Global existence and blowup of smooth solutions to the semilinear wave equations in FLRW spacetime

Abstract

We are interested in the semilinear wave equations evolving in the expanding spacetimes with Friedmann–Lemaître–Robertson–Walker (FLRW) metric. By the weighted energy estimate, we show that when the nonlinearity depends on the time derivative of the unknown, the equation admits a global smooth solution if the spacetime is undergoing accelerated expansion. While the solution will blowup in the sense of some averaged quantity if the expanding rate is not fast enough. When the nonlinearity depends on the space derivatives of the unknown or the unknown itself, we can show that the solution will blowup in finite time even though the expanding rate is fast enough (accelerated expansion). Our results show that the semilinear wave equations in FLRW spacetimes have different properties from the famous Glassey and Strauss conjectures in flat or asymptotically flat spacetimes.