For simplicity, consider solutions where $ u$ does not depend on $x, y$: 
$A u_t + B(t) u = 0$.  If $y^T A = 0$, that says $y^T B(t) u = 0$, so $u$ is
restricted to belong to a certain (possibly $t$-dependent) subspace.  The solutions
are usually **not** periodic in $t$.  Rather, the linear operator $u(0) \to u(2\pi)$  will have eigenvalues $\lambda$ corresponding to solutions where $u(2\pi) = \lambda u(0)$ (see Floquet theory).