On Periodic Solutions of a Nonlinear Reaction-Diffusion System

We consider a system of three parabolic partial differential equations of a special reaction-diffusion type. In this system, the terms that describe diffusion are identical and linear with constants coefficients, whereas reactions are described by homogenous polynomials of degree 3 that depend on three parameters. The desired functions are considered to be dependent on time and an arbitrary number of spatial variables (a multi-dimensional case). It has been shown that the reaction-diffusion system under study has a whole family of exact solutions that can be expressed via a product of the solution to the Helmholz equations and the solution to a system of ordinary differential equations with homogenous polynomials, taken from the original system, in the right-hand side. We give the two first integrals and construct a general solution to the system of three ordinary differential equations, which is represented by the Jacobi elliptic functions. It has been revealed that all particular solutions derived from the general solution to the system of ordinary differential equations are periodic functions of time with periods depending on the choice of initial conditions. Additionally, it has been shown that this system of ordinary differential equations has blow-up on time solutions that exist only on a finite time interval. The corresponding values of the first integrals and initial data are found through the equality conditions. A special attention is paid to a class of radially symmetric with respect to spatial variables solutions. In this case, the Helmholz equation degenerates into an non-autonomous linear second-order ordinary differential equation, which general solution is found in terms of the power functions and the Bessel functions. In a particular case of three spatial variables the general solution is expressed using trigonometric or hyperbolic functions.