Abstract. In this article, we provide existence results for a general class of nonlocal and nonlinear second-order parabolic equations. The main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way. Among applications, we present level-set equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo systems.

We are interested in nonlocal eikonal equations describing the evolution of interfaces moving with a nonlocal, non-monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts.

Abstract. We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.Mathematics Subject Classification. 53C44, 49Q15, 49L25, 28A75, 58A25.

We prove existence of a solution for a polymer crystal growth model describing the movement of a front (Γ(t)) evolving with a nonlocal velocity. In this model the nonlocal velocity is linked to the solution of a heat equation with source δΓ. The proof relies on new regularity results for the eikonal equation, in which the velocity is positive but merely measurable in time and with Hölder bounds in space. From this result, we deduce a priori regularity for the front. On the other hand, under this regularity assumption, we prove bounds and regularity estimates for the solution of the heat equation. 1991 Mathematics Subject Classification. 49L25, 35F25, 35A05, 35D05, 35B50, 45G10.We introduce the following set of assumptions, denoted by (A) in the rest of the paper. (A1) κ is a fixed real number (κ is positive in the case of a negative heat source and negative otherwise),ḡ : R N → R is Lipschitz continuous, bounded, and there exist A, B > 0 such that A ≤ḡ(z) ≤ B for all z ∈ R .(A2) v 0 : R N → R is Lipschitz continuous and bounded.

We prove integral formulations of the eikonal equation u t = c(x, t)|Du|, equivalent to the notion of viscosity solution in the framework of the set-theoretic approach to front propagation problems. We apply these integral formulations to investigate the regularity of the front: we prove that under regularity assumptions on the velocity c, the front has locally finite perimeter in {c = 0}, and we give a time-integral estimate of its perimeter.