Radial Symmetry and Strict Radial Decrease for Master Equations with Decreasing Radial Potentials

We study the master equation $ (\partial_t -Δ)^{s} u(x,t) = R(|x|)f(u(x,t))\quad\mbox{in}\ \mathbb{R}^n\times\mathbb{R}, $ where $s\in(0,1)$, $R$ is positive and strictly decreasing in the radial variable, and $f$ is positive, locally Lipschitz, and satisfies $f'(0)<0$. Existing symmetry results for