If $N_1(t) = \int_0^L(u(x,t)-u_0)^2\,dx$ is monotonically decreasing in time, what does that say about $u$? 
Consider the heat equation in a rod of length $L$ with fixed temperatures at the endpoints:
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2} \\
u(0,t) = u_0 \\
u(L,t) = u_0
\end{cases}
\end{equation}
Show that the norm
$$N_1(t) = \int_0^L (u(x,t)-u_0)^2\,dx$$
is monotonically decreasing in time for any solution that is nonuniform in $x$. Conclude
from this that the temperature $u$ decays to a uniform density if $t\to\infty$.

I have already shown that
$$\frac{d}{dt} N_1(t) \leq 0,$$
which means that $N_1$ is monotonically decreasing in $t$. What I don't understand is how to conclude from this that $u$ goes to a uniform density. Can I say that since the norm is non-negative but monotonically decreasing, if $t\to\infty$, we must have $N_1(t) \to 0$? And since $u$ is continuous from this it follows that $u\to u_0$? I feel that this is wrong, since maybe the norm goes to a constant $c\geq0$. But how can I then conclude anything about $u$?

In that case:
$$
\frac{\mathrm{d}}{\mathrm{d}t} N_1(t) = -2\kappa \int^L_0 \left( \partial_x u\right)^2~\mathrm{d}x  = -2\kappa \int^L_0 \left( \partial_x u - \partial_x u_0\right)^2~\mathrm{d}x 
$$
Since $u-u_0$ vanishes at $0$ and $L$, we can use Poincaré's inequality to conclude that there exists some $C>0$ s.t.:
$$
-2\kappa \int^L_0 \left( \partial_x u - \partial_x u_0\right)^2~\mathrm{d}x \leq -2\kappa C \int^L_0 (u-u_0)^2~\mathrm{d}x = -2\kappa CN_1(t)
$$
Gronwall's lemma gives us:
$$
N_1(t) \leq N_1(0)\exp\left(-2C\kappa t \right) 
$$
Now I am sure you see where convergence comes from.
But just $N_1$ being decreasing is not enough as an argument. The point is that the derivative is "negative enough".
