We characterize all the solutions of the heat equation that have their
(spatial) equipotential surfaces which do not vary with the time. Such
solutions are either isoparametric or split in space-time. The result gives a
final answer to a problem raised by M. S. Klamkin, extended by G. Alessandrini,
and that was named the Matzoh Ball Soup Problem by L. Zalcman. Similar results
can also be drawn for a class of quasi-linear parabolic partial differential
equations with coefficients which are homogeneous functions of the gradient
variable. This class contains the (isotropic or anisotropic) evolution
p-Laplace and normalized p-Laplace equations
