Exact solutions and qualitative features of nonlinear hyperbolic reaction-diffusion equations with delay

New classes of exact solutions to nonlinear hyperbolic reaction-diffusion equations with delay are described. All of the equations under consideration depend on one or two arbitrary functions of one argument, and the derived solutions contain free parameters (in certain cases, there can be any number of these parameters). The following solutions are found: periodic solutions with respect to time and space variable, solutions that describe the nonlinear interaction between a standing wave and a traveling wave, and certain other solutions. Exact solutions are also presented for more complex nonlinear equations in which delay arbitrarily depends on time. Conditions for the global instability of solutions to a number of reaction-diffusion systems with delay are derived. The generalized Stokes problem subject to the periodic boundary condition, which is described by a linear diffusion equation with delay, is solved.