Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion

2012 ◽  
Vol 82 (6) ◽  
pp. 1112-1132 ◽  
Author(s):  
G. Gambino ◽  
M.C. Lombardo ◽  
M. Sammartino
Keyword(s):  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Cross Diffusion ◽  
Traveling Fronts ◽  
Nonlinear Reaction

Unique continuation for a reaction-diffusion system with cross diffusion

2019 ◽  
Vol 27 (4) ◽  
pp. 511-525 ◽  
Author(s):  
Bin Wu ◽  
Ying Gao ◽  
Zewen Wang ◽  
Qun Chen
Keyword(s):  
Unique Continuation ◽  
Reaction Diffusion ◽  
Cauchy Data ◽  
Carleman Estimate ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Lateral Boundary ◽  
Strongly Coupled ◽  
Cross Diffusion ◽  
Coupled Reaction

Abstract This paper concerns unique continuation for a reaction-diffusion system with cross diffusion, which is a drug war reaction-diffusion system describing a simple dynamic model of a drug epidemic in an idealized community. We first establish a Carleman estimate for this strongly coupled reaction-diffusion system. Then we apply the Carleman estimate to prove the unique continuation, which means that the Cauchy data on any lateral boundary determine the solution uniquely in the whole domain.

Numerical investigation on oscillatory Turing patterns in a two-layer coupled reaction–diffusion system

2016 ◽  
Vol 30 (07) ◽  
pp. 1650085 ◽  
Author(s):  
Xin-Zheng Li ◽  
Zhan-Guo Bai ◽  
Yan Li ◽  
Kun Zhao
Keyword(s):  
Dimensional Space ◽  
Reaction Diffusion ◽  
Matching Condition ◽  
Turing Instability ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Square Lattice ◽  
Stripe Pattern ◽  
Brusselator Model ◽  
Dynamic Patterns

In this paper, various kinds of spontaneous dynamic patterns are investigated based on a two-layer nonlinearly coupled Brusselator model. It is found that, when the Hopf mode or supercritical Turing mode respectively plays major role in the short or long wavelength mode layer, the dynamic patterns appear under the action of nonlinearly coupling interactions in the reaction–diffusion system. The stripe pattern can change its symmetrical structure and form other graphics when influenced by small perturbations sourced from other modes. If two supercritical Turing modes are nonlinearly coupled together, the transition from Turing instability to Hopf instability may appear in the short wavelength mode layer, and the twinkling-eye square pattern, traveling and rotating pattern will be obtained in the two subsystems. If Turing mode and subharmonic Turing mode satisfy the three-mode resonance relation, twinkling-eye patterns are generated, and oscillating spots are arranged as square lattice in the two-dimensional space. When the subharmonic Turing mode satisfies the spatio-temporal phase matching condition, the traveling patterns, including the rhombus, hexagon and square patterns are obtained, which presents different moving velocities. It is found that the wave intensity plays an important role in pattern formation and pattern selection.

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