There is something I don't understand.
Imagine I want to solve : $$Min_{u}\int_{B(0,1)} \left| \nabla u(x) \right|^{2}+F(u(x))dx$$
It's L2 gradient flow is given by :
$$\partial_{t} u =2 \Delta u - F^{\prime}(u)$$
with $u=0$ on $\partial B(0,1)\times [0;T]$
If I have correctly understood solving the partial equation gives you the minimum of $$Min_{u}\int_{B(0,1)} \left| \nabla u(x) \right|^{2}+F(u(x))dx$$.
However the solution of the PDE depends (a priori) on time but the solution of the minimization doesn't. So I think there is something I don't understand.
Take a look at the post: What is the $L^2$ gradient flow?
The basic idea behind it is that the gradient flow allows you to start with an initial $u_0$ and consists of the evolution equation for this $u$, for which it eventually converges to a minimum. This evolution is parametrised by the time variable. The hope is that the minimum you reach is an stationary state, that is, independent of time.
It is always easier to think in the lowest dimensional analogue, and as said in the linked post, that corresponds to vector fields and their flow: given a real valued function, you might want to find the minimum of this function, and for this you first generate the associated gradient vector field and then its corresponding flow. The idea is that given any initial point, if you follow the flow of the vector field, you would eventually reach the minimum.