A variational approach to parabolic equations under general and p, q-growth conditions. (May 2020) - Marcellini, Paolo
Abstract: We consider variational solutions to the Cauchy-Dirichlet problem ∂ t u = div D ξ f ( x, u, D u ) − D u f ( x, u, D u ) in Ω T u = u 0 on ∂ par Ω T where the function f = f x, u, ξ, f : R n × R N × R N × n → [ 0, ∞ ), is 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 [45]  and to double phase problems [23], [24], [4], [5], [6] . In the parabolic context the notion of variational solution (see the references from [8] to [15] ) is compatible with the lack of the same polynomial growth from below and from above.