In this paper, we investigate the initial-boundary value problem for a fourth order evolutionary systems of the partial differential equations. The main part of the systems include different terms,  namely, the time derivative, the linear bilaplasian, the monotonous and nonmonotonous nonlinear terms with the variable exponents of the nonlinearity. These exponents depend on space and time variables. Moreover, the systems contain the nonlinear nonlocal terms with integrals with respect to the space variables. We seek a weak solution to the considered initial-boundary value problem. This solution belongs to the corresponding generalized Lebesque and Sobolev spaces. The main theorem is devoted to the existance of the weak solution to considered problem. To prove this theorem we use the Faedo-Galerkin method, the integral estimates and the Aubin theorem.

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