Non-Local Hamilton-Jacobi Equations Arising in Dislocation Dynamics

Abstract

We investigate a class of non-local Hamilton–Jacobi equations arising in dislocation dynamics. The class of Hamilton–Jacobi equations treated here is a variation of those studied by N. Forcadel, C. Imbert and R. Monneau in [Discrete Contin. Dyn. Syst. 23 (2009)(3), 785 – 826], and the new feature lies in the singularity at the origin of the kernel functions which describe non-local effects. For the class of Hamilton–Jacobi equations, we establish some stability properties of (viscosity) solutions, comparison theorems between subsolutions and supersolutions and existence theorems of solutions.