Reaction-Diffusion Models with Delay: Some Properties, Equations, Problems, and Solutions

The delay reaction-diffusion models used in thermal physics, chemistry, biochemistry, biology, ecology, biomedicine, and control theory were reviewed. New exact solutions were obtained for several classes of one- and three-dimensional nonlinear equations with distributed parameters, in which the kinetic functions involve a delay. The qualitative features of these equations related to nonsmoothness and potential instability of solutions (these features should be taken into account in the mathematical modeling of the corresponding processes) were discussed. The properties of delay reaction-diffusion equations were described, which allow exact solutions to be obtained and multiplied. The key principles of construction, selection, and use of the test problems of the reaction-diffusion type were formulated, which can be used for evaluating the accuracy of rough analytical and numerical methods for solving the delay equations.