scholarly journals Curved fronts of bistable reaction–diffusion equations with nonlinear convection

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hui-Ling Niu ◽  
Jiayin Liu
Keyword(s):  
Dimensional Space ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Asymptotically Stable ◽  
Two Dimensional ◽  
Nonlinear Convection ◽  
Globally Asymptotically Stable

Abstract This paper is concerned with traveling curved fronts of bistable reaction–diffusion equations with nonlinear convection in a two-dimensional space. By constructing super- and subsolutions, we establish the existence of traveling curved fronts. Furthermore, we show that the traveling curved front is globally asymptotically stable.

scholarly journals A three‐level time‐split MacCormack method for two‐dimensional nonlinear reaction‐diffusion equations

2020 ◽  
Vol 92 (12) ◽  
pp. 1681-1706 ◽  
Author(s):  
Eric Ngondiep ◽  
Nabil Kerdid ◽  
Mohammed Abdulaziz Mohammed Abaoud ◽  
Ibrahim Abdulaziz Ibrahim Aldayel
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Nonlinear Reaction ◽  
Maccormack Method

Stripes or spots? Nonlinear effects in bifurcation of reaction—diffusion equations on the square

1991 ◽  
Vol 434 (1891) ◽  
pp. 413-417 ◽  
Keyword(s):  
Pattern Formation ◽  
General Pattern ◽  
Reaction Diffusion ◽  
Nonlinear Effects ◽  
Neural Nets ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Linearized Model ◽  
Selection Of

Bifurcation to spatial patterns in a two-dimensional reaction—diffusion medium is considered. The selection of stripes versus spots is shown to depend on the nonlinear terms and cannot be discerned from the linearized model. The absence of quadratic terms leads to stripes but in most common models quadratic terms will lead to spot patterns. Examples that include neural nets and more general pattern formation equations are considered.

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