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 A Reaction-Diffusion Model for Interference in Meiotic Crossing Over

Genetics ◽  
2002 ◽  
Vol 161 (1) ◽  
pp. 365-372 ◽  
Author(s):  
Youhei Fujitani ◽  
Shintaro Mori ◽  
Ichizo Kobayashi
Keyword(s):  
Genetic Distance ◽  
Diffusion Model ◽  
Genetic Model ◽  
Initial Density ◽  
Reaction Diffusion ◽  
Crossover Point ◽  
Random Walker ◽  
Random Walkers ◽  
Reaction Diffusion Model ◽  
Two Parameters

Abstract One crossover point between a pair of homologous chromosomes in meiosis appears to interfere with occurrence of another in the neighborhood. It has been revealed that Drosophila and Neurospora, in spite of their large difference in the frequency of crossover points, show very similar plots of coincidence—a measure of the interference—against the genetic distance of the interval, defined as one-half the average number of crossover points within the interval. We here propose a simple reaction-diffusion model, where a “randomly walking” precursor becomes immobilized and matures into a crossover point. The interference is caused by pair-annihilation of the random walkers due to their collision and by annihilation of a random walker due to its collision with an immobilized point. This model has two parameters—the initial density of the random walkers and the rate of its processing into a crossover point. We show numerically that, as the former increases and/or the latter decreases, plotted curves of the coincidence vs. the genetic distance converge on a unique curve. Thus, our model explains the similarity between Drosophila and Neurospora without parameter values adjusted finely, although it is not a “genetic model” but is a “physical model,” specifying explicitly what happens physically.

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 GLOBAL EXISTENCE AND BLOW-UP FOR A DOUBLY DEGENERATE PARABOLIC EQUATION SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

2011 ◽  
Vol 54 (2) ◽  
pp. 309-324
Author(s):  
YONG-SHENG MI ◽  
CHUN-LAI MU ◽  
DENG-MING LIU
Keyword(s):  
Global Existence ◽  
Parabolic System ◽  
Parabolic Equations ◽  
Nonlinear Boundary ◽  
Blow Up ◽  
Nonlinear Boundary Conditions ◽  
Degenerate Parabolic Equations ◽  
Nonlinear Boundary Flux ◽  
Degenerate Parabolic ◽  
Boundary Flux

AbstractIn this paper, we deal with the global existence and blow-up of solutions to a doubly degenerative parabolic system with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of non-negative solutions, which extend the recent results of Zheng, Song and Jiang (S. N. Zheng, X. F. Song and Z. X. Jiang, Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), 308–324), Xiang, Chen and Mu (Z. Y. Xiang, Q. Chen, C. L. Mu, Critical curves for degenerate parabolic equations coupled via nonlinear boundary flux, Appl. Math. Comput. 189 (2007), 549–559) and Zhou and Mu (J. Zhou and C. L Mu, On critical Fujita exponents for degenerate parabolic system coupled via nonlinear boundary flux, Pro. Edinb. Math. Soc. 51 (2008), 785–805) to more general equations.

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