let $\Omega$ be a bounded and open subset of $\mathbb{R}^n$. Let $u \in L^2(0,T; H^1(\Omega)$ be a weak solution of the heat equation with zero right side $f$ and homogeneous Neumann boundary conditions, i.e. $u$ solves
$\begin{align*} (\partial_t u(t), v)_{L^2(\Omega)} + (\nabla u(t), \nabla v)_{L^2(\Omega)} = 0 \end{align*}$
for all $v \in H^1(\Omega)$ and $u_0 \in L^2(\Omega)$.
What does it mean in this context, that $u$ depends stably on $u_0$? Especially, in which norms do I have to show the dependence? Do I have to show that $u$ measured in $L^2(0,T; H^1(\Omega))$ is smaller than $u_0$ measured in $L^2(\Omega)$? Or do I need to show stability for $u$ also in $L^2(\Omega)$?