On the uniqueness of semi-wavefronts for non-local delayed reaction–diffusion equations

We establish the uniqueness of semi-wavefront solution for a non-local delayed reaction–diffusion equation. This result is obtained by using a generalization of the Diekmann–Kaper theory for a nonlinear convolution equation. Several applications to the systems of non-local reaction–diffusion equations with distributed time delay are also considered.     

Existence of fast positive wavefronts for a non-local delayed reaction-diffusion equation

We establish the existence of a continuous family of fast positive wavefronts u(t,x)=?(x+ct), ?(−∞)=0, ?(+∞)=κ, for the non-local delayed reaction-diffusion equation . Here 0 and κ>0 are fixed points of g∈C²(R₊,R₊) and the non-negative K is such that is finite for every real λ. We also prove that the fast wavefronts are non-monotone if .     

On uniqueness of semi-wavefronts

Motivated by the uniqueness problem for monostable semi-wave-fronts, we propose a revised version of the Diekmann and Kaper theory of a nonlinear convolution equation. Our version of the Diekmann?CKaper theory allows (1) to consider new types of models which include nonlocal KPP type equations (with either symmetric or anisotropic dispersal), nonlocal lattice equations and delayed reaction?Cdiffusion equations; (2) to incorporate the critical case (which corresponds to the slowest wavefronts) into the consideration; (3) to weaken or to remove various restrictions on kernels and nonlinearities. The results are compared with those of Schumacher (J Reine Angew Math 316: 54?C70, 1980), Carr and Chmaj (Proc Am Math Soc 132: 2433?C2439, 2004), and other more recent studies.     

On the minimal speed of traveling waves for a nonlocal delayed reaction–diffusion equation

In this note, we give constructive upper and lower bounds for the minimal speed of propagation of traveling waves for a nonlocal delayed reaction–diffusion equation.