Physical interpretation of L2 norm of heat equation solution

Question

For the heat equation

\begin{equation} u_t(t,x) = \nu u_{xx}(t,x) \end{equation}

for $x \in [0,1]$ with boundary conditions $u(t,0) = u(t,1) = 0$ and initial value $u(0,x) = u_0(x)$ it is easy to show that the "energy" defined as

\begin{equation} E(t) = \frac{1}{2} \int_{0}^{1} u^2(t,x)~dx \end{equation}

decays over time, that is $E(t) \leq E(0)$. I wonder if there is any physical interpretation of the quantity $E$?

In terms of units, $u$ is temperature in Kelvin while the thermal diffusivity $\nu$ has units $m^2/s$ and is composed from \begin{equation} \nu = \frac{k}{\rho c_p} \end{equation} where $k$ is the thermal conductivity in $W/(m*K)$, $\rho$ the density in $kg/m^3$ and $c_p$ the specific heat capacity in $J/(kg*K)$.

I figured out that $u$ is related to the internal energy in Joule per volume via \begin{equation} I = c_p \rho u. \end{equation}

But what is the interpretation of $E$, which is related to $u^2$?

Answer 1

In these slide there are some comments about the energy.

At pag 4 it focus on the fact that this energy is not a physical energy, but it is a mathematical tool.

At pag 8 it observes that:

From a physical point of view it seems reasonable that a the energy will decrease in a system without any heat source

And after this the author defines another kind of energy based on $u_x$. Also this energy is decreasing (with the same boundary conditions).