Weak solution $u(x,t)$ of heat equation converges as $t \in \infty$ Where can I find a proof that the weak solution $u \in L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$ of the heat equation
$$u_t -\Delta u = f$$
converges as $t \to \infty$ to the solution of the elliptic PDE
$$-\Delta u = f$$
??
Any references greatly appreciated.
 A: You can get an $L^2$ convergence result from basic energy methods.  It seems that you're assuming that $f = f(x)$, i.e. $f$ is not time-dependent.  Let's go ahead and take the slightly more general $f \in H^{-1}$.  I also assume that you mean $H_0^1(\Omega)$ and $H^{-1} = (H_0^1)^*$. 
Produce a weak solution $v \in H_0^1$ to the elliptic problem
$$
\begin{cases}
-\Delta v = f & \text{in }\Omega \\
 v = 0 &\text{on } \partial \Omega.
\end{cases}
$$
Next produce your weak solution $u \in L^2 H_0^1 \cap H^1 H^{-1}$ to
$$
\begin{cases}
\partial_t u -\Delta u = f & \text{in }\Omega \\
 u = 0 &\text{on } \partial \Omega \\
 u = u_0 &\text{for }t=0.
\end{cases}
$$
Consider $w = u-v$.  Since $v$ is time-independent, it's easy to see that $w \in L^2 H_0^1 \cap H^1 H^{-1}$ is a weak solution to 
$$
\begin{cases}
\partial_t w -\Delta w = 0 & \text{in }\Omega \\
 w = 0 &\text{on } \partial \Omega \\
 w = u_0 - v &\text{for }t=0.
\end{cases}
$$
Then use $w$ as a test function in the weak formulation:
$$
<\partial_t w,w>_* + (w,w)_1 = 0.
$$
Here $<,>_*$ denotes the dual pairing between $H_0^1$ and $H^{-1}$ and $(,)_1$ is the $H^1_0$ inner product.  From this we can deduce that
$$
\frac{d}{dt} \int_\Omega \frac{|w(x,t)|^2}{2}dx + \int_\Omega |\nabla w(x,t)|^2 dx =0.
$$
Poincare's inequality in $H_0^1$ says there exists $C >0$ such that
$$
C \int_\Omega | w(x,t)|^2 \le \int_\Omega |\nabla w(x,t)|^2.
$$
Plugging this in above shows
$$
\frac{d}{dt} \int_\Omega |w(x,t)|^2 dx + C \int_\Omega | w(x,t)|^2 \le 0
$$
(here we trivially bound $C \ge C/2$ and then multiply everything by $2$).
This is a differential inequality of the form $\dot{z}(t) + Cz(t) \le 0$. Multiply by $e^{Ct}$:
$$
\frac{d}{dt}(z(t) e^{Ct}) \le 0.
$$
Upon integrating, we find that $z(t) \le z(0)e^{-Ct}$ for all $t \ge 0$.
Using this above, we find that
$$
\int_\Omega |u(x,t) - v(x) |^2 dx = \int_\Omega |w(x,t)|^2 dx \le \exp(-Ct) \int_\Omega |u_0(x) -v(x)|^2 dx,
$$
from which we deduce that
$$
\lim_{t \to \infty}  || u(\cdot,t) - v||_{L^2(\Omega)} = 0.
$$
