We introduce a novel framework for the analysis of linear wave equations on nonstationary asymptotically flat spacetimes, under the assumptions of mode stability and absence of zero energy resonances for a stationary model operator. Our methods apply in all spacetime dimensions and to tensorial equations, and they do not require any symmetry or almost-symmetry assumptions on the spacetime metrics or on the wave type operators. Moreover, we allow for the presence of terms which are asymptotically scaling critical at infinity, such as inverse square potentials. For simplicity of presentation, we do not allow for normally hyperbolic trapping or horizons.
In the first part of the paper, we study stationary wave type equations, i.e. equations with time-translation symmetry, and prove pointwise upper bounds for their solutions. We establish a relationship between pointwise decay rates and weights related to the mapping properties of the zero energy operator. Under a nondegeneracy assumption, we prove that this relationship is sharp by extracting leading order asymptotic profiles at late times. The main tool is the analysis of the resolvent at low energies.
In the second part, we consider a class of wave operators without time-translation symmetry which settle down to stationary operators at a rate $t_*^{-δ}$ as an appropriate hyperboloidal time function $t_*$ tends to infinity. The main result is a sharp solvability theory for forward problems on a scale of polynomially weighted spacetime $L^2$-Sobolev spaces. The proof combines a regularity theory for the nonstationary operator with the invertibility of the stationary model established in the first part. The regularity theory is fully microlocal and utilizes edge-b-analysis near null infinity, as developed in joint work with Vasy, and 3b-analysis in the forward cone.
