= Solution
The standard <Shakura--Sunyaev thin disk> has three qualitative radial zones. Its hot inner part is dominated by <radiation pressure> and <electron-scattering opacity>; farther out, <gas pressure> overtakes radiation pressure while electron scattering can remain the main opacity; in the cool outer zone, gas pressure remains dominant and <free-free opacity> becomes important. The transition radii vary with $M$, $\dot M$, and $\alpha$.
For the inner zone, hold the <surface density of a disk>[surface density] $\Sigma$ fixed during a local thermal perturbation. Vertical <hydrostatic equilibrium> gives
$$
P_{\rm rad}\sim\Sigma\Omega_K^2H,
\qquad P_{\rm rad}\propto T_c^4,
$$
so $H\propto T_c^4/\Sigma$. The <alpha disk> prescription then yields
$$
Q^+\sim\nu\Sigma\Omega_K^2
\sim\alpha\Sigma\Omega_K^3H^2
\propto\frac{T_c^8}{\Sigma},
$$
whereas optically thick radiative diffusion with nearly constant electron-scattering opacity gives
$$
Q^-\propto\frac{T_c^4}{\kappa_{\rm es}\Sigma}.
$$
At equilibrium $Q^+=Q^-=Q_0$. For net cooling $\dot Q=Q^--Q^+$,
$$
\left.\frac{\partial\dot Q}{\partial T_c}\right|_\Sigma
=\frac{4Q_0}{T_c}-\frac{8Q_0}{T_c}
=-\frac{4Q_0}{T_c}<0.
$$
Thus the total-pressure alpha prescription predicts <thermal instability of a radiation-pressure-dominated alpha disk>: a temperature increase makes heating outrun cooling. The absence of ubiquitous, large-amplitude thermal limit cycles in luminous <active galactic nucleus>[AGN] light curves indicates that this local model omits stabilizing effects, plausibly magnetic pressure and stress, vertical advection, winds, or a stress law that does not simply track total pressure.
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