When does the heat equation have a solution with a sequence of similar level sets? (Q893366)

The authors consider an overdetermined Cauchy problem for the heat equation \[ \begin{alignedat}{2} \partial_t u & =\Delta u \quad & \text{ in } & \quad \mathbb R^N\times (0,+\infty),\\ u& =g\geq 0 \quad & \text{ on }& \quad \mathbb R^N\times \{0\},\end{alignedat} \] where \(N\geq 1 \) and \(g\) is a non-trivial bounded nonnegative function. It is proved that if the problem has a non-trivial nonnegative solution with a certain sequence of similar level sets, then the solution must be radially symmetric.