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For a diffusion equation with fixed homogeneous boundary conditions, a fundamental diffusion mode is an eigenfunction of its nonnegative decay operator belonging to the smallest decay eigenvalue ${\nu }_0$, when such an eigenvalue exists. It is the slowest decaying separated normal mode. With a discrete spectrum and an isolated lowest eigenvalue, initial data having a nonzero projection onto its eigenspace approach this decay law at late times. Strongly vanishing thermal diffusivity can destroy existence of a lowest mode, as illustrated by the loss of a spectral gap under strongly degenerate radial diffusion.

For the proposed trial profile, $R^{\prime }=-q{r}^{q-1}$. Its weighted energy isThe squared weighted norm isBoth are finite for $q>0$ and $n\ge 0$, and $R(1)=0$. Therefore the Rayleigh-Ritz variational principle givesWhen a fundamental diffusion mode exists, its decay rate is the infimum and is bounded above by this expression.

The existence qualification matters for the full range of $n$ printed in the question. For $n>2$, trial functions concentrated within radius $\epsilon$ have quotient proportional to ${\epsilon }^{n-2}$, proving the loss of a spectral gap under strongly degenerate radial diffusion. The infimum is zero and is not attained, because zero energy together with $R(1)=0$ forces the zero function. Thus there is no positive lowest-mode decay rate in that case. Even at $n=2$, the mode equation is $r^2{R}^{\prime \prime }+3r{R}^{\prime }+\nu R=0$: its characteristic powers are $r^{-1\pm \sqrt{1-\nu }}$, and for $\nu >0$ every nonzero solution is unbounded at the centre. Hence the asserted bounded fundamental mode cannot be assumed for all $n\ge 0$. The trial quotient remains valid, and the next part's $n=0$ case has the usual fundamental mode.