Time-fractional gradient flows for nonconvex energies in Hilbert spaces

TL;DR

This paper develops an abstract theory for time-fractional gradient flows in Hilbert spaces, establishing existence results for solutions to nonconvex energy-driven evolution equations with applications to subdiffusion PDEs.

Contribution

It introduces new fractional chain-rule and perturbation theories, enabling analysis of nonconvex energies with subdiffusive dynamics in Hilbert spaces.

Findings

Proved local and global existence of strong solutions.

Applied theory to p-Laplace subdiffusion equations with blow-up.

Developed tools for handling non-smooth energy evolutions.

Abstract

This article is devoted to presenting an abstract theory on time-fractional gradient flows for nonconvex energy functionals in Hilbert spaces. Main results consist of local and global in time existence of (continuous) strong solutions to time-fractional evolution equations governed by the difference of two subdifferential operators in Hilbert spaces. To prove these results, fractional chain-rule formulae, a Lipschitz perturbation theory for convex gradient flows and Gronwall-type lemmas for nonlinear Volterra integral inequalities are developed. They also play a crucial role to cope with the lack of continuity (in time) of energies due to the subdiffusive nature of the issue. Moreover, the abstract theory is applied to the Cauchy-Dirichlet problem for some $p$-Laplace subdiffusion equations with blow-up terms complying with the so-called Sobolev (sub)critical growth condition.