Nonlinear delay reaction–diffusion equations with varying transfer coefficients: Exact methods and new solutions

The paper deals with one-dimensional nonlinear delay reaction–diffusion equations with varying transfer coefficients of the form ut=[G(u)ux]x+F(u,ū), where u=u(x,t)u=u(x,t) and ū=u(x,t−τ), with ττ denoting the delay time. Generalized and functional separable solutions for this class of equations have been obtained and presented for the first time; these equations have not been known to have such solutions so far. To construct these solutions and solutions of some other delay PDEs, we developed a few exact methods that rely on using invariant subspaces for corresponding nonlinear differential operators. Many of the results are extendable to more complex nonlinear reaction–diffusion equations with several delay times, τ1,…,τmτ1,…,τm, and equations with time-varying delay, τ=τ(t)τ=τ(t). All of the equations considered involve several free parameters (or an arbitrary function) and so their solutions can be suitable for testing approximate analytical and numerical methods for nonlinear delay reaction–diffusion equations. The exact methods described may also be applied to other classes of nonlinear delay PDEs.