 Solution

Try $\rho =t^{a}f(\eta )$, $\eta =x/t^b$. The boundary data force $a=3$ and $f(0)=1$. The time derivative scales as $t^{a-1}$, advection as $t^{2a-b}$ and the source as $t^{a+c}$. Balancing all three gives $c=-1$ and $b=a+1=4$. Substitution then yieldsFor the pointwise initial trace at any fixed $x>0$, the far-profile requirement is $f(\eta )=o({\eta }^{3/4})$ as $\eta \to \infty$ on any smooth branch extending that far; a compact-support continuation also meets it. Solving or selecting a shock continuation was not required. The time-dependent source is singular at $t=0$, so the similarity reduction is stated on $t>0$.