= Solution
The instantaneous <kinematic viscosity> law is $\bar\nu(r,\Sigma)$, with no explicit constitutive memory. Expand its transported quantity at the possibly evolving background:
$$
\bar\nu(r,\Sigma_0+\Sigma_1)(\Sigma_0+\Sigma_1)
=\bar\nu_0\Sigma_0+\left.\partial_\Sigma(\bar\nu\Sigma)\right|_0\Sigma_1+O(\Sigma_1^2).
$$
The <viscous transport response exponent> satisfies $\partial_\Sigma(\bar\nu\Sigma)|_0=q\bar\nu_0$. Subtract the background equation and retain only linear terms to obtain
$$
\boxed{\partial_t\Sigma_1=\frac3r\partial_r\left[r^{1/2}\partial_r(r^{1/2}q\bar\nu_0\Sigma_1)\right].}
$$
This remains a <linear equation> with space- and time-dependent background coefficients; neither a steady background nor constant $q$ is required for this step. The sign of the response coefficient is the <negative-diffusion criterion for viscous disk instability>.
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