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.     

The asymptotic critical wave speed in a family of scalar reaction-diffusion equations

We study traveling wave solutions for the class of scalar reaction-diffusion equations

     

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.     

Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays

This paper deals with the existence of travelling wave fronts in reaction-diffusion systems with spatio-temporal delays. Our approach is to use monotone iterations and a nonstandard ordering for the set of profiles of the corresponding wave system. New iterative techniques are established for a class of integral operators when the reaction term satisfies different monotonicity conditions. Following this, the existence of travelling wave fronts for reaction-diffusion systems with spatio-temporal delays is established. Finally, we apply the main results to a single-species diffusive model with spatio-temporal delay and obtain some existence criteria of travelling wave fronts by choosing different kernels.     

Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications

This paper is concerned with the travelling wave fronts of nonlocal reaction-diffusion systems with delays. The existence of travelling wave fronts for nonlocal reaction-diffusion systems with delays is established by using Schauder’s fixed point theorem and upper-lower solution technique. Then these results are applied to the nonlocal delayed Logistic model and the delayed Belousov-Zhabotinskii reaction-diffusion system. Our results show that the time delay can reduce the minimal wave speed while the nonlocality can increase the minimal wave speed. Wan-Tong Li: Supported by NNSF of China (10571078) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China.     

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.     

Critical wave speeds for a family of scalar reaction-diffusion equations

We study the set of traveling waves in a class of reaction-diffusion equations for the family of potentials f_m(U) = 2U^m(1 − U). We use perturbation methods and matched asymptotics to derive expansions for the critical wave speed that separates algebraic and exponential traveling wave front solutions for m → 2 and m → ∞. Also, an integral formulation of the problem shows that nonuniform convergence of the generalized equal area rule occurs at the critical wave speed.     

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 of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method

The monotone iteration method is employed to establish the existence of traveling wave fronts in delayed reaction-diffusion systems with monostable nonlinearities.

     

On nonlinear coupled reaction-diffusion equations

This paper is concerned with the existence and uniqueness of solutions of reaction-diffusion equations arising in many applications and often coupled. Our approach to the problem is based on converting the coupled equations into a system of Volterra-Fredholm type integral equations by means of the Green's function representation and using the Banach fixed point theorem. An application to nonlinear second order evolution equations is also indicated.     

On the existence of traveling waves for delayed reaction-diffusion equations

We study the existence of traveling wave solutions for reaction-diffusion equations with nonlocal delay, where reaction terms are not necessarily monotone. The existence of traveling wave solutions for reaction-diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the Schauder fixed point theorem. The smoothness of upper and lower solutions is not required in this paper.     

Global stability of traveling wave fronts for non-local delayed lattice differential equations

This paper is concerned with the global stability of traveling wave fronts of a non-local delayed lattice differential equation. By the comparison principle together with the semi-discrete Fourier transform, we prove that, all noncritical traveling wave fronts are globally stable in the form of t^−1/αe^−μt for some constants μ>0 and 0<α≤2, and the critical traveling wave fronts are globally stable in the algebraic form of t^−1/α.     

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.     

Factorization of the reaction-diffusion equation,the wave equation,and other equations

We investigate equations of the form D _t u = Δu + ξ? _u for an unknown function u(t, x), t ∈ ?, x ∈ X, where D _t u = a ₀(u, t) + Σ _k=1 ^r a _k (t, u)? _t ^k u, Δ is the Laplace-Beltrami operator on a Riemannian manifold X, and ξ is a smooth vector field on X. More exactly, we study morphisms from this equation within the category PDE of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called geometric morphisms, which are given by maps of X to other smooth manifolds (of the same or smaller dimension). It is shown that a map f: X → Y defines a morphism from the equation D _t u = Δu + ξ? _u if and only if, for some vector field Ξ and a metric on Y, the equality (Δ + ξ?)f*v = f*(Δ + Ξ?)v holds for any smooth function v: Y → ?. In this case, the quotient equation is D _t v = Δv + Ξ?v for an unknown function v(t, y), y ∈ Y. It is also shown that, if a map f: X → Y is a locally trivial bundle, then f defines a morphism from the equation D _t u = Δu if and only if fibers of f are parallel and, for any path γ on Y, the expansion factor of a fiber translated along the horizontal lift γ to X depends on γ only.     

Laplace asymptotics for reaction-diffusion equations

Summary We obtain sharp (i.e. non logarithmic) asymptotics for the solution of non homogeneous Kolmogorov-Petrovski-Piskunov equation depending on a small parameter , for points ahead of the Freidlin-KPP front.     

Waveform relaxation for reaction-diffusion equations

We report a new waveform relaxation (WR) algorithm for general semi-linear reaction-diffusion equations. The superlinear rate of convergence of the new WR algorithm is proved, and we also show the advantages of the new approach superior to the classical WR algorithms by the estimation on iteration errors. The corresponding discrete WR algorithm for reaction-diffusion equations is presented, and further the parallelism of the discrete WR algorithm is analyzed. Moreover, the new approach is extended to handle the coupled reaction-diffusion equations. Numerical experiments are carried out to verify the effectiveness of the theoretic work.     

Long-term analysis of semilinear wave equations with slowly varying wave speed

A semilinear wave equation with slowly varying wave speed is considered in one to three space dimensions on a bounded interval, a rectangle or a box, respectively. It is shown that the action, which is the harmonic energy divided by the wave speed and multiplied with the diameter of the spatial domain, is an adiabatic invariant: it remains nearly conserved over long times, longer than any fixed power of the time scale of changes in the wave speed in the case of one space dimension, and longer than can be attained by standard perturbation arguments in the two- and three-dimensional cases. The long-time near-conservation of the action yields long-time existence of the solution. The proofs use modulated Fourier expansions in time.