BIFURCATIONS OF ONE-DIMENSIONAL REACTION–DIFFUSION EQUATIONS

2001 ◽  
Vol 11 (05) ◽  
pp. 1295-1306 ◽  
Author(s):  
CHANGPIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Boundary Value ◽  
Reaction Diffusion ◽  
Singularity Theory ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
One Dimensional ◽  
Boundary Value Condition

Bifurcations of a class of one-dimensional reaction–diffusion equations of the form u″+μu-uk=0, where μ is a parameter, 2≤k∈Z+, with boundary value condition u(0)=u(π)=0, are investigated. Using the singularity theory based on the Liapunov–Schmidt reduction, some characterization results are obtained.

Generic properties of equilibria of reaction-diffusion equations with variable diffusion

1985 ◽  
Vol 101 (1-2) ◽  
pp. 45-55 ◽  
Author(s):  
Carlos Rocha
Keyword(s):  
Parabolic Equations ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Equilibrium Solutions ◽  
Generic Properties ◽  
One Dimensional ◽  
Variable Diffusion ◽  
Node Type

SynopsisIt is shown that, generically, scalar one-dimensional parabolic equations ut = (a2(x)ux)x + f(u), x ∈ [0, 1], with Neumann boundary conditions, have all the equilibrium solutions hyperbolic.Moreover, the bifurcations of these equilibria are generically of the saddle-node type.

A stochastic simulation for solving scalar reaction–diffusion equations

1990 ◽  
Vol 22 (01) ◽  
pp. 88-100
Author(s):  
B. Chauvin ◽  
Rouault
Keyword(s):  
Monte Carlo ◽  
Limit Theorem ◽  
Branching Processes ◽  
Martingale Problem ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Convergence Problem ◽  
One Dimensional ◽  
Spaces Of Measures

A recent Monte Carlo method for solving one-dimensional reaction–diffusion equations is considered here as a convergence problem for a sequence of spatial branching processes with interaction. The martingale problem is studied and a limit theorem is proved by embedding spaces of measures in Sobolev spaces.

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