= Solution
Assume again positive <diffusion> $\gamma>0$. Remove the drift with $A(x,t)=e^{Ux/(2\gamma)-i\omega t}f(x)$. Differentiating explicitly cancels the first <derivative> and gives
$$
\gamma f''+\left[\mu_0-\frac{U^2}{4\gamma}-\epsilon\lambda x+i\omega\right]f=0.
$$
Balance the second <derivative> with the linear confinement by taking $x=O(\epsilon^{-1/3})$, so the requested exponent is \b[$\sigma=1/3$]. More precisely, put
$$
\xi=\left(\frac{\epsilon\lambda}{\gamma}\right)^{1/3}x,\qquad
\Lambda=(\epsilon^2\gamma\lambda^2)^{1/3},\qquad
c=\frac{\mu_0-U^2/(4\gamma)+i\omega}{\Lambda}.
$$
The <eigenvalue problem> reduces to $f_{\xi\xi}=(\xi-c)f$. Its two solutions are <Airy functions>. The growing $\operatorname{Bi}(\xi-c)$ is excluded by the condition at infinity: its growth proportional to $e^{2\xi^{3/2}/3}$ dominates any linear-in-$x$ drift factor. The decaying $\operatorname{Ai}(\xi-c)$ remains admissible even after multiplication by $e^{Ux/(2\gamma)}$. The wall imposes $\operatorname{Ai}(-c)=0$, so $-c=z_n$, giving the <Airy global modes of a linearly confined Ginzburg-Landau equation>:
$$
\boxed{\omega_n=i\left[\mu_0-\frac{U^2}{4\gamma}+\Lambda z_n\right]},\qquad
A_n(x,t)=C_n e^{Ux/(2\gamma)-i\omega_nt}\operatorname{Ai}(\xi+z_n).
$$
The least negative zero, $z_1\simeq-2.338107$, gives the largest temporal <growth rate>. Hence
$$
\boxed{\text{global instability: }\mu_0>\frac{U^2}{4\gamma}+|z_1|(\epsilon^2\gamma\lambda^2)^{1/3}}.
$$
At equality the leading global <eigenmode> is neutral, and below it all these <eigenmodes> decay. In contrast, freezing the coefficients locally gives <absolute hydrodynamic instability> wherever $\mu_0-\epsilon\lambda x>U^2/(4\gamma)$. Such a pocket exists iff $\mu_0>U^2/(4\gamma)$ and extends to $x_a=[\mu_0-U^2/(4\gamma)]/(\epsilon\lambda)$. Thus global instability implies a locally absolutely unstable pocket, while the converse fails: a pocket shorter than $|z_1|(\gamma/(\epsilon\lambda))^{1/3}$ does not overcome the wall and confinement losses. The additional threshold is order $\epsilon^{2/3}$, precisely the loss produced by the <Airy function> spatial scale.
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