The Neumann problem for one-dimensional parabolic equations with linear growth Lagrangian: evolution of singularities

Abstract In this paper, we obtain existence and uniqueness of strong solutions to the inhomogenous Neumann initial-boundary problem for a parabolic PDE which arises as generalization time-dependent minimal surface equation. Existence regularity in time solution are proved by means suitable pseudoparabolic relaxed approximation equation corresponding passage limit. Our main result is monotonicity positive negative singular parts distributional space derivative bounded variation initial data. Sufficient conditions instantaneous $$L^1$$ L 1 - $$W^{1,1}_{{\mathrm{loc}}}$$ W loc , or BV regularizing effects also discussed.