Affiliations: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy | Department of Mathematics, MIT, Cambridge, MA 02139, USA | Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada
Note: [] The first author was partially supported by M.U.R.S.T. project: Analisi e controllo di equazioni di evoluzione deterministiche e stocastiche.
Abstract: This paper is concerned with the asymptotic behavior as ε→0 of the solutions of nonlocal reaction–diffusion equations of the form ut−Δu+ε−2f(u,ε∫0u)=0 in O×(0,T) associated with nonlinear oblique derivative boundary conditions. We show that such behavior is described in terms of an interface evolving with normal velocity depending not only on its curvature but also on the measure of the set it encloses. To this purpose we introduce a weak notion of motion of hypersurfaces with nonlocal normal velocities depending on the volume they enclose, which extends the geometric definition of generalized motion of hypersurfaces in bounded domains introduced by G. Barles and the first author to solve a similar problem with local normal velocities depending on the normal direction and the curvature of the front. We also establish comparison and existence theorems of viscosity solutions to initial‐boundary value problems for some singular degenerate nonlocal parabolic pde's with nonlinear Neumann‐type boundary conditions.
