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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).