Large amplitude solutions of spatially non-homogeneous non-local reaction diffusion equations

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.

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Non-local reaction-diffusion equations modelling predator-prey coevolution

Publicacions Matemàtiques ◽

10.5565/publmat_38294_04 ◽

1994 ◽

Vol 38 ◽

pp. 315-325 ◽

Cited By ~ 31

Author(s):

A. Calsina ◽

C. Perelló ◽

J. Saldaña

Keyword(s):

Local Reaction ◽

Reaction Diffusion ◽

Reaction Diffusion Equations ◽

Diffusion Equations ◽

Predator Prey ◽

Non Local

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Travelling waves for delayed reaction–diffusion equations with global response

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1554 ◽

2005 ◽

Vol 462 (2065) ◽

pp. 229-261 ◽

Cited By ~ 111

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.

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Time-dependent attractors for non-autonomous non-local reaction–diffusion equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210517000348 ◽

2018 ◽

Vol 148 (5) ◽

pp. 957-981

Author(s):

Tomás Caraballo ◽

Marta Herrera-Cobos ◽

Pedro Marín-Rubio

Keyword(s):

Existence And Uniqueness ◽

Local Reaction ◽

Reaction Diffusion ◽

Strong Solutions ◽

Reaction Diffusion Equations ◽

Time Dependent ◽

Reaction Diffusion Equation ◽

Pullback Attractors ◽

Non Local ◽

Bounded Sets

In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.

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