= Square-root decay before characteristic crossing
{title2=$\|u(t)\|_\infty\le Kt^{-1/2}$}
Under the <inverse-flux quadratic bounds>, smooth compactly supported data in a <scalar conservation law> satisfy
$$
\|u(t,\cdot)\|_\infty\le-2k_-\sqrt{\frac{2\|u_0\|_1}{-k_+}}\,t^{-1/2}\qquad(0<t<t_*).
$$
At the maximizing foot $x_0$, the <maximum representation for a concave conservation law> bounds $U$ below by $-\|u_0\|_1$ and above by $\|u_0\|_1+k_+(x-x_0-ct)^2/t$. This bounds the deviation of the <characteristic speed> from $c$; the linear inverse-flux bound then controls $u$. The time interval ends at the first <characteristic crossing>.
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