Nonlinear Stability of Traveling Waves to a Hyperbolic-Parabolic System Modeling Chemotaxis

2010 ◽  
Vol 70 (5) ◽  
pp. 1522-1541 ◽  
Author(s):  
Tong Li ◽  
Zhi-an Wang
Keyword(s):  
Nonlinear Stability ◽  
Parabolic System ◽  
Traveling Waves ◽  
System Modeling

NONLINEAR STABILITY OF LARGE AMPLITUDE VISCOUS SHOCK WAVES OF A GENERALIZED HYPERBOLIC–PARABOLIC SYSTEM ARISING IN CHEMOTAXIS

2010 ◽  
Vol 20 (11) ◽  
pp. 1967-1998 ◽  
Author(s):  
TONG LI ◽  
ZHI-AN WANG
Keyword(s):  
Nonlinear Stability ◽  
Parabolic System ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Wave Band ◽  
Nonlinear Kinetics ◽  
Diffusive Limit ◽  
Viscous Shock Waves ◽  
The Stability

Traveling wave (band) behavior driven by chemotaxis was observed experimentally by Adler1,2 and was modeled by Keller and Segel.15 For a quasilinear hyperbolic–parabolic system that arises as a non-diffusive limit of the Keller–Segel model with nonlinear kinetics, we establish the existence and nonlinear stability of traveling wave solutions with large amplitudes. The numerical simulations are performed to show the stability of the traveling waves under various perturbations.

scholarly journals On a nonlinear parabolic system-modeling chemical reactions in rivers

2005 ◽  
Vol 4 (4) ◽  
pp. 889-899 ◽  
Author(s):  
Wenxiong Chen ◽  
◽  
Congming Li ◽  
Eric S. Wright ◽  
◽  
...  
Keyword(s):  
Chemical Reactions ◽  
Parabolic System ◽  
System Modeling ◽  
Nonlinear Parabolic System ◽  
Nonlinear Parabolic

scholarly journals Hyperbolic traveling waves driven by growth

2014 ◽  
Vol 24 (06) ◽  
pp. 1165-1195 ◽  
Author(s):  
Emeric Bouin ◽  
Vincent Calvez ◽  
Grégoire Nadin
Keyword(s):  
Nonlinear Stability ◽  
Traveling Waves ◽  
Hyperbolic Model ◽  
Kpp Equation ◽  
Transition Parameter ◽  
Minimal Speed ◽  
Traveling Fronts ◽  
Traveling Front ◽  
Maximal Speed ◽  
Existence And Stability

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed ϵ-1 (ϵ > 0), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter ϵ: for small ϵ the behavior is essentially the same as for the diffusive Fisher-KPP equation. However, for large ϵ the traveling front with minimal speed is discontinuous and travels at the maximal speed ϵ-1. The traveling fronts with minimal speed are linearly stable in weighted L2 spaces. We also prove local nonlinear stability of the traveling front with minimal speed when ϵ is smaller than the transition parameter.

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