= Solution
The original PDF supplies the intended profiles $\rho=\rho_1f(\eta)$, $p=\rho_1\dot R^2g(\eta)$ and $u=\dot Rh(\eta)$; the TeX has corrupted the first two. At fixed $r$, $\partial_t\eta=-\eta\dot R/R$, and at fixed $t$, $\partial_r\eta=1/R$. The spherical <continuity equation> is therefore
$$
0=\partial_t\rho+\frac1{r^2}\partial_r(r^2\rho u)=\frac{\rho_1\dot R}{R}\left[-\eta f'+\frac1{\eta^2}(\eta^2fh)'\right].
$$
Since $\dot R/R=3/(5t)$ is nonzero, the <continuity equation for a superbubble similarity solution> is
$$
\boxed{-\eta f'+\frac1{\eta^2}(\eta^2fh)'=0,\qquad\text{or}\qquad(h-\eta)f'+fh'+\frac{2fh}{\eta}=0.}
$$
The ambient gas is at rest. In the locally stationary <shock frame>, its velocity is $-\dot R$; the downstream velocity is $u-\dot R$. Using the <shock compression ratio> from part (a) converts the downstream speed back to the laboratory frame and gives the inner shock boundary conditions
$$
\boxed{f(1^-)=\frac{\gamma+1}{\gamma-1},\qquad g(1^-)=\frac2{\gamma+1},\qquad h(1^-)=\frac2{\gamma+1}.}
$$
Immediately outside the shock, the corresponding unperturbed profiles are $f=1$, $g=0$ and $h=0$, with $g=0$ understood in the negligible-ambient-pressure limit.
The inner boundary must supply the injected power. If the source is idealized as a point and the profiles extend all the way inward, <central energy input in a similarity solution> requires
$$
\lim_{r\downarrow0}4\pi r^2u\left(\frac{\rho u^2}{2}+\frac{\gamma p}{\gamma-1}\right)=\dot E,
$$
or equivalently $\lim_{\eta\downarrow0}\eta^2h(fh^2/2+\gamma g/(\gamma-1))=125/(108\pi\alpha^5)$. A finite source region or an inner wind/contact region supplies a different matching description. One cannot impose a regular zero-flux centre on a source-free interior and still inject nonzero power.
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