Spatial infinity

A boundary condition at spatial infinity specifies limiting behaviour as the spatial coordinate leaves every bounded set, such as ${\lim }_{x\to +\infty }u(x,t)=g(t)$ on a half-line. A pointwise limit for each time is different from uniform control over a time interval. Maximum-principle and uniqueness proofs on unbounded domains must use a suitable uniform boundedness, growth restriction or barrier hypothesis to justify passage from finite domains to infinity.