Damping threshold for a simple wave
= Damping threshold for a simple wave
{title2=$\alpha_c=b\max(-u_0')$}
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 $x=c_0t+\xi+(b/\alpha)(1-e^{-\alpha t})u_0(\xi)$. If $m=\max[-u_0']>0$, 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.