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.