Blow-Up of Nonlinear Wave Equations with Memory Terms in High-Dimensional Schwarzschild Spacetime

This paper is concerned with the blow-up of solutions to the Cauchy problems of nonlinear wave equations with memory terms in high-dimensional Schwarzschild spacetime. The blow-up results and upper-bound estimates of the lifespans of the solutions to initial-value problems of nonlinear wave equations are obtained by using the appropriate test function method related to the Riemann–Liouville fractional derivative. For the wave equation with combined nonlinear memory terms, we improve the upper-bound estimate of the lifespan of the solution in the existing literature. The lifespan estimates of the solutions are obtained without assuming that the support of the initial values should be far away from a black hole. It is worth mentioning that the crucial difficulty is the setting of the test function framework for the different types of nonlinear memory terms. To the best of our knowledge, the results in Theorems – are new.