Energy functional - potential energy? Consider
$$
u_t=u_{xx}+f(u), x\in\mathbb{R}.
$$
We can write
$$
u_t=\nabla_{L^2}E(u)
$$
for the energy functional
$$
E(u):=\int_{\mathbb{R}}\frac{1}{2}u_x^2+F(u)\, dx, F(s):=\int_0^s f(s).
$$
Here, $\nabla_{L^2}E(u)
$ is the first variation of $E(u)$, playing the role of a Gradient, and we can determine it explicitly to be $u_{xx}+f(u)$ by computing
$$
\langle\nabla_{L^2}E(u),v\rangle_{L^2}.
$$
Another notation which is widely used for the first Variation is $\delta E(u)$. 
One says that the PDE has variational structure.
My question concern the interpretation of the "physics":
It is said that E(u) is the energy (functional). 
(1) What kind of energy does it represent: Potential energy, kinetic energy? Both?
My guess would be it is the potential energy since this reminds on something: Up to my knowledge, if we can write a force field $F(r)$ as $F(r)=\nabla \phi(r)$, then we call $\phi$ the potential (energy). Here, the situation looks similar, hence I would guess that $E(u)$ is the potential (energy)?
(2) I am no physicist, but isn't $E(u)$ in fact a work integral (integrating some force, here $\frac{1}{2}u_x^2+F(u)$, with respect to space)? So, can we interprete $u_t$ as work instead of energy? I guess this males no big difference since often one uses the terms work and energy quite equivalently.
Or does it make no sense to think about this 'energy (functional' in these physical terms?
 A: Energy functionals arise in many different contexts, but often describe the total energy of the state of a system. As such, E[u] will be a function of kinetic energy, potential energy, and other relevant forms of energy to the system (vibrational, etc.). To answer your question, E[u] will be the potential energy only if that is the only relevant form of energy to the system being studied.
Often, energy functionals are written for variational analysis of physical systems. At equilibrium points of the system, these functionals are minimized. It is these functionals that typically represent the potential energy of the system, as kinetic energy vanishes for dynamical equilibrium.  These minimizing (or at least extremizing) configurations correspond to functions for which the functional gradient that you mentioned vanishes. These functions are solutions of the Euler-Lagrange equations, in physics. If your problem included the second variation, this would no longer be so.
(It looks like in your case, the statement of your question is very similar to notation in discrete variational derivative method, but I'm not sure.)
