Time-fractional nonlinear evolution equations with time-dependent constraints

TL;DR

This paper develops an abstract theory for time-fractional gradient flow equations with time-dependent convex functionals, establishing existence of solutions and fractional chain-rule formulae, and applies results to fractional parabolic equations on moving domains.

Contribution

It introduces fractional chain-rule formulae for subdifferentials of time-dependent convex functionals and proves existence of solutions for time-fractional evolution equations with moving domain applications.

Findings

Established fractional chain-rule formulae under Kenmochi condition.

Proved existence of strong solutions for time-fractional evolution equations.

Applied abstract theory to fractional parabolic equations on moving domains.

Abstract

This article is devoted to developing an abstract theory of time-fractional gradient flow equations for time-dependent convex functionals in real Hilbert spaces. The main results concern the existence of strong solutions to time-fractional abstract evolution equations governed by subdifferential operators of time-dependent convex functionals. In the classical theory of gradient flow equations, chain-rule formulae play a crucial role in various analyses, and such formulae for subdifferentials of time-dependent functionals are also known in the case of first-order time derivatives. In contrast, in the present setting, the presence of time-fractional derivatives prevents the direct use of the usual chain-rule. To overcome this difficulty, fractional chain-rule formulae for subdifferentials of time-dependent convex functionals are established under a nonlocal variant of the so-called Kenmochi condition. Moreover, Gronwall-type lemmas for nonlinear Volterra integral inequalities are developed. Finally, the abstract results obtained are applied to initial-boundary value problems for time-fractional degenerate parabolic equations on moving domains.