Spatial critical points not moving along the heat flow II: The centrosymmetric case

We consider solutions of initial-boundary value problems for the hear equat ion on bounded domains in R-N, and their spatial critical points as in the previous paper [MS]. In Dirichlet, Neumann, and Robin homogeneous initial-b oundary value problems on bounded domains, it is proved that if the origin is a spatial critical point never moving for sufficiently many compactly su pported initial data being centrosymmetric with respect to the origin, then the domain must be centrosymmetric with respect to the origin. Furthermore , we consider spatial zero points instead of spatial critical points, and p rove some similar symmetry theorems. Also, it is proved that these symmetry theorems hold for initial-boundary value problems for the wave equation.