The paper considers the following partial neutral functional differential equation \[ \begin{aligned} & \frac{\partial}{\partial t}\left[u(x,t)-b\,u(x,t-r)\right]=K\,\frac{\partial^2}{\partial x^2}\left[u(x,t)-b\,u(x,t-r)\right]\\ & +f(u(x,t)-b\,u(x,t-r),u(x,t),u(x,t-r)),\quad x\in\mathbb R, t>0, \end{aligned}\tag{NDDE} \] where \(K>0, r>0\) and \(0<b<1\) are constants. The function \(f(s_1,s_2,s_3)\in C^1(\mathbb R^3,\mathbb R)\) is such that the corresponding real-valued function \(F(s):=f((1-b)s,s,s), s\geq0,\) has exactly two equilibria \(s=0\) and \(s=s^*>0\) with \(F(s)>0\) for \(s\in(0,s^*)\) and \(F(s)<0\) for \(s\in(s^*,\infty)\). A range of additional assumptions is imposed on \(f(s_1,s_3,s_3)\) including the monotonicity, positivity, and bound or growth estimates in some of the three variables (they are listed as assumptions (H1) through (H5) in Section 2).
The existence of monotone traveling wave solutions is established for equation (NDDE). A solution of the form \(u(x,t)=w(x+ct)\) is called a traveling wave solution, with the function \(w\) termed as wave’s profile and the constant \(c\) as the speed of propagation of the wave. The existence of the traveling wave solutions is established for all \(c\geq c_*\), while it is showed that there are no wavefronts for any \(0<c<c_*\), for some \(c_*>0\) (\(c_*\) is known as the critical speed). The approach that has been used in the paper is to first establish analogous results for related and somewhat simpler retarded partial differential equations with finite and infinite number of delays. Then those results are carried over to the neutral equation (NDDE) through a finite delay approximation and limiting transition arguments.