On the existence of periodic solutions of a fourth-order nonlinear system of parabolic equations

In this paper, we consider a fourth-order system of two nonlinear parabolic equations with the double Laplace operator. We propose two approaches to constructing a solution, which allow separating components depending on time and spatial variables. The original problem is reduced to solving systems of algebraic and ordinary differential equations. We obtain explicit expressions for the exact solutions in terms of elementary or harmonic functions. Also, we identify the cases of solutions that are periodic in time and anisotropic in spatial variables.