SOME RESULTS FOR THE LARGE TIME BEHAVIOR OF HAMILTON-JACOBI EQUATIONS WITH CAPUTO TIME DERIVATIVE

Olivier Ley (1) , Erwin Topp (2) , Miguel Yangari (3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 DMCC - Departamento de Matemática y Ciencia de la Computación [Santiago de Chile]
3 Escuela Politécnica Nacional, Quito

We obtain some Hölder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order α ∈ (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 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.

Keywords: Integro-Differential Equations; Nonlinear Partial Differential Equations; Caputo derivative; Regularity; Large Time Behavior

Domains: Analysis of PDEs [math.AP]

Olivier Ley, Erwin Topp, Miguel Yangari. SOME RESULTS FOR THE LARGE TIME BEHAVIOR OF HAMILTON-JACOBI EQUATIONS WITH CAPUTO TIME DERIVATIVE. Discrete and Continuous Dynamical Systems - Series A, 2021, 41 (8), pp.3555-3577. ⟨10.3934/dcds.2021007⟩. ⟨hal-02167760⟩