# Some results for the large time behavior of Hamilton-Jacobi Equations with Caputo Time Derivative

## TL;DR

This paper investigates the large time behavior of solutions to Hamilton-Jacobi equations with Caputo fractional time derivatives, establishing regularity estimates and partial convergence results under certain geometric conditions.

## Contribution

It provides the first regularity estimates and convergence insights for Hamilton-Jacobi equations with fractional Caputo derivatives, extending classical results to fractional time derivatives.

## Key findings

- Hölder regularity estimates independent of time
- Counterexample showing non-convergence with nonpositive Caputo derivative
- Partial convergence results under geometric assumptions

## Abstract

We obtain some H\"older regularity estimates for an Hamilton-Jacobi with fractional time derivative of order $\alpha \in (0,1)$ cast by a Caputo derivative. The H\"older 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.