# Show that, given spherically symmetric initial data, a solution to the heat equation is spherically symmetric

Let $\phi \colon \mathbb{R}^n \to \mathbb{R}$ be continuous with compact support. Furthermore, suppose that $\phi$ is spherically symmetric. That is, suppose that $\phi(Tx) = \phi(x)$ for every orthogonal transformation $T\colon\mathbb{R}^n \to \mathbb{R}^n$. Prove that a solution $f\in C^2(\mathbb{R}^n)$ to the heat equation

$\cases{ \Delta f – f_t &$(x,t) \in \mathbb{R}^n \times (0, \infty)$,\cr f(x, 0) = \phi(x) &$ x \in \mathbb{R}^n$, }$

must also have spherical symmetry in the variable $x \in \mathbb{R}^n$.

This question is from an old PDE qual that I’m studying. In my PDE course, I encountered a similar problem where I proved that, given spherically symmetric initial data, the solution to the wave equation is spherically symmetric. That proof utilized the uniqueness of the solution to the wave equation, and the fact (which we proved in class) that the Laplacian “commutes” with orthogonal transformations (i.e., $\Delta f(Tx) = \Delta (f \circ T)(x)$ for $f \in C^2(\mathbb{R}^n)$ and orthogonal $T\colon\mathbb{R}^n \to \mathbb{R}^n$) .

However, my proof of this previous problem does not translate to the question above, because I know that the heat equation does not have a unique solution unless solutions are required to satisfy a certain growth estimate.