I work on a bounded domain in $\mathbb{R}^n$. Let $u \in H^1(0,T;H^{-1})\cap L^2(0,T;H^1)$ be a solution of the heat equation: $$\langle u', v \rangle + \int \nabla u \nabla v = 0$$ for each test function $v$. The solution has the property $$\int_\Omega u(t) = 0$$ for each $t$.

Is it possible to conclude from this information that $u \equiv 0$?