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.

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关于具有Caputo时间导数的Hamilton-Jacobi方程的大时间行为的一些结果

我们获得了由Caputo导数转换的阶次时间导数为\ \ alpha \ in（0，1）$的汉密尔顿-雅各比的一些Hölder正则估计。Hölder半范数与时间无关，这使得可以研究解的长时间行为。我们关注Namah-Roquejoffre设置，其典型示例是Eikonal方程。与经典时间导数情况$ \ alpha = 1 $相反，解决方案在所谓的投影Aubry集上的收敛不是直截了当的，这是捕获较大时间行为的重要一步。确实，在所有时间都具有非正数Caputo导数的函数不一定会收敛。我们提供了这样的反例。但是，我们在某些几何假设下建立了收敛的部分结果。