Let $u$ and $v$ be the weak solutions of $$u_t - \Delta u = f$$ $$u(0)=u_0$$ and $$-\Delta v = f$$ $$|\Omega|^{-1}\int_\Omega v =0$$ on a bounded domain $\Omega$, where $u$ and $v$ satisfy homogeneous Neumann BCs. Here we may take $f$ and $u_0$ to have spacial mean values zero.

I'm trying to prove an estimate of the form $$\lVert{u(t)-v}\rVert_{L^1(\Omega)} \leq C(t)\lVert{u_0-v}\rVert_{L^1(\Omega)}$$ where $C(t) \to 0$ as $t \to \infty$.

I can prove this if instead of $L^1$ norms we had $L^2$, but I need $L^1$ on both sides. Does anyone know how to achieve this? The $L^2$ case can be done with testing the weak form satisfied by $u-v$ with $u-v$, Poincare's inequality and actually we will find $C(t) = e^{-Kt}$ after using it as an integrating factor.