Nonlinear delay reaction-diffusion equations with varying transfer coefficients: generalized and functional separable solutions

We present a number of new simple separable, generalized separable, and functional separable solutions to one-dimensional nonlinear delay reaction-diffusion equations with varying transfer coefficients of the form $u_t=[G(u)u_x]_x+F(u,w)$, where $w = u(x,t)$ and $w = u(x,t-\tau)$, with $\tau$ denoting the delay time. All of the equations considered contain one, two, or three arbitrary functions of a single argument. The generalized separable solutions are sought in the form $u=\sum_{n=1}^N\varphi_n(x)\psi_n(t)$, with $\varphi_n(x)$ and $\psi_n(t)$ to be determined in the analysis using a new modification of the functional constraints method. Some of the results are extended to nonlinear delay reaction-diffusion equations with time-varying delay $\tau=\tau(t)$. We also present exact solutions to more complex, three-dimensional delay reactiondiffusion equations of the form $u_t=\mathrm{div}[G(u)\nabla u]+F(u,w)$. Most of the solutions obtained involve free parameters, so they may be suitable for solving certain problems as well as testing approximate analytical and numerical methods for non-linear delay PDEs.