Blow-up of solutions to the wave equations with memory terms in Schwarzschild spacetime

Abstract

The main goal of this work is to discuss formation of singularities of solutions to the Cauchy problems for single wave equation and coupled system of wave equations with nonlinear memory terms in Schwarzschild spacetime. For the initial value problem of nonlinear wave equations, blow-up results of sign changing solutions are derived by employing the suitable test function technique which is related to the Riemann-Liouville fractional derivative and contradiction argument. Additionally, upper bound lifespan estimates of solutions to the small initial value problem of wave equations are established by imposing certain conditions on the initial values. It is worth to mention that crucial difficulty is the setting of test function framework, which has not been applied to the Schwarzschild spacetime in the previous literatures. To the best of our knowledge, the results in Theorems 1.1-1.4 are new.