A geometric approach to front propagation problems in bounded domains with Neumann-type boundary conditions
Summary: We are interested in the asymptotic behavior of the solutions of scaled reaction-diffusion equations 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 establish such results rigourously, we modify and adapt the ``geometrical approach'' introduced by P. E. Souganidis and the first author for solving problems 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.
scientific article; zbMATH DE number 2052701