  We are interested in nonlocal Eikonal Equations arising in the study of the
dynamics of dislocations lines in crystals. For these nonlocal but also non
monotone equations, only the existence and uniqueness of Lipschitz and
local-in-time solutions were available in some particular cases. In this paper,
we propose a definition of weak solutions for which we are able to prove the
existence for all time. Then we discuss the uniqueness of such solutions in
several situations, both in the monotone and non monotone case.
