In dimensions one to three, the fundamental solution to the free wave equation is positive. Therefore, there exists a simple positivity criterion for solutions. We use this to obtain large global solutions to two well-studied energy-supercritical semilinear wave equations, as well as some new results in the subcritical and critical cases.
Beceanu, Marius and Soffer, Avy, "A Positivity Criterion for the Wave Equation and Global Existence of Large Solutions" (2016). Mathematics and Statistics Faculty Scholarship. 11.