Propagation speed of travelling fronts in non local reaction–diffusion equations

2005 ◽  
Vol 60 (5) ◽  
pp. 797-819 ◽  
Author(s):  
Jérôme Coville ◽  
Louis Dupaigne
Keyword(s):  
Local Reaction ◽  
Propagation Speed ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Non Local ◽  
Travelling Fronts

scholarly journals Propagation direction of the bistable travelling wavefront for delayed non-local reaction diffusion equations

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180898
Author(s):  
Manjun Ma ◽  
Jiajun Yue ◽  
Chunhua Ou
Keyword(s):  
Population Biology ◽  
Wave Speed ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Kernel Functions ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Non Local ◽  
Selection Mechanisms

For delayed non-local reaction–diffusion equations arising from population biology, selection mechanisms of the speed sign for the bistable travelling wavefront have not been found. In this paper, based on the theory of asymptotic speeds of spread for monotone semiflows, we firstly provide an interval of values of wave speed and a novel general condition for determining the speed sign by applying the comparison principle and the globally asymptotic stability of the bistable travelling wave. Moreover, through constructing novel upper/lower solutions, we give explicit conditions for the speed sign to be positive or negative. The obtained results are efficiently applied to three classical forms of the kernel functions.

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.

Travelling waves for delayed reaction–diffusion equations with global response

2005 ◽  
Vol 462 (2065) ◽  
pp. 229-261 ◽  
Author(s):  
Teresa Faria ◽  
Wenzhang Huang ◽  
Jianhong Wu
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Index Formula ◽  
Delayed Response ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Non Local

We develop a new approach to obtain the existence of travelling wave solutions for reaction–diffusion equations with delayed non-local response. The approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain Banach space, coupled with an index formula of the associated Fredholm operator and some careful estimation of the nonlinear perturbation. The general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation, and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non-local and delayed response for the matured population.

Stability of fronts and transient behaviour in KPP systems

2010 ◽  
Vol 466 (2118) ◽  
pp. 1769-1788 ◽  
Author(s):  
Anna Ghazaryan ◽  
Peter Gordon ◽  
Alexander Virodov
Keyword(s):  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Travelling Wave Solutions ◽  
Transient Behaviour ◽  
Stable Pattern ◽  
Wave Solutions ◽  
Travelling Fronts ◽  
Pressure Driven

We consider a system of two reaction diffusion equations with the Kolmogorov–Petrovsky–Piskunov (KPP) type nonlinearity which describes propagation of pressure-driven flames. It is known that the system admits a family of travelling wave solutions parameterized by their velocity. In this paper, we show that these travelling fronts are stable under the assumption that perturbations belong to an appropriate weighted L 2 space. We also discuss an interesting meta-stable pattern the system exhibits in certain cases.

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