Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations

Guy Barles (1) , Pierre Cardaliaguet (2) , Olivier Ley (1, 3) , Aurélien Monteillet (2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 LM-Brest - Laboratoire de Mathématiques
3 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Nonlocal Hamilton-Jacobi Equations; existence results; dislocation dynamics; Fitzhugh-Nagumo system; nonlocal front propagation; viscosity solutions; $L^1-$dependence in time

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Pierre Cardaliaguet, Olivier Ley, Aurélien Monteillet. Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations. Nonlinear Analysis: Theory, Methods and Applications, 2009, 71 (7-8), pp.2801-2810. ⟨10.1016/j.na.2009.01.156⟩. ⟨hal-00287372⟩