Let $T>0$, let $n\in\mathbb N$, set $Q=(0,T)\times\mathbb R^n$ and $\overline Q=[0,T]\times\mathbb R^n$, and let $\lambda>0$. Let $H:[0,T]\times\mathbb R^n\times\mathbb R^n\to\mathbb R$ be continuous. Assume that for every $R>0$ there exists $L_R>0$ such that, for all $t\in[0,T]$, all $x,y\in\overline{B}(0,R)$, and all $p,q\in\mathbb R^n$,

\begin{align*} |H(t,x,p)-H(t,y,q)|\le L_R\bigl((1+|p|+|q|)|x-y|+|p-q|\bigr). \end{align*}

Let $g\in C_b(\mathbb R^n)$ be uniformly continuous, and let $\mathcal C$ be a class of functions $u:\overline Q\to\mathbb R$ such that each $u\in\mathcal C$ is bounded, continuous, uniformly continuous on $\overline Q$, is a viscosity solution on $Q$ of

\begin{align*} \lambda u(t,x)-\partial_t u(t,x)+H(t,x,\nabla u(t,x))=0, \end{align*}

and satisfies $u(T,x)=g(x)$ for every $x\in\mathbb R^n$. Assume moreover that $\mathcal C$ satisfies the following pairwise spatial-infinity compatibility condition: for every $u,v\in\mathcal C$,

\begin{align*} \limsup_{R\to\infty}\ \sup\{u(t,x)-v(t,x):t\in[0,T],\ |x|\ge R\}\le 0 \end{align*}

and

\begin{align*} \limsup_{R\to\infty}\ \sup\{v(t,x)-u(t,x):t\in[0,T],\ |x|\ge R\}\le 0. \end{align*}

Then $\mathcal C$ contains at most one function. Equivalently, if $u,v\in\mathcal C$, then $u(t,x)=v(t,x)$ for every $(t,x)\in\overline Q$.