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

  • * Corresponding author: Erwin Topp

    * Corresponding author: Erwin Topp 

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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  • 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.

    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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