Let $u$ be a solution of $$u' - \Delta u = 0 \quad\text{on $\Omega$}$$ $$\partial_\nu u = 0\quad\text{in $\partial \Omega$}$$ $$u(t=0)=u_0\quad\text{on $\Omega$}$$ where $\Omega$ is a bounded Lipschitz domain.

Here $u_0 \in H^1(\Omega)$ can be as regular as needed.

Is it possible to obtain a bound on $\lVert u \rVert_{L^2((0,\infty);L^2(\Omega)}$?

The infinite time interval is the problem -- Gronwall doesn't appear to help. One can bound the gradient of $u$ though in the desired space, and also one can bound $u$ in $L^\infty((0,T);L^2(\Omega))$. I've looked in books but all of the ones I've seen rely on Dirichlet BCs and the Poincare inequality in $H^1_0(\Omega)$, which I don't have.

Let $u$ be a solution of $$u' - \Delta u = 0 \quad\text{on $\Omega$}$$ $$\partial_\nu u = 0\quad\text{in $\partial \Omega$}$$ $$u(t=0)=u_0\quad\text{on $\Omega$}$$ where $\Omega$ is a bounded Lipschitz domain.

Here $u_0 \in H^1(\Omega)$ can be as regular as needed, but it must satisfy $$u_0 \geq 0 \text{ a.e., and } \int_\Omega u_0 > 0.$$

Is it at all possible to choose $u_0$ satisfying the above conditions such that we can obtain a bound on $\lVert u \rVert_{L^2((0,\infty);L^2(\Omega)}$?

The infinite time interval is the problem -- Gronwall doesn't appear to help. One can bound the gradient of $u$ though in the desired space, and also one can bound $u$ in $L^\infty((0,T);L^2(\Omega))$.

Is this even possible? Could it be achieved if there were a non-positive source term?