When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable?
The author considers heat equations on a bounded domain under the Neumann boundary condition. He classifies the solutions whose level surfaces are invariant with respect to the time variable. He extends earlier results of Alessandrini who studied the problem for the Dirichlet boundary condition. In the latter case, the invariance is possible only if the solution is radially symmetric (and the domain is a ball) or if it coincides with an eigenfunction of the Laplace operator. In the Neumann case, there are additional possibilities. Also there are extra difficulties to be dealt with. The author also addresses the classification problem for some nonlinear diffusion equations, such as the porous medium equation.
scientific article; zbMATH DE number 1369962
scientific article