Time-fractional nonlinear evolution equations with time-dependent constraints

TL;DR

This work develops a robust abstract framework for time-fractional gradient flows driven by time-dependent convex functionals on a real Hilbert space. It introduces time-dependent fractional chain-rule formulas and Gronwall-type lemmas for nonlinear Volterra inequalities, enabling existence and uniqueness of strong solutions to $\partial_{t}[k*(u-u_{0})](t) + \partial \varphi^{t}(u(t)) \ni f(t)$ under the Kenmochi-type conditions (A1)–(A2) and kernels $(k,\ell)\in PC$. The results cover both $f\in W^{1,2}(0,T;H)$ and $f\in L^{2}(0,T;H)$, providing energy estimates such as $\varphi^{\bullet}(u(\bullet)) \in L^{\infty}(0,T)$ and $\ell*\|\partial_t[k*(u-u_{0})]\|_{H}^{2}\in L^{\infty}(0,T)$, and include an explicit convergence scheme via Moreau–Yosida approximations. An application to time-fractional nonlinear parabolic equations on moving domains is given, including a $p$-Laplacian subdiffusion problem with moving boundaries, thus validating the abstract theory for practical problems with evolving geometries.

Abstract

This article is devoted to presenting an abstract theory of time-fractional gradient flow equations for time-dependent convex functionals in real Hilbert spaces. The main results are concerned with the existence of strong solutions to time-fractional abstract evolution equations governed by time-dependent subdifferential operators. To prove these results, Gronwall-type lemmas for nonlinear Volterra integral inequalities and fractional chain-rule formulae are developed. Moreover, the obtained abstract results are applied to the initial-boundary value problem for time-fractional nonlinear parabolic equations on moving domains.

Theorems & Definitions (6)

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