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In this paper, the existence of an $\omega$-periodic weak solution of a parabolic equation (1.1) with the boundary conditions (1.2) and (1.3) is proved. The real functions $f(t,r),h\left(t\right),a\left(t\right)$ are assumed to be $\omega$-periodic in $t,f\in L_2(S,H),a,h$ such that $a^{\text{'}}\in L_{\infty }\left(R\right),h^{\text{'}}\in L_{\infty }\left(R\right)$ and they fulfil (3). The solution $u$ belongs to the space $L_2(S,V)\cap L_{\infty }(S,H)$, has the derivative $u^{\text{'}}\in L_2(S,H)$ and satisfies the equations (4.1) and (4.2). In the proof the Faedo-Galerkin method is employed.

The existence of a periodic solution of a nonlinear equation $z^{\text{'}}+A_{0}z+B_{0}z=F$ is proved. The theory developed may be used to prove the existence of a periodic solution of the variational formulation of the Navier-Stokes equations or the equations of magnetohydrodynamics. The proof of the main existence theorem is based on Rothe method in combination with the Galerkin method, using the Brouwer fixed point theorem.