Consider the boundary problem
$$ \dfrac{\partial u}{\partial t}+ A^2 u+Af(u)+\lambda g(u)=0\ \ \ \ \ \mbox{ in }\ \ \ \Omega. $$ $$ \dfrac{\partial u}{\partial \nu}=\dfrac{\partial \Delta u}{\partial \nu}=0\ \ \ \ \ \ \mbox{ on }\ \ \Gamma=(\mbox{boundary}). $$ where here $A=-\Delta$ and $\lambda$ are positive parameters.
Rewritting the system in the equivalent form $$ A^{-1}\dfrac{\partial \bar{u}}{\partial t}+A\bar{u}+f(\bar{u}+\langle u\rangle)+\lambda g(\bar{u}+\langle u \rangle)=0, $$ where, here $\bar{u}=u-\langle u \rangle$ and $\langle u \rangle=\displaystyle \int_{\Omega} u \,dx$.
I was trying to write the variational formulation of previous equation and define the system using Galerkin scheme (first define the orthonormal base of $A$ and the eigenvalues) using the form of $u_m$ like $$ u_m(t)=\sum_{i=1}^{m} \bar{u}_i \omega_j,\ \ \bar{u}_i=\bar{u}_i(t),\ i=1,\cdots, m, $$ where $\{w_j\}_{j=1, \cdots, m}$ is the orthonormal base of $A$, then I get $$ \dfrac{\,d}{\,dt}((A^{-1} u_m, \omega_j))+((A u_m, w_j))+((f(u_m+\langle u\rangle), \omega_j))+\lambda ((g(u_m+\langle u\rangle), \omega_j))=0 $$
Is this correct? In general, I see that it is okay, but am stuck in the term $\langle u\rangle$ (the spatial average) how we can define it using this scheme? If it is not correct, how I must write it? and how I can define each term in this formulation, so that I can obtain an ordinary differential equation?