Decay Estimate of Maxwell-Higgs System on Schwarzschild Black Holes

In this paper, we prove the decay estimate of Maxwell-Higgs system on four dimensional Schwarzschild spacetimes. We show that if the field equations support a Morawetz type estimate supported around the trapped surface, the uniform decay properties in the entire exterior of the Schwarzschild black holes can be obtained by using Sobolev inequalities and energy estimates. Our results also consider various forms of physical potential such as the mass terms, $\phi^4$-theory, sine Gordon potential, and Toda potential.