The Lifespan of Smooth Solutions to Semilinear Wave Equations in Schwarzschild Space-Time

Abstract

This paper considers the Cauchy problem of the semilinear wave equations with small initial data in the Schwarzschild space-time, $□_gu = |u_t | ^p ,$ where $g$ denotes the Schwarzschild metric. When $1< p<2$ and the initial data are supported far away from the black hole, we can prove that the lifespan of the spherically symmetric solution obtains the same order as the semilinear wave equation evolving in the Minkowski space-time by introducing an auxiliary function.