= Solution
Both profiles have integrable central cusps. Directly integrating the <surface density of a disk> and its <specific angular momentum> $j=\sqrt{GMr}$ gives
$$
M_d=2\pi\int_0^R\Sigma r\,dr=\pi\sigma R^2\frac{a}{2-a},
\qquad
J_d=2\pi\sqrt{GM}\int_0^R\Sigma r^{3/2}dr
=\frac{8\pi a}{5(5-2a)}\sqrt{GM}\,\sigma R^{5/2}.
$$
For $a=1$, substitution of the similarity functions gives
$$
\boxed{M_d=\frac{\pi C^2}{15A}t^{-1/5},\qquad
J_d=\frac{8\pi\sqrt{GM}C^{5/2}}{225A}=\mathrm{constant}}.
$$
The inward <accretion rate> is $\dot M(r)=-2\pi r\Sigma\bar u_r$. At the origin it tends to $M_d/(5t)=-dM_d/dt$. The outward <viscous torque in an accretion disk> is
$$
\mathcal G=-2\pi r^2W=3\pi\bar\nu\Sigma j
=3\pi A\sqrt{GM}\,\sigma^2r^{5/2}(X-1)^2.
$$
For this profile $\mathcal G\to0$ as $r\to0$, and the advected angular-momentum flux $\dot Mj$ also vanishes there. Both outer fluxes vanish at $R$. Thus disk mass decreases through central accretion while disk angular momentum is conserved. Including the accreted central mass restores total mass conservation for the disk-plus-sink system.
For $a=5/4$, instead,
$$
\boxed{M_d=\frac{\pi C^2}{9A}=\mathrm{constant},\qquad
J_d=\frac{4\pi\sqrt{GM}C^{5/2}}{75A}t^{1/4}}.
$$
Here the mass flux tends to zero at the origin, despite positive radial velocity everywhere else. The inner torque has the nonzero limit
$$
\boxed{\mathcal G(0,t)=\frac{\pi\sqrt{GM}C^{5/2}}{75A}t^{-3/4}=\frac{dJ_d}{dt}}.
$$
Thus the <nonaccreting similarity disk supplied by an inner torque> conserves disk mass but gains angular momentum from an external central torque. Its vanishing edge flux cannot supply that gain. The disk is not globally isolated in angular momentum; the combined disk and torque source may of course conserve it.
These profiles illustrate different boundary-controlled similarity classes of the <Keplerian viscous diffusion equation>. The $a=1$ solution is compatible with a <zero-torque inner boundary condition> and conserved finite angular momentum, and can describe a late-time spreading accretion disk after the detailed initial profile has been forgotten. Its mass diverges as $t\downarrow0$, so the formal zero-radius origin is not a finite-mass initial disk. The $a=5/4$ solution has finite initial mass concentrated at the origin and initially zero disk angular momentum, but requires the specified inner torque for $t>0$; that torque's $t^{-3/4}$ singularity is time-integrable. It may describe a spreading nonaccreting disk for compatible torque-driven boundary conditions, but cannot be an isolated angular-momentum-conserving disk's generic asymptotic state. Similarity scaling suggests their possible asymptotic roles; it does not establish convergence for arbitrary initial and boundary data.
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