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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.

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So this basically amounts to showing that if $g(x,0) = 0$ and $g_t = 0$ whenever $g(\cdot, t) = 0$, that $g$ is identically zero. – user7530 Dec 23 '12 at 18:02
Since your initial condition is continuous with compact support, my guess is that by a solution it's meant the convolution with the fundamental solution. – Jose27 Dec 23 '12 at 18:04
I agree with @Jose27: the author of the problem had in mind the physically relevant solution, not the rapidly growing ones. Otherwise you can give a counterexample with $\phi\equiv 0$ by taking any nonzero solution with zero initial data (if it happens to be spherically symmetric, translate it by a fixed vector in $x$ space). – user53153 Dec 24 '12 at 0:33
Thank you for the comments. I agree that the author expects us to show that the physical solution must be spherically symmetric, given symmetric initial data. – JZShapiro Dec 24 '12 at 5:24

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