Entire solutions in monostable reaction-advection-diffusion equations in infinite cylinders

This paper is concerned with entire solutions of a monostable reaction-advection-diffusion equation in infinite cylinders without the condition f^′(u)≤f^′(0). By constructing a quasi-invariant manifold, we prove that there exist two classes of entire solutions. Furthermore, we show that one class of such entire solutions is unique up to space and time translation.     

Entire solutions of reaction-advection-diffusion equations with bistable nonlinearity in cylinders

This paper deals with entire solutions and the interaction of traveling wave fronts of bistable reaction-advection-diffusion equation with infinite cylinders. Assume that the equation admits three equilibria: two stable equilibria 0 and 1, and an unstable equilibrium θ. It is well known that there are different wave fronts connecting any two of those three equilibria. By considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions, we establish three different types of entire solutions. Finally, we analyze a model for shear flows in cylinders to illustrate our main results.     

Travelling fronts in integrodifferential equations

We provide results of the existence, uniqueness and asymptotic behavior of travelling-wave solutions for convolution equations involving different kinds of nonlinearities (bistable, ignition and monostable). We recover for these equations most of the known results about the standard equation $\text{?u}\text{?t}\text{+u″+f(u)=0}$. Some min–max formulas are also given. To cite this article: J. Coville, L. Dupaigne, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Entire solutions in reaction-advection-diffusion equations with bistable nonlinearities in heterogeneous media

This paper is concerned with entire solutions ( t ∈ R) for bistable reaction-advection-diffusion equations in heterogeneous media. By using traveling curved fronts connecting a constant unstable stationary state and a stable stationary state, we proved that there exist entire solutions behaving as two traveling curved fronts coming from opposite directions, and approaching each other. Furthermore, we prove that such an entire solution is unique and Liapunov stable. The key technique is to characterize the asymptotic behavior of solutions at infinity in term of appropriate subsolutions and supersolutions.     

Pulsating type entire solutions of monostable reaction-advection-diffusion equations in periodic excitable media

We establish the existence of pulsating type entire solutions of reaction-advection-diffusion equations with monostable nonlinearities in a periodic framework. Here the nonlinearities include the classic KPP case. The pulsating type entire solutions are defined in the whole space and for all time t∈R. By studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function, we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions, and approaching each other. The key techniques are to characterize the asymptotic behavior of the solutions as t→−∞ in terms of appropriate subsolutions and supersolutions.     

Generalized fronts in reaction-diffusion equations with mono-stable nonlinearity

We consider transition fronts (generalized traveling fronts) of mono-stable reaction-diffusion equations with spatially inhomogeneous nonlinearity. By constructing a cutoff function and using an approximate method, we establish the existence of transition fronts of the equation. Furthermore, we give the uniform non-degeneracy estimates of the solutions, such as a lower bound on the time derivative on some level sets, as well as an upper bound on the spatial derivative.     

Generalized fronts in reaction-diffusion equations with bistable nonlinearity

In this paper, we first study the existence of transition fronts (generalized traveling fronts) for reaction-diffusion equations with the spatially heterogeneous bistable nonlinearity. By constructing sub-solution and super-solution we then show that transition fronts are globally exponentially stable for the solutions of the Cauchy problem. Furthermore, we prove that transition fronts are unique up to translation in time by using the monotonicity in time and the exponential decay of such transition fronts.     

Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations

For a balanced bistable reaction-diffusion equation, an axisymmetric traveling front has been well known. This paper proves that an axially asymmetric traveling front with any positive speed does exist in a balanced bistable reaction-diffusion equation. Our method is as follows. We use a pyramidal traveling front for an unbalanced reaction-diffusion equation whose cross section has a major axis and a minor axis. Preserving the ratio of the major axis and a minor axis to be a constant and taking the balanced limit, we obtain a traveling front in a balanced bistable reaction-diffusion equation. This traveling front is monotone decreasing with respect to the traveling axis, and its cross section is a compact set with a major axis and a minor axis when the constant ratio is not 1.     

Stationary internal layers in a reaction-advection-diffusion integro-differential system

A class of singularly perturbed nonlinear integro-differential problems with solutions involving internal transition layers (contrast structures) is considered. An asymptotic expansion of these solutions with respect to a small parameter is constructed, and the stability of stationary solutions to the associated integro-parabolic problems is investigated. The asymptotics are substantiated using the asymptotic method of differential inequalities, which is extended to the new class of problems. The method is based on well-known theorems about differential inequalities and on the idea of using formal asymptotics for constructing upper and lower solutions in singularly perturbed problems with internal and boundary layers.     

Convergence to periodic fronts in a class of semilinear parabolic equations

A class of scalar autonomous parabolic equations, nonlinear with respect to the unknown and its gradient, is investigated. The main topic of this paper is the convergence of the solutions of the Cauchy Problem towards solutions which exhibit, modulo a linear growth in time, periodic spatiotemporal oscillations. Different generalizations are discussed. Received October 23, 1995; Revised version received June 25, 1996     

Existence of traveling wave fronts of delayed lattice differential equations

Existence of traveling wave fronts for delayed lattice differential equations is established by Schauder fixed point theorem. The main result is applied to a delayed and discretely diffusive model for the population of Daphnia magna.     

Traveling wave fronts for generalized Fisher equations with spatio-temporal delays

We study the existence of traveling wave fronts for a reaction-diffusion equation with spatio-temporal delays and small parameters. The equation reduces to a generalized Fisher equation if small parameters are zero. We present two results. In the first one, we deal with the equation with very general kernels and show the persistence of Fisher wave fronts for all sufficiently small parameters. In the second one, we deal with some particular kernels, with which the nonlocal equation can be reduced to a system of singularly perturbed ODEs, and we are then able to apply the geometric singular perturbation theory and phase plane arguments to this system to show the existence of the minimal wave speed, the existence of a continuum of wave fronts, and the global uniqueness of the physical wave front with each wave speed.     

Existence and global stability of traveling curved fronts in the Allen-Cahn equations

This paper is concerned with existence and stability of traveling curved fronts for the Allen-Cahn equation in the two-dimensional space. By using the supersolution and the subsolution, we construct a traveling curved front, and show that it is the unique traveling wave solution between them. Our supersolution can be taken arbitrarily large, which implies some global asymptotic stability for the traveling curved front.     

Multidimensional stability of traveling fronts in monostable reaction-diffusion equations with complex perturbations

This paper studies the multidimensional stability of traveling fronts in monostable reaction-difusion equations,including Ginzburg-Landau equations and Fisher-KPP equations.Eckmann and Wayne(1994)showed a one-dimensional stability result of traveling fronts with speeds c c(the critical speed)under complex perturbations.In the present work,we prove that these traveling fronts are also asymptotically stable subject to complex perturbations in multiple space dimensions(n=2,3),employing weighted energy methods.     

Multidimensional stability of traveling fronts in monostable reaction-difusion equations with complex perturbations

This paper studies the multidimensional stability of traveling fronts in monostable reaction-difusion equations,including Ginzburg-Landau equations and Fisher-KPP equations.Eckmann and Wayne(1994)showed a one-dimensional stability result of traveling fronts with speeds c c(the critical speed)under complex perturbations.In the present work,we prove that these traveling fronts are also asymptotically stable subject to complex perturbations in multiple space dimensions(n=2,3),employing weighted energy methods.     

Stability of planar shock fronts for multidimensional systems of relaxation equations

We investigate stability of multidimensional planar shock profiles of a general hyperbolic relaxation system whose equilibrium model is a system, under the necessary assumption of spectral stability and a standard set of structural conditions that are known to hold for many physical systems. Our main result, generalizing the work of Kwon and Zumbrun in the scalar relaxation case, is to establish the bounds on the Green?s function for the linearized equation and obtain nonlinear L² asymptotic behavior/sharp decay rate of perturbed weak shock profiles. To establish Green?s function bounds, we use the semigroup approach in the low-frequency regime, and use the energy method for the high-frequency bounds, separately. For the system equilibrium case, the analysis of the linearized equation is complicated due to glancing phenomena. We treat this difficulty similarly as in the inviscid and viscous systems, under the constant multiplicity condition.     

The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations

This paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three-dimensional whole space. For a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity. For this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces. Moreover we characterize the pyramidal traveling front in another way, that is, we write it as a combination of two-dimensional V-form waves on the edges. This characterization also uniquely determines a pyramidal traveling front.