Let $M$ be a smooth compact closed manifold. 
Let $u \in H^1(0,T;H^{-1}(M)) \cap L^2(0,T;H^1(M))$ be a solution of
$$u_t - \Delta u - u = 0$$
$$u(0)=u(T)$$
satisfying $\int_M u(t) = 0$ for all $t$. Is there any way to show that $u$ must be zero (i.e. solutions are unique)?

The problem is the $-u$ term. We have Poincare's inequality in this but it does not help much.
I would appreciate not using a method to do with eigenvalue problems since this PDE is a simplified version of what I am working on.