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Some results for the large time behavior of Hamilton-jacobi equations with caputo time derivative

Abstract:

We obtain some Hölder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order-2 (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-Roquejo re setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case α = 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.