Existence of traveling wave solutions for a reaction-diffusion equation with distributed delays

Abstract In this paper, we consider a reaction–diffusion equation with distributed delays. Special attention is paid to the existence of traveling wave solutions. For special delay kernels, using the linear chain trick and the geometric singular perturbation theory, we investigate a natural connection between the existence of traveling wave solutions for the reaction–diffusion equation with distributed delays and the existence of traveling wave solutions for the corresponding undelayed reaction–diffusion equation. We show that if the corresponding undelayed reaction–diffusion equation has a traveling wave solution, then, for any sufficiently small delay, the delayed reaction–diffusion equation also has a traveling wave solution.

[1]  Vitaly Volpert,et al.  Traveling-wave solutions of parabolic systems with discontinuous nonlinear terms , 2002 .

[2]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[3]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[4]  Maoan Han,et al.  Travelling wavefronts in the diffusive single species model with Allee effect and distributed delay , 2004, Appl. Math. Comput..

[5]  Shiwang Ma,et al.  Traveling Wavefronts for Delayed Reaction-Diffusion Systems via a Fixed Point Theorem , 2001 .

[6]  Steven R. Dunbar,et al.  Travelling wave solutions of diffusive Lotka-Volterra equations , 1983 .

[7]  Robert A. Gardner.  Existence and stability of travelling wave solutions of competition models: A degree theoretic approach , 1982 .

[8]  C. Conley,et al.  An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model , 1984 .

[9]  R. Gardner,et al.  Existence of Travelling Wave Solutions of Predator–Prey Systems via the Connection Index , 1984 .

[10]  Van Vuuren,et al.  The existence of travelling plane waves in a general class of competition-diffusion systems , 1995 .

[11]  A. Volpert,et al.  EXISTENCE OF MULTIDIMENSIONAL TRAVELLING WAVES AND SYSTEMS OF WAVES , 2001 .

[12]  Mark A. J. Chaplain,et al.  Travelling fronts in a food-limited population model with time delay , 2002, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[13]  Xingfu Zou,et al.  Traveling Wave Fronts of Reaction-Diffusion Systems with Delay , 2001 .

[14]  X. Zou,et al.  Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method , 1997 .

[15]  Xingfu Zou,et al.  Traveling waves for the diffusive Nicholson's blowflies equation , 2001, Appl. Math. Comput..

[16]  Shigui Ruan,et al.  Stability of steady states and existence of travelling waves in a vector-disease model , 2004, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[17]  Klaus W. Schaaf.  Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations , 1987 .

[18]  S. A. Gourley.  Travelling fronts in the diffusive Nicholson's blowflies equation with distributed delays , 2000 .