共查询到20条相似文献,搜索用时 609 毫秒
1.
In this paper, we study the pulsating fronts of reaction–advection-diffusion equations with two types of nonlinear term in periodic excitable media. Firstly, for the case with combustion nonlinearity, the unique front is proved to decay exponentially when it approaches the unstable limiting state. Secondly, for the degenerate monostable type nonlinearity, it is shown that the front with critical speed is unique, monotone and decays exponentially at negative end, while the fronts of noncritical speeds decay to zero non-exponentially. 相似文献
2.
We study flow-induced enhancement of the speed of pulsating traveling fronts for reaction-diffusion equations, and quenching of reaction by fluid flows. We prove, for periodic flows in two dimensions and any combustion-type reaction, that the front speed is proportional to the square root of the (homogenized) effective diffusivity of the flow. We show that this result does not hold in three and more dimensions. We also prove conjectures from Audoly, Berestycki and Pomeau (2000) [1], Berestycki (2003) [3], Fannjiang, Kiselev and Ryzhik (2006) [11] for cellular flows, concerning the rate of speed-up of fronts and the minimal flow amplitude necessary to quench solutions with initial data of a fixed (large) size. 相似文献
3.
Mohammad Ibrahim El Smaily 《Annali di Matematica Pura ed Applicata》2010,189(1):47-66
This paper is concerned with some nonlinear propagation phenomena for reaction–advection–diffusion equations in a periodic framework. It deals with travelling wave solutions of the equation $u_t =\nabla\cdot(A(z)\nabla u)\;+q(z)\cdot\nabla u+\,f(z,u),\qquad t\in\mathbb{R},\quad z\in\Omega,$ propagating with a speed c. In the case of a “combustion” nonlinearity, the speed c exists and it is unique, while the front u is unique up to a translation in t. We give a min–max and a max–min formula for this speed c. On the other hand, in the case of a “ZFK” or a “KPP” nonlinearity, there exists a minimal speed of propagation c*. In this situation, we give a min–max formula for c*. Finally, we apply this min–max formula to prove a variational formula involving eigenvalue problems for the minimal speed c* in the “KPP” case. 相似文献
4.
This paper deals with front propagation for discrete periodic monostable equations. We show that there is a minimal wave speed
such that a pulsating traveling front solution exists if and only if the wave speed is above this minimal speed. Moreover,
in comparing with the continuous case, we prove the convergence of discretized minimal wave speeds to the continuous minimal
wave speed. 相似文献
5.
Henri Berestycki François Hamel Nikolai Nadirashvili 《Comptes Rendus Mathematique》2004,339(3):163-168
This Note is devoted to the analysis of some propagation phenomena for reaction–diffusion–advection equations with Fisher or Kolmogorov–Petrovsky–Piskunov (KPP) type nonlinearities. Some formulæ for the speed of propagation of pulsating fronts in periodic domains are given. These allow us to describe the influence of the various terms in the equation or of geometry on propagation. We also derive results for propagation speed in more general domains without periodicity. To cite this article: H. Berestycki et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004). 相似文献
6.
罗操 《数学物理学报(B辑英文版)》2013,33(1):75-83
The paper is concerned with the long-time behaviour of the travelling fronts of the damped wave equation αutt+ut=uxx V’(u) on R.The long-time asymptotics of the solutions of this equation are quite similar to those of the corresponding reaction-diffusion equation ut=uxxV’(u).Whereas a lot is known about the local stability of travelling fronts in parabolic systems,for the hyperbolic equations it is only briefly discussed when the potential V is of bistable type.However,for the combustion or monostable type of V,the problem is much more complicated.In this paper,a local stability result for travelling fronts of this equation with combustion type of nonlinearity is established.And then,the result is extended to the damped wave equation with a case of monostable pushed front. 相似文献
7.
This paper is concerned with the existence and stability of periodic traveling curved fronts for reaction-diffusion equations with time-periodic bistable nonlinearity in two-dimensional space. By constructing supersolution and subsolution, we prove the existence of periodic traveling wave fronts. Furthermore, we show that the front is globally stable. 相似文献
8.
This paper is concerned with the analysis of speed-up of reaction-diffusion-advection traveling fronts in infinite cylinders with periodic boundary conditions. The advection is a shear flow with a large amplitude and the reaction is nonnegative, with either positive or zero ignition temperature. The unique or minimal speeds of the traveling fronts are proved to be asymptotically linear in the flow amplitude as the latter goes to infinity, solving an open problem from Berestycki (Nonlinear PDEs in condensed matter and reactive flows, Kluwer, Doordrecht, 2003). The asymptotic growth rate is characterized explicitly as the unique or minimal speed of traveling fronts for a limiting degenerate problem, and the convergence of the regular traveling fronts to the degenerate ones is proved for positive ignition temperatures under an additional Hörmander-type condition on the flow. 相似文献
9.
This paper is devoted to the study of pulsating fronts and pulsating front-like entire solutions for a reaction–advection–diffusion model of two competing species in a periodic habitat. Under certain assumptions, the competition system admits a leftward and a rightward pulsating fronts in the bistable case. In this work we construct some other types of entire solutions by interacting the leftward and rightward pulsating fronts. Some of these entire solutions behave as the two pulsating fronts approaching each other from both sides of the x-axis, which turn out to be unique and Liapunov stable 2-dimensional manifolds of solutions, furthermore, the leftward and rightward pulsating fronts are on the boundary of these 2-dimensional manifolds. The others behave as the two pulsating fronts propagating from one side of the x-axis, the faster one then invades the slower one as . These kinds of pulsating front-like entire solutions then provide some new spreading ways other than pulsating fronts for two strongly competing species interacting in a heterogeneous habitat. 相似文献
10.
This Note deals with reaction–diffusion equations in periodic media arising in the modelling of persistence and invasions of biological species. We investigate the influence of the distribution of heterogeneities on the existence and uniqueness of nonzero stationary states and on the existence of invading pulsating travelling fronts. To cite this article: H. Berestycki et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004). 相似文献
11.
This paper examines a simplified active combustion model in which the reaction influences the flow. We consider front propagation in a reactive Boussinesq system in an infinite vertical strip. Nonlinear stability of planar fronts is established for narrow domains when the Rayleigh number is not too large. Planar fronts are shown to be linearly unstable with respect to long‐wavelength perturbations if the Rayleigh number is sufficiently large. We also prove uniform bounds on the bulk burning rate and the Nusselt number in the KPP reaction case. © 2003 Wiley Periodicals, Inc. 相似文献
12.
《Nonlinear Analysis: Real World Applications》2003,4(3):503-524
Travelling waves are natural phenomena ubiquitously for reaction–diffusion systems in many scientific areas, such as in biophysics, population genetics, mathematical ecology, chemistry, chemical physics, and so on. It is pretty well understood for a diffusing Lotka–Volterra system that there exist travelling wave solutions which propagate from an equilibrium point to another one. In this paper, we prove there exists, at least, a wave front—the monotone travelling wave—with its minimal speed. 相似文献
13.
We investigate the large time behavior of solutions of reaction–diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic holes, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one. 相似文献
14.
In this work we study the existence of wave solutions for a scalar reaction-diffusion equation of bistable type posed in a multi-dimensional periodic medium. Roughly speaking our result states that bistability ensures the existence of waves for both balanced and unbalanced reaction term. Here the term wave is used to describe either pulsating travelling wave or standing transition solution. As a special case we study a two-dimensional heterogeneous Allen–Cahn equation in both cases of slowly varying medium and rapidly oscillating medium. We prove that bistability occurs in these two situations and we conclude to the existence of waves connecting \(u = 0\) and \(u = 1\). Moreover in a rapidly oscillating medium we derive a sufficient condition that guarantees the existence of pulsating travelling waves with positive speed in each direction. 相似文献
15.
The effect of turbulence on mixing in prototype reaction‐diffusion systems is analyzed here in the special situation where the turbulence is modeled ideally with two separated scales consisting of a large‐scale mean flow plus a small‐scale spatiotemporal periodic flow. In the limit of fast reaction and slow diffusion, it is rigorously proved that the turbulence does not contribute to the location of the mixing zone in the limit and that this mixing zone location is determined solely by advection of the large‐scale velocity field. This surprising result contrasts strongly with earlier work of the authors that always yields a large‐scale propagation speed enhanced by small‐scale turbulence for propagating fronts. The mathematical reasons for these differences are pointed out. This main theorem rigorously justifies the limit equilibrium approximations utilized in non‐premixed turbulent diffusion flames and condensation‐evaporation modeling in cloud physics in the fast reaction limit. The subtle nature of this result is emphasized by explicit examples presented in the fast reaction and zero‐diffusion limit with a nontrivial effect of turbulence on mixing in the limit. The situation with slow reaction and slow diffusion is also studied in the present work. Here the strong stirring by turbulence before significant reaction occurs necessarily leads to a homogenized limit with the strong mixing effects of turbulence expressed by a rigorous turbulent diffusivity modifying the reaction‐diffusion equations. Physical examples from non‐premixed turbulent combustion and cloud microphysics modeling are utilized throughout the paper to motivate and interpret the mathematical results. © 2000 John Wiley & Sons, Inc. 相似文献
16.
17.
We consider a simple scalar reaction‐advection‐diffusion equation with ignition‐type nonlinearity and discuss the following question: What kinds of velocity profiles are capable of quenching any given flame, provided the velocity's amplitude is adequately large? Even for shear flows, the answer turns out to be surprisingly subtle. If the velocity profile changes in space so that it is nowhere identically constant, (or if it is identically constant only in a region of small measure) then the flow can quench any initial data. But if the velocity profile is identically constant in a sizable region, then the ensuing flow is incapable of quenching large enough flames, no matter how much larger is the amplitude of this velocity. The constancy region must be wider across than a couple of laminar propagating front‐widths. The proof uses a linear PDE associated to the nonlinear problem and quenching follows when the PDE is hypoelliptic. The techniques used allow the derivation of new, nearly optimal bounds on the speed of traveling wave solutions. © 2000 John Wiley & Sons, Inc. 相似文献
18.
Grgoire Nadin 《Journal de Mathématiques Pures et Appliquées》2009,92(3):232-262
This paper is concerned with the existence of pulsating traveling fronts for the equation:(1)
∂tu−(A(t,x)u)+q(t,x)u=f(t,x,u),