Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times (Q2187877)

This research investigates the Cauchy problem to a system of semi linear wave equations on manifolds: \[ \Box _gu_i=F_{p_i}(u_j)\text{ for }(t,x)\in M, \] \[ (u_i,\partial_tu_t)|_{t=0}=(f_i,g_i),\quad i=1,2\text{ and }j=3-i. \] \((M,g)\) denotes a \((1+3)\)-dimensional asymptotically flat space-time manifold, initial data are assumed small , \(F_{p_i}\) are of power nonlinearity type and \(p_i\geq 0\), \(i=1,2\). Under a set of quite restrictive assumptions (space-time assumption, local energy assumption, stationary and split metric assumption), a result on the existence of a solution is proved. Different aspects related to the lower bound of the lifespan, compactly supported initial data, size of initial data, as well as a uniqueness result are shown. The proofs require some weighted Strichartz estimates.