scholarly journals Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction–diffusion equations

2012 ◽  
Vol 386 (2) ◽  
pp. 542-558 ◽  
Author(s):  
Peter De Maesschalck ◽  
Nikola Popović
Keyword(s):  
Wave Speed ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations

EXTERNAL NOISE AND FRONT PROPAGATION IN REACTION-TRANSPORT SYSTEMS WITH INERTIA: THE MEAN SPEED OF FISHER WAVES

2002 ◽  
Vol 02 (04) ◽  
pp. R109-R124 ◽  
Author(s):  
WERNER HORSTHEMKE
Keyword(s):  
Wave Speed ◽  
Reaction Diffusion ◽  
Front Propagation ◽  
External Noise ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Transport Systems ◽  
The Mean ◽  
Reaction Transport ◽  
Fisher Waves

We review the effect of spatiotemporal noise, white in time and colored in space, on front propagation in systems of reacting and dispersing particles, where the particle motion displays inertia or persistence. We discuss the three main approaches that have been developed to describe transport with inertia, namely hyperbolic reaction-diffusion equations, reaction-Cattaneo systems or reaction-telegraph equations, and reaction random walks. We focus on the mean speed of Fisher waves in these systems and study in particular reaction random walks, which are the most natural generalization of reaction-diffusion equations. Hyperbolic reaction-diffusion equations account for inertia in the transport process in an ad hoc way, whereas the other reaction-transport systems have a proper macroscopic or microscopic foundation. For the former, external noise affects neither the mean wave speed nor the region in parameter space for which Fisher waves exist. For the latter, external noise increases the mean wave speed of Fisher waves and decreases the upper limit for the characteristic time of the transport process, below which propagating fronts exist.

scholarly journals Minimum wave speeds in monostable reaction–diffusion equations: sharp bounds by polynomial optimization

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200450
Author(s):  
Jason J. Bramburger ◽  
David Goluskin
Keyword(s):  
Lower Bounds ◽  
Wave Speed ◽  
Polynomial Optimization ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Upper And Lower Bounds ◽  
Wave Speeds ◽  
Minimum Wave Speed

Many monostable reaction–diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. We present methods for finding upper and lower bounds on minimum wave speed. They rely on constructing trapping boundaries for dynamical systems whose heteroclinic connections correspond to the travelling waves. Simple versions of this approach can be carried out analytically but often give overly conservative bounds on minimum wave speed. When the reaction–diffusion equations being studied have polynomial nonlinearities, our approach can be implemented computationally using polynomial optimization. For scalar reaction–diffusion equations, we present a general method and then apply it to examples from the literature where minimum wave speeds were unknown. The extension of our approach to multi-component reaction–diffusion systems is then illustrated using a cubic autocatalysis model from the literature. In all three examples and with many different parameter values, polynomial optimization computations give upper and lower bounds that are within 0.1% of each other and thus nearly sharp. Upper bounds are derived analytically as well for the scalar reaction-diffusion equations.

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 Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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