Asymptotic behaviour, uniqueness and stability of coexistence states of a non-cooperative reaction–diffusion model of nuclear reactors

2010 ◽  
Vol 140 (1) ◽  
pp. 189-201 ◽  
Author(s):  
Rui Peng ◽  
Dong Wei ◽  
Guoying Yang
Keyword(s):  
Asymptotic Behaviour ◽  
Diffusion Model ◽  
Blow Up ◽  
Nuclear Reactors ◽  
Reaction Diffusion ◽  
Fast Neutrons ◽  
Fuel Temperature ◽  
Reaction Diffusion Model ◽  
Coexistence States ◽  
Spatial Dimensions

We investigate a non-cooperative reaction-diffusion model arising in the theory of nuclear reactors and are concerned with the associated steady-state problem. We determine the asymptotic behaviour of the coexistence states near the point of bifurcation from infinity, which exhibits the following very interesting spatial blow-up pattern: when the fuel temperature reaches a certain value, the free fast neutrons undergoing nuclear reaction will blow up in each spatial point of the interior of the reactor. Without any restriction on spatial dimensions, we also discuss the uniqueness and stability of the coexistence states. Our results complement and sharpen those derived in two recent works by Arioli and Lóopez-Gómez.

scholarly journals CRITICAL EXPONENTS FOR A REACTION–DIFFUSION MODEL WITH ABSORPTIONS AND COUPLED BOUNDARY FLUX

2005 ◽  
Vol 48 (1) ◽  
pp. 241-252 ◽  
Author(s):  
Sining Zheng ◽  
Fengjie Li
Keyword(s):  
Diffusion Model ◽  
Critical Exponents ◽  
Nonlinear Boundary ◽  
Blow Up ◽  
Reaction Diffusion ◽  
Nonlinear Boundary Conditions ◽  
Reaction Diffusion Model ◽  
Precise Analysis ◽  
Two Parameters ◽  
Boundary Flux

AbstractThis paper deals with a reaction–diffusion model with inner absorptions and coupled nonlinear boundary conditions of exponential type. The critical exponents are described via a pair of parameters that satisfy a certain matrix equation containing all the six nonlinear exponents of the system. Whether the solutions blow up or not is determined by the signs of the two parameters. A more precise analysis, depending on the geometry of $\varOmega$ and the absorption coefficients, is proposed for the critical sign of the parameters.AMS 2000 Mathematics subject classification: Primary 35K55; 35B33

scholarly journals Synchronization of the Glycolysis Reaction-Diffusion Model via Linear Control Law

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1516
Author(s):  
Adel Ouannas ◽  
Iqbal M. Batiha ◽  
Stelios Bekiros ◽  
Jinping Liu ◽  
Hadi Jahanshahi ◽  
...  
Keyword(s):  
Diffusion Model ◽  
Reaction Diffusion ◽  
Linear Control ◽  
Control Law ◽  
Synchronization Problem ◽  
Reaction Diffusion Model ◽  
Spatial Dimensions ◽  
Error System ◽  
The Stability ◽  
Complex Formations

The Selkov system, which is typically employed to model glycolysis phenomena, unveils some rich dynamics and some other complex formations in biochemical reactions. In the present work, the synchronization problem of the glycolysis reaction-diffusion model is handled and examined. In addition, a novel convenient control law is designed in a linear form and, on the other hand, the stability of the associated error system is demonstrated through utilizing a suitable Lyapunov function. To illustrate the applicability of the proposed schemes, several numerical simulations are performed in one- and two-spatial dimensions.

Asymptotic behaviour of a reaction–diffusion model with a quiescent stage

2007 ◽  
Vol 463 (2080) ◽  
pp. 1029-1043 ◽  
Author(s):  
Kate Fang Zhang ◽  
Xiao-Qiang Zhao
Keyword(s):  
Asymptotic Behaviour ◽  
Diffusion Model ◽  
Wave Speed ◽  
Global Attractivity ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Model ◽  
Asymptotic Speed ◽  
Minimal Wave Speed ◽  
Quiescent Stage

This paper is devoted to the investigation of the asymptotic behaviour for a reaction–diffusion model with a quiescent stage. We first establish the existence of the asymptotic speed of spread and show that it coincides with the minimal wave speed for monotone travelling waves. Then we obtain a threshold result on the global attractivity of either zero or positive steady state in the case where the spatial domain is bounded.

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