A variational approach to parabolic equations under general and p, q-growth conditions. (May 2020)

• Record Type: Journal Article
• Authors: Marcellini, Paolo
• Is Part Of: Nonlinear analysis. Volume 194(2020)
• Publication Date: 2020-05
• Subjects: primary 35B65 49N60 -- secondary 35K10 35K55; Elliptic equations and systems -- p, q-growth conditions -- Parabolic equations and systems -- Variational solutions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.02.010
• Languages: English
• ISSNs: 0362-546X

Abstract

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.