Exact solutions for the general reaction-diffusion equation with variable delay

Construct exact solutions for the general nonlinear reaction-diffusion partial differential equation with variable delay ut = k u_xx + F(u, ū), where ū = u(x, t − τ(x,t)) for a given continuous delay function τ(x,t) > 0, and F is an arbitrary real function of two variables.

Background

Following their statement about the absence of non-traveling-wave exact solutions for the constant-delay case, the authors emphasize that the situation is even more challenging for reaction-diffusion equations with variable delay. This underscores the need for analytical methods capable of handling variable-delay structures in reaction-diffusion models, where the delay may depend on space and/or time.