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.     

Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity

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.     

Entire solutions in nonlocal dispersal equations with bistable nonlinearity

We consider entire solutions of nonlocal dispersal equations with bistable nonlinearity in one-dimensional spatial domain. A two-dimensional manifold of entire solutions which behave as two traveling wave solutions coming from both directions is established by an increasing traveling wave front with nonzero wave speed. Furthermore, we show that such an entire solution is unique up to space-time translations and Liapunov stable. A key idea is to characterize the asymptotic behaviors of the solutions as t→−∞ in terms of appropriate subsolutions and supersolutions. We have to emphasize that a lack of regularizing effect occurs.     

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

Traveling wave front for a two-component lattice dynamical system arising in competition models

We study traveling front solutions for a two-component system on a one-dimensional lattice. This system arises in the study of the competition between two species with diffusion (or migration), if we divide the habitat into discrete regions or niches. We consider the case when the nonlinear source terms are of Lotka–Volterra type and of monostable case. We first show that there is a positive constant (the minimal wave speed) such that a traveling front exists if and only if its speed is above this minimal wave speed. Then we show that any wave profile is strictly monotone. Moreover, under some conditions, we show that the wave profile is unique (up to translations) for a given wave speed. Finally, we characterize the minimal wave speed by the parameters in the system.     

Traveling wave solutions for reaction-diffusion systems

This paper is concerned with traveling waves of reaction-diffusion systems. The definition of coupled quasi-upper and quasi-lower solutions is introduced for systems with mixed quasimonotone functions, and the definition of ordered quasi-upper and quasi-lower solutions is also given for systems with quasimonotone nondecreasing functions. By the monotone iteration method, it is shown that if the system has a pair of coupled quasi-upper and quasi-lower solutions, then there exists at least a traveling wave solution. Moreover, if the system has a pair of ordered quasi-upper and quasi-lower solutions, then there exists at least a traveling wavefront. As an application we consider the delayed system of a mutualistic model.     

Traveling wave solutions of competitive–cooperative Lotka–Volterra systems of three species

This paper is concerned with the existence of traveling front solutions for competitive–cooperative Lotka–Volterra systems of three species. By converting the system into a monotone system, we show that under certain assumptions on the parameters appearing in the system, traveling front solutions exist. Also, exact traveling front solutions, which are polynomials in the hyperbolic tangent function, are given explicitly in certain parameter regimes.     

Propagation and its failure in a lattice delayed differential equation with global interaction

We study the existence, uniqueness, global asymptotic stability and propagation failure of traveling wave fronts in a lattice delayed differential equation with global interaction for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. In the bistable case, under realistic assumptions on the birth function, we prove that the equation admits a strictly monotone increasing traveling wave front. Moreover, if the wave speed does not vanish, then the wave front is unique (up to a translation) and globally asymptotic stable with phase shift. Of particular interest is the phenomenon of “propagation failure” or “pinning” (that is, wave speed c = 0), we also give some criteria for pinning in this paper.     

Existence,uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems

In the one-dimensional space, traveling wave solutions of parabolic differential equations have been widely studied and well characterized. Recently, the mathematical study on higher-dimensional traveling fronts has attracted a lot of attention and many new types of nonplanar traveling waves have been observed for scalar reaction-diffusion equations with various nonlinearities. In this paper, by using the comparison argument and constructing appropriate super- and subsolutions, we study the existence, uniqueness and stability of threedimensional traveling fronts of pyramidal shape for monotone bistable systems of reaction-diffusion equations in R³. The pyramidal traveling fronts are characterized as either a combination of planar traveling fronts on the lateral surfaces or a combination of two-dimensional V-form waves on the edges of the pyramid. In particular, our results are applicable to some important models in biology, such as Lotka-Volterra competition-diffusion systems with or without spatio-temporal delays, and reaction-diffusion systems of multiple obligate mutualists.     

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.     

Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay

This paper is concerned with the existence, uniqueness and globally asymptotic stability of traveling wave fronts in the quasi-monotone reaction advection diffusion equations with nonlocal delay. Under bistable assumption, we construct various pairs of super- and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique nondecreasing traveling wave front (up to translation), which is monotonically increasing and globally asymptotically stable with phase shift. The influence of advection on the propagation speed is also considered. Comparing with the previous results, our results recovers and/or improves a number of existing ones. In particular, these results can be applied to a reaction advection diffusion equation with nonlocal delayed effect and a diffusion population model with distributed maturation delay, some new results are obtained.     

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.     

Traveling waves of a curvature flow in almost periodic media

In a plane media with almost periodic vertical striations, we study a curvature flow and construct two kinds of traveling waves, one having a straight line like profile and the other having a V shaped profile. For each of the first-kind traveling waves, its profile is the graph of a function whose derivative is almost periodic. For each of the second-kind traveling waves, its profile is like a pulsating cone, whose two tails approach asymptotically the profiles of the first-kind traveling waves. Also we consider a homogenization problem and provide an explicit formula for the homogenized traveling speed.     

Stability of post-fertilization traveling waves

This paper studies the stability of a family of traveling wave solutions to the system proposed by Lane et al. [D.C. Lane, J.D. Murray, V.S. Manoranjan, Analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post-fertilization waves on eggs, IMA J. Math. Appl. Med. Biol. 4 (4) (1987) 309-331], to model a pair of mechanochemical phenomena known as post-fertilization waves on eggs. The waves consist of an elastic deformation pulse on the egg's surface, and a free calcium concentration front. The family is indexed by a coupling parameter measuring contraction stress effects on the calcium concentration. This work establishes the spectral, linear and nonlinear orbital stability of these post-fertilization waves for small values of the coupling parameter. The usual methods for the spectral and evolution equations cannot be applied because of the presence of mixed partial derivatives in the elastic equation. Nonetheless, exponential decay of the directly constructed semigroup on the complement of the zero eigenspace is established. We show that small perturbations of the waves yield solutions to the nonlinear equations decaying exponentially to a phase-modulated traveling wave.     

Simple Diffraction Processes for Nagumo- and Fisher-type Diffusion Fronts

The diffraction of a diffusion front by concave and convex wedges is studied for Nagumo and Fisher's equations on the limit of fast reaction and small diffusion, using both the asymptotic theory and full numerical solutions. For the case of a convex corner, the full numerical solution confirms that the front evolves according to the asymptotic theories. On the other hand, for the concave corner, it is shown numerically that the diffraction produces at the corner a region of low values of the solution for both the Nagumo and Fisher's equations. Moreover, in both cases, the front eventually evolves, leaving behind a cavity. In the case of the Nagumo equation, it is shown that the long-term behavior of the diffraction front is just a traveling front, bent at the sloping wall. The bent region maintains its size as the front travels. This behavior is predicted by an exact traveling wave solution of the asymptotic equation for the front propagation. Good agreement is found between the numerical and the asymptotic solutions. On the other hand, behavior of the diffracted front for Fisher's equation is different. In this case, the front is bent at the sloping wall, but, as time passes, the bend becomes smaller and moves toward the sloping wall. This behavior is, again, predicted by the asymptotic solution. The numerics strongly suggest that the final state for the concave corner is a steady cavity-like solution with low values at the corner and high values away from it. This solution has an angular dependence that varies with the angle of the sloping wall.     

Long time behavior of classical solutions to the generalized Goursat problem of quasilinear hyperbolic systems

We investigate the existence of a global classical solution to the generalized Goursat problem. Under some degenerate assumptions of boundary conditions, we prove that the solution approaches a combination of Lipschitz continuous and a piecewise C¹ traveling wave solution.     

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.     

A Stefan problem in combustion

This paper deals with a mathematical model of a condensed two-phase combustion process which describes combustion of solid materials in which melting occurs. The paper shows the existence of a weak solution of the resulting differential equations system and, furthermore, shows that the phase change set (the set where the temperature is equal to the given constant melting temperature) is not a front but a whole mushy region. For this mushy region an estimate in measure is given.This work has been supported by the Deutsche Forschungsgemeinschaft.     

Monotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage

This paper is concerned with the monotonicity and uniqueness of traveling waves for a reaction-diffusion model with quiescent stage. We first obtain the exponential decay rate of wave profiles, and then we show that any profile is strictly monotone by using the strong comparison principle. Furthermore, we prove the uniqueness (up to translation) of all traveling waves including even the waves with minimal speed.