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Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative

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Abstract

We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton–Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional Hamilton–Jacobi equation obtained via nonlinear adjoint method and sharp estimates in Sobolev and Hölder spaces for the corresponding linear problem.

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Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second-named author wishes to thank R. Schnaubelt and R. Zacher for useful discussions and references, the Department SBAI, Sapienza University of Rome and the Department of Mathematics of the University of Padova for the hospitality during the preparation of the paper.

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  1. SBAI, Università di Roma “La Sapienza”, Via A. Scarpa 14, 00161, Roma, Italy

    Fabio Camilli

  2. Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100, L’Aquila, Italy

    Alessandro Goffi

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  1. Fabio Camilli
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Camilli, F., Goffi, A. Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative. Nonlinear Differ. Equ. Appl. 27, 22 (2020). https://doi.org/10.1007/s00030-020-0624-0

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