Asymptotic pattern for a partial neutral functional differential equation

In the present article, the asymptotic speed of spread and traveling wavefronts of a partial neutral functional differential equation (PNFDE) are investigated. A variable transform yields to a partial retarded functional differential equation (PRFDE) with infinite number of constant delays. The relation between the above PNFDE and PRFDE is studied and an equivalence in a certain meaning of these two equations is verified. Using this equivalence, we in fact obtain the existence and uniqueness of solution for the initial value problem of the PNFDE. We then consider the truncated system of the above PRFDE, which is a partial retarded functional differential equation with finite number of constant delays. We establish the existence of the traveling wave solution and the spreading speed for this truncated system. A limit argument and the method of finite delay approximation are used to obtain the existence of the traveling wave solutions and the spreading speed for the PRFDE with infinite number of constant delays. This leads to the existence of the traveling wave solution and the spreading speed for the considered PNFDE by the above equivalence. At last, we confirm that the spreading speed is coincidental with the minimal wave speed for all equations. An example is given to illustrate the application of our results.