 Gradient flow with diagonal mobility

Positive mobility coefficients $m_r(x)$ give $dU/dt={\sum }_r{m}_r(x)({\partial }_{r}U{)}^2\ge 0$. With $m_r={\kappa }_r{x}_r$, the substitution $z_r=2\sqrt{x_r/{\kappa }_r}$ turns the equation into ordinary gradient ascent for the transformed potential on positive coordinates.