Generalized gradient flows for time-dependent energies and applications to PDEs involving variable exponents

The present paper presents an abstract theory for proving (local-in-time) existence of energy solutions to some doubly-nonlinear evolution equations of gradient flow type involving time-dependent subdifferential operators with a quantitative estimate for the local-existence time. Furthermore, the abstract theory is employed to obtain an optimal existence result for some doubly-nonlinear parabolic equations involving space-time variable exponents, which are (possibly) non-monotone in time. More precisely, global-in-time existence of solutions is proved for the parabolic equations.