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We prove existence and asymptotic behaviour of a weak solutions of a mixed problem for $$\begin{array}{cc}& u^{\text{'}\text{'}}+Au-\Delta u^{\text{'}}+{\left|v\right|}^{\rho +2}{\left|u\right|}^{\rho }u=f_1\hfill \\ & v^{\text{'}\text{'}}+Av-\Delta v^{\text{'}}+{\left|u\right|}^{\rho +2}{\left|v\right|}^{\rho }v=f_2*\hfill \end{array}$$ where $A$ is the pseudo-Laplacian operator.

In this paper we prove existence of periodic solutions to a nonlinear evolution system of second order partial differential equations involving the pseudo-Laplacian operator. To show the existence of periodic solutions we use Faedo-Galerkin method with a Schauder fixed point argument.