I will first prove that, if the solutions of certain Cauchy and Cauchy-Dirichlet problems for the heat equation possess one invariant spatial level surface, then they must be symmetric. In order to prove this result, short-times asymptotic estimates for the heat content of balls touching the boundary of the domain are needed. Secondly, I will show how these estimates and symmetry results can be extended to relevant nonlinear settings such as the porous media and the evolutionary p-Laplace equations. Finally, I will show some connections to some related problems for the Helmholtz equation.