A variational approach to parabolic equations under general and p,q-growth conditions

We consider variational solutions to a Cauchy-Dirichlet problem, related to a function f = f (x, u, ξ), convex with respect to (u, ξ) and coercive in ξ ∈ R^(N×n), but it not necessarily satisfies a growth condition from above. A motivation to consider a class of such energy functions f can be also easily found in the stationary case, where a large literature in the calculus of variations is devoted to the minimization of p,q-growth problems