Weighted Energy-Dissipation approach to doubly nonlinear problems on the half line

• DOI: 10.1007/s00028-017-0390-6
• Publication Date: 2017-04-09T03:21:21Z
• Authors: Goro Akagi; Stefano Melchionna; Ulisse Stefanelli
• Keywords: Regularization; Real line; Line (geometry); Minification

We discuss a variational approach to abstract doubly nonlinear evolution systems defined on the time half line (Formula presented.). This relies on the minimization of weighted energy-dissipation (WED) functionals, namely a family of (Formula presented.)-dependent functionals defined over entire trajectories. We prove WED functionals admit minimizers and that the corresponding Euler-Lagrange system, which is indeed an elliptic-in-time regularization of the original problem, is strongly solvable. Such WED minimizers converge, up to subsequences, to a solution of the doubly nonlinear system as (Formula presented.). The analysis relies on a specific estimate on WED minimizers, which is specifically tailored to the unbounded time interval case. In particular, previous results on the bounded time interval are extended and generalized. Applications of the theory to classes of nonlinear PDEs are also presented.