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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2601.12352 |
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| _version_ | 1866910011966357504 |
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| author | Nakajima, Yoshihito |
| author_facet | Nakajima, Yoshihito |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12352 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Time-fractional nonlinear evolution equations with time-dependent constraints Nakajima, Yoshihito Analysis of PDEs 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. |
| title | Time-fractional nonlinear evolution equations with time-dependent constraints |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.12352 |