When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable?

We consider solutions of the initial-Neumann problem for the heat equation on bounded Lipschitz domains in ℝ^N; and classify the solutions whose spatial level surfaces are invariant with respect to the time variable. (Of course, the values of each solution on its spatial level surfaces vary with time.) The prototype of such classification is a result of Alessandrini, which proved a conjecture of Klamkin. He considered the initial-Dirichlet problem for the heat equation on bounded domains and showed that if all the spatial level surfaces of the solution are invariant with respect to the time variable under the homogeneous Dirichlet boundary condition, then either the initial data is an eigenfunction or the domain is a ball and the solution is radially symmetric with respect to the space variable. His proof is restricted to the initial-Dirichlet problem for the heat equation. In the present paper, in order to deal with the initial-Neumann problem, we overcome this obstruction by using the invariance condition of spatial level surfaces more intensively with the help of the classification theorem of isoparametric hypersurfaces in Euclidean space of Levi-Civita and Segre. Furthermore, we can deal with nonlinear diffusion equations, such as the porous medium equation.