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".