Influence of obstacles on the propagation speeds of nonlinear waves driven by curvature

We consider nonlinear waves driven by curvature in the presence of periodically distributed obstacles. When the obstacles are small, Matano et al. (2006) proved rigorously that the propagation speed of a wave mainly depends on the opening angle of the obstacles. In this paper we first explain in a formal way that, when the obstacles are small, the propagation behavior of a wave near the obstacles has four different stages in one spatial period, they can be described by traveling waves, self-similar solutions, or curvature flows without driving force. When the obstacles are large, by numerical simulation we show that obstacles always decrease the propagation of waves; the propagation speed is decreasing in the opening angles and the sizes of the obstacles; but it is not monotone in the homogenization degree of the obstacles, the periods of obstacles in a row, or the isthmus widths between two rows.