Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative

Olivier Ley; Erwin Topp; Miguel Yangari

doi:10.3934/dcds.2021007

Article Contents

2021, Volume 41, Issue 8: 3555-3577. Doi: 10.3934/dcds.2021007

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Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative

1.
Université de Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
2.
Departamento de Matemática y C.C., Universidad de Santiago de Chile, Casilla 307, Santiago, Chile
3.
Departamento de Matemática, Escuela Politécnica Nacional, Ladrón de Guevara E11-253, P.O. Box 17-01-2759, Quito, Ecuador

^* Corresponding author: Erwin Topp
^* Corresponding author: Erwin Topp

Received: February 2020

Early access: January 2021

Published: August 2021

Part of this work was done during a visit of E.T. and M.Y. to the Institut de Recherche Math´ematique de Rennes. They acknowledge the hospitality of the Institut.

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Abstract

We obtain some Hölder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order $ \alpha \in (0, 1) $ cast by a Caputo derivative. The Hölder seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejoffre setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case $ \alpha = 1 $, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive Caputo derivative for all time does not necessarily converge; we provide such a counterexample. However, we establish partial results of convergence under some geometrical assumptions.

Keywords:

Integro-Differential Equations,
Nonlinear Partial Differential Equations,
Caputo derivative,
Regularity,
Large Time Behavior.

Mathematics Subject Classification: Primary: 35K55, 35R09, 47G20, 35B40; Secondary: 33B20.

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Figure 1. Behavior of $ f = f_1f_2 $

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References

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