviewed. The focus is on local-in-time existence results for general systems with normally elliptic diffusion operators, due to Amann, and global-in-time existence theorems by Lepoutre, Moussa, and co-workers for cross-diffusion systems with an additional Laplace structure. The boundedness-by-entropy method allows for global bounded weak solutions to certain diffusion systems. Furthermore, a partial result on the uniqueness of weak...

Dynamics and patterns of an activator-inhibitor model with cubic polynomial source

Yanqiu Li, Juncheng Jiang (2019)

Applications of Mathematics

The dynamics of an activator-inhibitor model with general cubic polynomial source is investigated. Without diffusion, we consider the existence, stability and bifurcations of equilibria by both eigenvalue analysis and numerical methods. For the reaction-diffusion system, a Lyapunov functional is proposed to declare the global stability of constant steady states, moreover, the condition related to the activator source leading to Turing instability is obtained in the paper. In addition, taking the...

Finite-time blow-up in a two-species chemotaxis-competition model with single production

Masaaki Mizukami, Yuya Tanaka (2023)

Archivum Mathematicum

This paper is concerned with blow-up of solutions to a two-species chemotaxis-competition model with production from only one species. In previous papers there are a lot of studies on boundedness for a two-species chemotaxis-competition model with productions from both two species. On the other hand, finite-time blow-up was recently obtained under smallness conditions for competitive effects. Now, in the biological view, the production term seems to promote blow-up phenomena; this implies that the...

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