Exact solutions and qualitative features of nonlinear hyperbolic reaction—diffusion equations with delay

2015 ◽  
Vol 49 (5) ◽  
pp. 622-635 ◽  
Author(s):  
A. D. Polyanin ◽  
V. G. Sorokin ◽  
A. V. Vyazmin
Keyword(s):  
Exact Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Equations With Delay

scholarly journals Uniqueness of fast travelling fronts in reaction–diffusion equations with delay

2008 ◽  
Vol 464 (2098) ◽  
pp. 2591-2608 ◽  
Author(s):  
Maitere Aguerrea ◽  
Sergei Trofimchuk ◽  
Gabriel Valenzuela
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Delayed Reaction ◽  
Birth Function ◽  
Scale Of Banach Spaces ◽  
Large Velocity ◽  
Travelling Fronts ◽  
Equations With Delay

We consider positive travelling fronts, u ( t ,  x )= ϕ ( ν . x + ct ), ϕ (−∞)=0, ϕ (∞)= κ , of the equation u t ( t ,  x )=Δ u ( t ,  x )− u ( t ,  x )+ g ( u ( t − h ,  x )), x ∈ m . This equation is assumed to have exactly two non-negative equilibria: u 1 ≡0 and u 2 ≡ κ >0, but the birth function g ∈ C 2 ( ,  ) may be non-monotone on [0, κ ]. We are therefore interested in the so-called monostable case of the time-delayed reaction–diffusion equation. Our main result shows that for every fixed and sufficiently large velocity c , the positive travelling front ϕ ( ν . x + ct ) is unique (modulo translations). Note that ϕ may be non-monotone. To prove uniqueness, we introduce a small parameter ϵ =1/ c and realize a Lyapunov–Schmidt reduction in a scale of Banach spaces.

scholarly journals On the Long Time Simulation of Reaction-Diffusion Equations with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Dongfang Li ◽  
Chengjian Zhang
Keyword(s):  
Numerical Method ◽  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Open Problems ◽  
Time Simulation ◽  
Long Time ◽  
Continuous Problem ◽  
Equations With Delay

For a consistent numerical method to be practically useful, it is widely accepted that it must preserve the asymptotic stability of the original continuous problem. However, in this study, we show that it may lead to unreliable numerical solutions in long time simulation even if a classical numerical method gives a larger stability region than that of the original continuous problem. Some numerical experiments on the reaction-diffusion equations with delay are presented to confirm our findings. Finally, some open problems on the subject are proposed.

scholarly journals Exact Solutions of Coupled Multispecies Linear Reaction–Diffusion Equations on a Uniformly Growing Domain

PLoS ONE ◽  
2015 ◽  
Vol 10 (9) ◽  
pp. e0138894 ◽  
Author(s):  
Matthew J. Simpson ◽  
Jesse A. Sharp ◽  
Liam C. Morrow ◽  
Ruth E. Baker
Keyword(s):  
Exact Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Growing Domain ◽  
Linear Reaction
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