Take the 2-minute tour ×  Given a hyperbolic PDE, the domain of
influence of a spacetime point $x$, say $I_x$ though $x$ could
be replaced by any set, can be defined in two ways. Lets call
one of them geometric ($I_x^G$) and the other analytical
($I_x^A$). In Lorentzian geometry, the geometric domain of
influence consists of the interior of the cone of null geodesics
emanating from $x$ (let me not bother about whether to include
the boundary in the definition of $I_x^G$ or not). In general, a
similar definition can be given using characteristic cones
instead of null cones. The analytical domain of influence can be
defined as the set of all spacetime points $y$ such that for
every neighborhood neighborhood $O$ of $x$ there exist two
solutions $u_1$ and $u_2$ satisfying the condition
$u_1(x')=u_2(x')$, for all $x'$ on a Cauchy surface passing
through $x$ except for $x'\in O$, and also the condition
$u_1(y)\ne u_2(y)$. The latter one is the definition used in
Lax's book on Hyperbolic PDEs.  Similar defintions ca