We consider the initial-boundary value problem for the standard quasilinear wave equation:
where is an exterior domain in is a function like and is a nonnegative function. Under two types of hypotheses on we prove existence theorems of global small amplitude solutions. We note that is required to be effective only in localized area and no geometrical condition is imposed on the boundary .
Mitsuhiro NAKAO. "On global smooth solutions to the initial-boundary value problem for quasilinear wave equations in exterior domains." J. Math. Soc. Japan 55 (3) 765 - 795, July, 2003. https://doi.org/10.2969/jmsj/1191419002