A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions

Abstract

In this article, we are interested in the asymptotic behavior of the solutions of scaled reaction-diffusion equations, set in bounded domains, associated with Neumann type boundary conditions, and more precisely in cases when such behavior is described in terms of moving interfaces. A typical example is the case of the Allen–Cahn Equation associated with an oblique derivative boundary condition where the generation of a front moving by mean curvature with an angle boundary condition is shown. In order to rigourously establish such results, we modify and adapt the “geometrical approach” introduced by P.E. Souganidis and the first author for solving problems set in ${\mathbb{R}}^N:$ we provide a new definition of weak solution for the global-in-time motion of fronts with curvature dependent velocities and with angle boundary conditions, which turns out to be equivalent to the level-set approach when there is no fattening phenomenon. We use this definition to obtain the asymptotic behavior of the solutions of a large class of reaction-diffusion equations, including the case of quasilinear ones and $(x,t)-$ dependent reaction terms, but also with any, possibly nonlinear, Neumann boundary conditions.