Time-fractional gradient flows for nonconvex energies in Hilbert spaces

TL;DR

The paper develops an abstract theory for time-fractional gradient flows in real Hilbert spaces with nonconvex energies, formulated as a Cauchy problem involving the difference of two subdifferentials and a nonlocal time derivative $(k*(u-u_0))'$. It introduces a robust toolkit—fractional chain-rule formulae, a continuous-representation criterion for convolutions, and Lipschitz perturbation theory—to overcome the lack of a semigroup and the time-continuity challenges, proving local and global existence of strong solutions under precise structural and growth conditions. The framework is then applied to Cauchy–Dirichlet problems for time-fractional $p$-Laplacian subdiffusion with blow-up terms, yielding local existence for Sobolev-subcritical exponents and global results for small data, with extensions to critical and large-data regimes under additional assumptions. This work extends Brézis–Otani–Ishii–Otani theory to time-fractional, nonconvex settings, providing a versatile analytical tool for nonlocal diffusion problems and their blow-up behavior in Hilbert spaces.

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.

Theorems & Definitions (14)

• proof
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• proof : Proof of Theorem \ref{['T:LE']}
• proof
• proof : Proof of Theorem \ref{['T:SDGE']}
• proof : Proof of Corollary \ref{['C:SDGEc']}
• proof : Proof of Theorem \ref{['T:GE']}