# Jamming exponent identity — H4

Let \(d,r:[0,\infty)\to\mathbb R\) be continuous, with \(d>0\) and \(r\ge0\). Let \(v:[0,\infty)\times\mathbb R\to\mathbb R\) be continuous and locally Lipschitz in space, and satisfy \(|v(\tau,h)|\le C_T(1+|h|)\) for \(0\le\tau\le T\), for each finite \(T\).
Suppose \(F\in C([0,\infty)\times\mathbb R)\cap C^{1,2}((0,\infty)\times\mathbb R)\) is bounded on each strip \([0,T]\times\mathbb R\) and satisfies
\[
F_\tau=d(\tau)F_{hh}+v(\tau,h)F_h+r(\tau)F(1-F),\qquad
0\le F(0,h)\le1\quad(h\in\mathbb R).
\]

Then \(0\le F(\tau,h)\le1\) for every \(\tau\ge0\) and \(h\in\mathbb R\).