Before getting to the question, some background. Let $u(x,t)$ be the temperature in a laterally insulated rod of length $L$, at position $x$ and time $t$. The temperature satisfies the heat equation
$\partial_t u = c \, \partial^2_x u$,
where $c > 0$, and let us assume $u(x,0) = f(x)=$ some given initial temperature distribution and assume $u(0,t) = 0$ and $u(L,t) = 0$ for all $t$. Thus, the ends of the rods are held at zero temperature.
To prove uniqueness of the solution, assume $u_1$ and $u_2$ satisfy the above conditions and let $u = u_1 - u_2$, which satisfies $\partial_t u = c\, \partial^2_x u$, $u(x,0) = 0$ and $u(0,t) = 0$ and $u(L,t) = 0$. The way everyone proves uniqueness (without using maximum principles) is to consider the integral
$V(t) = \int_0^L u(x,t)^2 dx$.
One then shows that $dV/dt \leq 0$ (this is easy). Thus, $V(t)$ is nondecreasing and hence $0 \leq V(t) \leq V(0)$ for all $t$. Since $V(0) = 0$, we get $V(t) = 0$ for all $t$ and hence $u_1 = u_2$.
Now, my question: What does $V(t)$ represent physically???
Not a single reference has ever explained what $V(t)$ represents, they simply introduce $V(t)$ "out of a hat", then show $dV/dt \leq 0$ without ever explaining what $V(t)$ physically represents. Thus, it seems the authors consider $V(t)$ as simply a "trick", but there should be some physical meaning of it.
What is the physical meaning of $V(t)$?