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Register a new userAdmissible speeds in spatially periodic bistable reaction-diffusion equations
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Spatially periodic reaction-diffusion equations typically admit pulsating waves which describe the transition from one steady state to another. Due to the heterogeneity, in general such an equation is not invariant by rotation and therefore the speed of the pulsating wave may a priori depend on its direction. However, little is actually known in the literature about whether it truly does: surprisingly, it is even known in the one-dimensional monostable Fisher-KPP case that the speed is the same in the opposite directions despite the lack of symmetry. Here we investigate this issue in the bistable case and show that the set of admissible speeds is actually rather large, which means that the shape of propagation may indeed be asymmetrical. More precisely, we show in any spatial dimension that one can choose an arbitrary large number of directions , and find a spatially periodic bistable type equation to achieve any combination of speeds in those directions, provided those speeds have the same sign. In particular, in spatial dimension 1 and unlike the Fisher-KPP case, any pair of (either nonnegative or nonpositive) rightward and leftward wave speeds is admissible. We also show that these variations in the speeds of bistable pulsating waves lead to strongly asymmetrical situations in the multistable equations.
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This paper is devoted to the study of the large time dynamics of bounded solutions of reaction-diffusion equations with unbounded initial support in R N. We first prove a general Freidlin-G{a}rtner type formula for the spreading speeds of the solutions in any direction. This formula holds under general assumptions on the reaction and for solutions emanating from initial conditions with general unbounded support, whereas most of earlier results were concerned with more specific reactions and compactly supported or almost-planar initial conditions. We also prove some results of independent interest on some conditions guaranteeing the spreading of solutions with large initial support and the link between these conditions and the existence of traveling fronts with positive speed. Furthermore, we show some flattening properties of the level sets of the solutions if initially supported on subgraphs. We also investigate the special case of asymptotically conical-shaped initial conditions. For Fisher-KPP equations, we prove some asymptotic one-dimensional symmetry properties for the elements of the $Omega$-limit set of the solutions, in the spirit of a conjecture of De Giorgi for stationary solutions of Allen-Cahn equations. Lastly, we show some logarithmicin-time estimates of the lag of the position of the solutions with respect to that of a planar front with minimal speed, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. The proofs use a mix of ODE and PDE methods, as well as some geometric arguments. The paper also contains some related conjectures and open problems.
Several stains of the intracellular parasitic bacterium Wolbachia limit severely the competence of the mosquitoes Aedes aegypti as a vector of dengue fever and possibly other arboviroses. For this reason, the release of mosquitoes infected by this bacterium in natural populations is presently considered a promising tool in the control of these diseases. Following works by M. Turelli [4] and subsequently M. Strugarek et al. [21, 22], we consider a simple scalar reaction-diffusion model describing the evolution of the proportion of infected mosquitoes, sufficient to reveal the bistable nature of the Wolbachia dynamics. A simple distributed feedback law is proposed, whose application on a compact domain during finite time is shown to be sufficient to invade the whole space. The corresponding stabilization result is established for any space dimension.
Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class of stochastic reaction-diffusion partial differential equations with cubic nonlinearity. Dividing space into overlapping finite elements, a special coupling condition between neighbouring elements preserves the self-adjoint dynamics and controls interelement interactions. When the interelement coupling parameter is small, an averaging method and an asymptotic expansion of the slow modes show that the macroscopic discrete model will be a family of coupled stochastic ordinary differential equations which describe the evolution of the grid values. This modelling shows the importance of subgrid scale interaction between noise and spatial diffusion and provides a new rigourous approach to constructing semi-discrete approximations to stochastic reaction-diffusion partial differential equations.
We investigate spreading properties of solutions of a large class of two-component reaction-diffusion systems, including prey-predator systems as a special case. By spreading properties we mean the long time behaviour of solution fronts that start from localized (i.e. compactly supported) initial data. Though there are results in the literature on the existence of travelling waves for such systems, very little has been known-at least theoretically-about the spreading phenomena exhibited by solutions with compactly supported initial data. The main difficulty comes from the fact that the comparison principle does not hold for such systems. Furthermore, the techniques that are known for travelling waves such as fixed point theorems and phase portrait analysis do not apply to spreading fronts. In this paper, we first prove that spreading occurs with definite spreading speeds. Intriguingly, two separate fronts of different speeds may appear in one solution-one for the prey and the other for the predator-in some situations.
We study a family of reaction-diffusion equations that present a doubly nonlinear character given by a combination of the $p$-Laplacian and the porous medium operators. We consider the so-called slow diffusion regime, corresponding to a degenerate behaviour at the level 0, ormalcolor in which nonnegative solutions with compactly supported initial data have a compact support for any later time. For some results we will also require $pge2$ to avoid the possibility of a singular behaviour away from 0. Problems in this family have a unique (up to translations) travelling wave with a finite front. When the initial datum is bounded, radially symmetric and compactly supported, we will prove that solutions converging to 1 (which exist, as we show, for all the reaction terms under consideration for wide classes of initial data) do so by approaching a translation of this unique traveling wave in the radial direction, but with a logarithmic correction in the position of the front when the dimension is bigger than one. As a corollary we obtain the asymptotic location of the free boundary and level sets in the non-radial case up to an error term of size $O(1)$. In dimension one we extend our results to cover the case of non-symmetric initial data, as well as the case of bounded initial data with supporting sets unbounded in one direction of the real line. A main technical tool of independent interest is an estimate for the flux. Most of our results are new even for the special cases of the porous medium equation and the $p$-Laplacian evolution equation.
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