On Neumann problems for nonlocal Hamilton-Jacobi equations with dominating gradient terms

TL;DR

This paper establishes well-posedness and analyzes the large-time behavior of solutions for nonlocal Hamilton-Jacobi equations with Neumann boundary conditions, considering jump processes and dominant gradient terms.

Contribution

It proves a comparison principle and existence-uniqueness results for nonlocal Hamilton-Jacobi equations with complex boundary conditions and dominant gradient growth.

Findings

Comparison principle for bounded solutions

Existence and uniqueness of continuous solutions

Large time behavior analysis

Abstract

We are concerned with the well-posedness of Neumann boundary value problems for nonlocal Hamilton-Jacobi equations related to jump processes in general smooth domains. We consider a nonlocal diffusive term of censored type of order less than 1 and Hamiltonians both in coercive form and in noncoercive Bellman form, whose growth in the gradient make them the leading term in the equation. We prove a comparison principle for bounded sub-and supersolutions in the context of viscosity solutions with generalized boundary conditions, and consequently by Perron's method we get the existence and uniqueness of continuous solutions. We give some applications in the evolutive setting, proving the large time behaviour of the associated evolutive problem under suitable assumptions on the data.