I have to solve the following exercise:
Consider the heat conduction in a rod of length $L$ described by $$\partial_tu = k\partial_x^2u$$ Suppose that the rod has fixed temperatures at the end points:
$$u(0,t) = u(L,t) = u_0, \forall t.$$
Show that the norm $$N_1(t) = \int_0^L(u(x,t) - u_0)^2dx$$ 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 $t\to\infty$.
How do I solve this exercise? I have no clue what I should do really. I've tried computing the norm $N_1(t)$ but this didn't really get me anywhere.
Take derivative of norm to see how it changes over time $$\begin{align} \frac{d}{dt}N_1(t)&=\frac{d}{dt}\int_0^L\left(u-u_0\right)^2=\int_0^L\frac{d}{dt}\left(u-u_0\right)^2=2\int_0^L\left(u-u_0\right)\partial_tu\\ &=2k\int_0^L(u-u_0)\partial_x^2u=\left.2k(u-u_0)\partial_xu\right|_0^L-\int_0^L\left(\partial_xu\right)^2\\ &=-\int_0^L\left(\partial_xu\right)^2 \end{align}$$ The very last term is always negative. This means the change in the norm over time is decreasing which is what we wanted to show.
Because the norm cannot be negative and because it is decreasing, it must decay to a uniform density with $u_0$ at the boundaries as $t\to\infty$.