Nonlinear reaction–diffusion systems with a non-constant diffusivity: Conditional symmetries in no-go case

Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction–diffusion systems with non-constant diffusivities are studied. The work is a natural continuation of our paper “Conditional symmetries and exact solutions of nonlinear reaction–diffusion systems with non-constant diffusivities” (Cherniha and Davydovych, 2012) [1] in order to extend the results on so-called no-go case. Using the notion of Q-conditional symmetries of the first type, an exhaustive list of reaction–diffusion systems admitting such symmetry is derived. The results obtained are compared with those derived earlier. The symmetries for reducing reaction–diffusion systems to two-dimensional dynamical systems (ODE systems) and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for nonlinear reaction–diffusion systems with an arbitrary power-law diffusivity are constructed and their properties for possible applicability are established.