 Diffusion from a gradient energy

The formal $L^2$ gradient flow of this energy is $u_t=\mathrm{div}({\Phi }^{\prime }(r)\nabla u/r)$, with a Neumann boundary condition for no flux. The radial Hessian eigenvalues of the gradient integrand are ${\Phi }^{\prime \prime }(r)$ and ${\Phi }^{\prime }(r)/r$. Their signs distinguish convex forward diffusion from formal backward diffusion. The heat equation uses $\Phi (r)=r^2/2$, whereas total variation flow uses $\Phi (r)=r$ with a subgradient interpretation at zero.