be continuous. Assume that $H$ is locally Lipschitz in the gradient variable: for every compact set $K \subset [0,T] \times \mathbb{R}^n$, there exists a constant $L_K > 0$ such that
for all $(t,x) \in K$ and all $p,q \in \mathbb{R}^n$.
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Assume also the Crandall-Lions doubled-variable continuity condition: for every compact set $K \subset [0,T] \times \mathbb{R}^n$, there exists a modulus of continuity $\omega_K: [0,\infty) \to [0,\infty)$, with $\omega_K(0)=0$ and $\omega_K(r) \to 0$ as $r \downarrow 0$, such that, whenever $t,s \in [0,T]$, $x,y \in \mathbb{R}^n$, $(t,x),(s,y) \in K$, and $\varepsilon > 0$,
be bounded and lower semicontinuous. Suppose that $u$ is a viscosity subsolution and $v$ is a viscosity supersolution of this equation on $[0,T) \times \mathbb{R}^n$, where test functions at $t=0$ are understood relative to the topology of $[0,T) \times \mathbb{R}^n$.
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Assume the terminal ordering
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\begin{align*}
u(T,x) \leq v(T,x)
\end{align*}
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for every $x \in \mathbb{R}^n$, and assume the spatial-infinity ordering