  For a wide variety of initial and boundary conditions, adiabatic one
dimensional flows of an ideal gas approach self-similar behavior when the
characteristic length scale over which the flow takes place, $R$, diverges or
tends to zero. It is commonly assumed that self-similarity is approached since
in the $R\to\infty(0)$ limit the flow becomes independent of any characteristic
length or time scales. In this case the flow fields $f(r,t)$ must be of the
form $f(r,t)=t^{\alpha_f}F(r/R)$ with $R\propto(\pm t)^\alpha$. We show that
requiring the asymptotic flow to be independent only of characteristic length
scales imply a more general form of self-similar solutions,
$f(r,t)=R^{\delta_f}F(r/R)$ with $\dot{R}\propto R^\delta$, which includes the
exponential ($\delta=1$) solutions, $R\propto e^{t/\tau}$. We demonstrate that
the latter, less restrictive, requirement is the physically relevant one by
showing that the asymptotic behavior of accelerating blast-waves, driven by the
release of energy at the center of a cold gas sphere of initial density
$\rho\propto r^{-\omega}$, changes its character at large $\omega$: The flow is
described by $0\le\delta<1$, $R\propto t^{1/(1-\delta)}$, solutions for
$\omega<\omega_c$, by $\delta>1$ solutions with $R\propto (-t)^{1/(\delta-1)}$
diverging at finite time ($t=0$) for $\omega>\omega_c$, and by exponential
solutions for $\omega=\omega_c$ ($\omega_c$ depends on the adiabatic index of
the gas, $\omega_c\sim8$ for $4/3<\gamma<5/3$). The properties of the new
solutions obtained here for $\omega\ge\omega_c$ are analyzed, and self-similar
solutions describing the $t>0$ behavior for $\omega>\omega_c$ are also derived.
