I'm looking for a reference that can explain under which conditions the variational problem
$$ E = \int_{\Omega} \mathcal{L}d\Omega $$
can be solved using the evolution equation
$$ \mathcal{\partial_t} u = -\nabla \mathcal{L} $$
by reading through Chapter 4 I've seen that we can construct a newton iteration for such problem, but there's no PDE involved. I've also looked up some youtube videos and it seems people just use that method, I'd like to understand what kind of conditions need to be met in order to allow to use such PDE to solve a variational problem.
Is there any reference you can suggest? I've found some references for the Sobolev gradient, but I found them quite heavy.