What is meant by 'variational structure'? Consider 
$$
u_t=D\Delta u+f(x,u,\nabla u)~~(*).
$$
In a script it is said:


We first consider the variational structure which makes $(*)$ an $L^2$-gradient flow.


I have two questions to this:
1.) What is meant with variational structure?
2.) What is an $L^2$-gradient flow and in which sense is $(*)$ an $L^2$-gradient flow?
 A: Finding a variational structure here means finding a functional such that the PDE is the gradient flow of that functional. 
The gradient flow of a functional $\psi :V\to \mathbb {R} $ is the differential equation 
$$ 
u_t=-\text{grad}_u\psi.
$$
That is, at each point of time the gradient flow changes $ u $ infinitesimally to achieve a minimization of $\psi $ (remember that the gradient gives the direction of steepest ascent,  so the negative gradient gives the direction of steepest descent)
For example, when $\psi $ is linear, the solution of the gradient flow has constant velocity, because the gradient is constant. 
A more useful example is $ V $ being the Sobolev space $ H^1(\mathbb{R}^n)$ and $\psi(u)=\int |\nabla u|^2$ (nonlinear! ). You can calculate using Greens theorem that the gradient flow here is the heat equation. However,  there are technicalities involved: when you say "direction of steepest descent" you imply that you are considering directions in a certain normed space. If you considered the Sobolev space here,  then you would get $ u_t=-u $. You get the heat equation only if you consider the space $ L^2$ as space of allowed directions and use formal calculations, but strictly speaking the functional is not defined for $ u+hv $ when $ u $ is a Sobolev function, $ v\in L^2$ and $ h> 0$.
