= Solution
For isentropic gas with $P=K\rho^\gamma$, $\gamma>1$, the <sound speed> obeys $c^2=P'(\rho)$. The one-dimensional mass and momentum equations are
$$
\rho_t+u\rho_x+\rho u_x=0,\qquad u_t+uu_x+\frac{c^2}{\rho}\rho_x=0.
$$
Since $c\propto\rho^{(\gamma-1)/2}$, these imply
$$
c_t+uc_x+\frac{\gamma-1}2cu_x=0,\qquad
u_t+uu_x+\frac2{\gamma-1}cc_x=0.
$$
Adding and subtracting yields the <Riemann invariants>
$$
\boxed{R_\pm=u\pm\frac{2(c-c_0)}{\gamma-1},\qquad
(\partial_t+(u\pm c)\partial_x)R_\pm=0.}
$$
A <simple wave> has one invariant constant throughout. For a right-moving wave into a resting state with <sound speed> $c_0$, $R_-=0$, so $c=c_0+(\gamma-1)u/2$ and the right characteristic speed is $c_0+bu$, $\boxed{b=(\gamma+1)/2}$.
For the damped scalar equation, characteristics from $\xi$ satisfy
$$
\boxed{u=e^{-\alpha t}u_0(\xi),\qquad
x=c_0t+\xi+\frac b\alpha(1-e^{-\alpha t})u_0(\xi).}
$$
Assume smooth bounded initial data with bounded <derivative>, and the stated maximal compressive slope exists. Then
$$
J=\frac{\partial x}{\partial\xi}=1+\frac b\alpha(1-e^{-\alpha t})u_0'(\xi),\qquad
u_x=\frac{e^{-\alpha t}u_0'(\xi)}J.
$$
For the <damping threshold for a simple wave>, if $m=\max_\xi[-u_0'(\xi)]>0$, put $\boxed{\alpha_c=bm}$. For $0<\alpha<\alpha_c$, a <gradient blow-up> first occurs at
$$
\boxed{t_s=-\frac1\alpha\log\left(1-\frac\alpha{\alpha_c}\right).}
$$
For $\alpha>\alpha_c$, $J$ stays uniformly positive. At $\alpha=\alpha_c$ it still satisfies $J\geq e^{-\alpha t}>0$ for every finite time. At a point attaining the minimum slope, $u_x=u_0'$ for all time, rather than blowing up. Hence the precise conclusion is \b[no finite-time shock occurs when $\alpha\geq\alpha_c$], not the printed strict “if and only if $\alpha>\alpha_c$”. Equality only makes the limiting characteristic map degenerate as $t\to\infty$. The profile $u_0(\xi)=-\sin\xi$ directly verifies this endpoint distinction. If there is no compressive slope, take $\alpha_c=0$; no finite-time shock occurs for positive damping. A nonattained maximal slope is interpreted with a supremum and the corresponding infimum of breakdown times. The gas interpretation also requires $c_0+(\gamma-1)u/2>0$.
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