= Solution
First remove the observation time from the spectral amplitude. For any fixed horizon $H>t$, replace $G_0(k,t)$ by $G_0(k,H)$ in the formula of part (d). The difference in the boundary integral is proportional to
$$
\int_{\partial D_+}e^{ikx}(3k^2+1)\int_t^H e^{w(k)(s-t)}g_0(s)\,ds\,dk.
$$
For every $s>t$, this integrand is holomorphic inside $D_+$, where $\operatorname{Re}w<0$, and decays on its large closing arcs. The <Cauchy integral theorem> makes the difference zero. This <finite-horizon extension of a boundary transform> therefore gives the time-independent spectra
$$
F(k),\qquad \mathcal B_H(k)=\mathcal TF(k)+(3k^2+1)\int_0^H e^{w(k)s}g_0(s)\,ds.
$$
All observation-time dependence is now in $e^{ikx-wt}$.
When the boundary history is sufficiently decaying, the natural long-time version is
$$
\boxed{\mathcal B_\infty(k)=\mathcal TF(k)+(3k^2+1)\int_0^\infty e^{w(k)s}g_0(s)\,ds.}
$$
The integral converges in $\operatorname{Re}w<0$ for bounded boundary data and on its boundary for integrable histories; otherwise one must state an Abel prescription or a suitable temporal growth assumption. Arbitrary smooth $g_0$ need not have such an infinite-time transform. The finite-horizon formula remains exact without that extra assumption, and the source alone does not prescribe a universal long-time limit.
A useful further decomposition, if $g_0(s)=g_\infty+r(s)$ with integrable $r$, is
$$
\int_0^\infty e^{ws}g_0(s)ds=-\frac{g_\infty}{w}+R(w),\qquad R(w)=\int_0^\infty e^{ws}r(s)ds,\qquad \operatorname{Re}w<0.
$$
This separates the persistent forcing from the transient spectrum. The pole at $k=i$, where $w=0$, accounts for the decaying stationary profile $g_\infty e^{-x}$; that profile also follows directly from $q_{xxx}-q_x=0$, spatial decay and its boundary value. For the remaining oscillatory integrals the phase is $i[kx+(k^3+k)t]$. At a fixed ray $x/t=\xi$, its stationary points satisfy $3k^2+1+\xi=0$, or $k=\pm i\sqrt{(1+\xi)/3}$. These time-independent amplitudes expose precisely the poles and saddle points used in the <method of steepest descent>; allowable deformations must also respect the analyticity strip of the initial transform.
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