The paper is devoted to a reaction-diffusion equation in an infinite two-dimensional strip with nonlinear boundary conditions. The existence of travelling waves is proved in the bistable case by the Leray-Schauder method. It is based on a topological degree for elliptic problems in unbounded domains and on a priori estimates of solutions.

We study a nonlinear diffusion equation ut = uxx + f(u) with Robin boundary condition at x = 0 and with a free boundary condition at x = h(t), where h(t) > 0 is a moving boundary representing the expanding front in ecology models. For any f ∈ C1 with f(0) = 0, we prove that every bounded positive solution of this problem converges to a stationary one. As applications, we use this convergence result to study diffusion equations with monostable and combustion types of nonlinearities. We obtain dichotomy results and sharp thresholds for the asymptotic behavior of the solutions.

Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviour than for the usual Fisher equation. A striking numerical observation is that a traveling wave with minimal speed can connect a dynamically unstable steady state 0 to a Turing unstable steady state 1, see [12]. This is proved in [1, 6] in the case where the speed is far from minimal, where we expect the wave to be monotone.

Here we introduce a simplified nonlocal Fisher equation for which we can build simple analytical traveling wave solutions that exhibit various behaviours. These traveling waves, with minimal speed or not, can (i) connect monotonically 0 and 1, (ii) connect these two states non-monotonically, and (iii) connect 0 to a wavetrain around 1. The latter exist in a regime where time dynamics converges to another object observed in [3, 8]: a wave that connects 0 to a pulsating wave around 1.

The notion of inside dynamics of traveling waves has been introduced in the recent paper [14]. Assuming that a traveling wave u(t,x) = U(x − c   t) is made of several components υi ≥ 0 (i ∈ I ⊂ N), the inside dynamics of the wave is then given by the spatio-temporal evolution of the densities of the components υi. For reaction-diffusion equations of the form ∂tu(t,x) = ∂xxu(t,x) + f(u(t,x)), where f is of monostable or bistable type, the results in [14] show that traveling waves can be classified into two main classes: pulled waves and pushed waves. Using the same framework, we study the pulled/pushed nature of the traveling wave solutions of delay equations

             ∂tu(t,x) = ∂xxu(t,x) + F(u(t −τ,x),u(t,x))

We begin with a review of the latest results on the existence of traveling wave solutions of such equations, for several classical reaction terms. Then, we give analytical and numerical results which describe the inside dynamics of these waves. From a point of view of population ecology, our study shows that the existence of a non-reproductive and motionless juvenile stage can slightly enhance the genetic diversity of a species colonizing an empty environment.

This paper is devoted to the study of spreading speeds and traveling waves for a class of reaction-diffusion equations with distributed delay. Such an equation describes growth and diffusion in a population where the individuals enter a quiescent phase exponentially and stay quiescent for some arbitrary time that is given by a probability density function. The existence of the spreading speed and its coincidence with the minimum wave speed of monostable traveling waves are established via the finite-delay approximation approach. We also prove the existence of bistable traveling waves in the case where the associated reaction system admits a bistable structure. Moreover, the global stability and uniqueness of the bistable waves are obtained in the case where the density function has zero tail

This paper is concerned with entire solutions of a class of bistable delayed lattice differential equations with nonlocal interaction. Here an entire solution is meant by a solution defined for all (n,t) ∈ ℤ × ℝ. Assuming that the equation has an increasing traveling wave front with nonzero wave speed and using a comparison argument, we obtain a two-dimensional manifold of entire solutions. In particular, it is shown that the traveling wave fronts are on the boundary of the manifold. Furthermore, uniqueness and stability of such entire solutions are studied.

This paper is concerned with the existence and stability of travelling front solutions for more general autocatalytic chemical reaction systems ut = duxx − uf(v), vt = vxx + uf(v) with d > 0 and d ≠ 1, where f(v) has super-linear or linear degeneracy at v = 0. By applying Lyapunov-Schmidt decomposition method in some appropriate exponentially weighted spaces, we obtain the existence and continuous dependence of wave fronts with some critical speeds and with exponential spatial decay for d near 1. By applying special phase plane analysis and approximate center manifold theorem, the existence of traveling waves with algebraic spatial decay or with some lower exponential decay is also obtained for d > 0. Further, by spectral estimates and Evans function method, the wave fronts with exponential spatial decay are proved to be spectrally or linearly stable in some suitable exponentially weighted spaces. Finally, by adopting the main idea of proof in [12] and some similar arguments as in [21], the waves with critical speeds or with non-critical speeds are proved to be locally exponentially stable in some exponentially weighted spaces and Lyapunov stable in Cunif(ℝ) space, if the initial perturbation of the waves is small in both the weighted and unweighted norms; the perturbation of the waves also stays small in L1(ℝ) norm and decays algebraically in Cunif(ℝ) norm, if the initial perturbation is in addition small in L1 norm.

An overview of recently obtained authors’ results on traveling wave solutions of some classes of PDEs is presented. The main aim is to describe all possible travelling wave solutions of the equations. The analysis was conducted using the methods of qualitative and bifurcation analysis in order to study the phase-parameter space of the corresponding wave systems of ODEs. In the first part we analyze the wave dynamic modes of populations described by the “growth - taxis - diffusion" polynomial models. It is shown that “suitable" nonlinear taxis can affect the wave front sets and generate non-monotone waves, such as trains and pulses, which represent the exact solutions of the model system. Parametric critical points whose neighborhood displays the full spectrum of possible model wave regimes are identified; the wave mode systematization is given in the form of bifurcation diagrams. In the second part we study a modified version of the FitzHugh-Nagumo equations, which model the spatial propagation of neuron firing. We assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, besides giving rise to the typical fast travelling wave solution exhibited in the original “diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where “normal" impulse propagation is possible. In the third part we describe all possible wave solutions for a class of PDEs with cross-diffusion, which fall in a general class of the classical Keller-Segel models describing chemotaxis. Conditions for existence of front-impulse, impulse-front, and front-front traveling wave solutions are formulated. In particular, we show that a non-isolated singular point in the ODE wave system implies existence of free-boundary fronts.

We study a reaction diffusion system that models the dynamics of two species that display inter-species competition and intra-species cooperation. We find that there are between three and six different equilibrium states and a variety of possible travelling wave solutions that can connect them. After examining the travelling waves that are generated in three different ecologically-relevant initial value problems, we construct asymptotic solutions in the limit λ ≪ 1 (fast diffusion, slow reaction for the second species relative to the first).

We carried out a computational study of propagation speeds of reaction-diffusion-advection fronts in three dimensional (3D) cellular and Arnold-Beltrami-Childress (ABC) flows with Kolmogorov-Petrovsky-Piskunov(KPP) nonlinearity. The variational principle of front speeds reduces the problem to a principal eigenvalue calculation. An adaptive streamline diffusion finite element method is used in the advection dominated regime. Numerical results showed that the front speeds are enhanced in cellular flows according to sublinear power law O(δp), p ≈ 0.13, δ the flow intensity. In ABC flows however, the enhancement is O(δ) which can be attributed to the presence of principal vortex tubes in the streamlines. Poincaré sections are used to visualize and quantify the chaotic fractions of ABC flows in the phase space. The effect of chaotic streamlines of ABC flows on front speeds is studied by varying the three parameters (a,b,c) of the ABC flows. Speed enhancement along x direction is reduced as b (the parameter controling the flow variation along x) increases at fixed (a,c) > 0, more rapidly as the corresponding ABC streamlines become more chaotic.

We perform a computationl study of front speeds of G-equation models in time dependent cellular flows. The G-equations arise in premixed turbulent combustion, and are Hamilton-Jacobi type level set partial differential equations (PDEs). The curvature-strain G-equations are also non-convex with degenerate diffusion. The computation is based on monotone finite difference discretization and weighted essentially nonoscillatory (WENO) methods. We found that the large time front speeds lock into the frequency of time periodic cellular flows in curvature-strain G-equations similar to what occurs in the basic inviscid G-equation. However, such frequency locking phenomenon disappears in viscous G-equation, and in the inviscid G-equation if time periodic oscillation of the cellular flow is replaced by time stochastic oscillation.

The model we consider treats the cell as a viscoelastic medium filling one of two kinds of thin domains (“shapes” of cells): the thin slab being a caricature of a tissue and the thin circular cylinder mimicking a long cell. This enables us to simplify the system of mechano-chemical equations. We construct abundant classes of explicit, but approximate, formulae for heteroclinic solutions to these equations.