Damping threshold for a simple wave (${\alpha }_c=b\max (-u_0^{\prime })$)

For $u_t+(c_0+bu)u_x=-\alpha u$, $b,\alpha >0$, the characteristic curves are $u=e^{-\alpha t}{u}_0(\xi )$ and $x=c_{0}t+\xi +(b/\alpha )(1-e^{-\alpha t})u_0(\xi )$. If $m=\max [-u_0^{\prime }]>0$, finite-time gradient blow-up occurs precisely for $\alpha <bm$, at $t_s=-{\alpha }^{-1}\log (1-\alpha /(bm))$. Equality $\alpha =bm$ still gives a positive Jacobian at each finite time; only the limiting map degenerates.