scholarly journals Hopf Bifurcation with the Spatial Average of an Activator in a Radially Symmetric Free Boundary Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-18
Author(s):  
YoonMee Ham
Keyword(s):  
Hopf Bifurcation ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Interface Problem ◽  
Spatial Average ◽  
Radially Symmetric Solutions ◽  
Dimensional Ball

An interface problem derived by a bistable reaction-diffusion system with the spatial average of an activator is studied on ann-dimensional ball. We analyze the existence of the radially symmetric solutions and the occurrence of Hopf bifurcation as a parameter varies in two and three-dimensional spaces.

A HOPF BIFURCATION IN A FREE BOUNDARY PROBLEM DEPENDING ON THE SPATIAL AVERAGE OF AN ACTIVATOR

2003 ◽  
Vol 13 (10) ◽  
pp. 3135-3145 ◽  
Author(s):  
YOONMEE HAM
Keyword(s):  
Hopf Bifurcation ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Spatial Average ◽  
Inhibitor System ◽  
A Free Boundary Problem ◽  
Dynamics Of Interfaces ◽  
Activator Inhibitor

We shall consider an activator–inhibitor system proposed by Radehaus [1990]. In this system, the activator is inhibited by not only the inhibitor but also its own spatial average. The purpose of this paper is to analyze the dynamics of interfaces in an interfacial problem which is reduced from the system in order to examine how this problem is different from an activator–inhibitor system [Ham-Lee et al., 1994].

An evolutional free-boundary problem of a reaction–diffusion–advection system

2017 ◽  
Vol 147 (3) ◽  
pp. 615-648 ◽  
Author(s):  
Ling Zhou ◽  
Shan Zhang ◽  
Zuhan Liu
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Ecological Model ◽  
Reaction Diffusion ◽  
Spreading Speed ◽  
Heterogeneous Environment ◽  
Long Run ◽  
Strong Competition ◽  
Asymptotic Spreading

In this paper we consider a system of reaction–diffusion–advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. The evolution of the free-boundary problem is discussed, which is an extension of the results of Du and Lin (Discrete Contin. Dynam. Syst. B19 (2014), 3105–3132). Precisely, when u is an inferior competitor, we prove that (u, v) → (0, V) as t→∞. When u is a superior competitor, we prove that a spreading–vanishing dichotomy holds, namely, as t→∞, either h(t)→∞ and (u, v) → (U, 0), or limt→∞h(t) < ∞ and (u, v) → (0, V). Moreover, in a weak competition case, we prove that two competing species coexist in the long run, while in a strong competition case, two species spatially segregate as the competition rates become large. Furthermore, when spreading occurs, we obtain some rough estimates of the asymptotic spreading speed.

Travelling wave solutions of a reaction—infiltration problem and a related free boundary problem

1994 ◽  
Vol 5 (3) ◽  
pp. 255-265 ◽  
Author(s):  
John Chadam ◽  
Xinfu Chen ◽  
Elena Comparini ◽  
Riccardo Ricci
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Suitable Norm ◽  
Formal Limit ◽  
A Free Boundary Problem

We consider travelling wave solutions of a reaction–diffusion system arising in a model for infiltration with changing porosity due to reaction. We show that the travelling wave solution exists, and is unique modulo translations. A small parameter ε appears in this problem. The formal limit as ε → 0 is a free boundary problem. We show that the solution for ε > 0 tends, in a suitable norm, to the solution of the formal limit.

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