Price's Law states that linear perturbations of a Schwarzschild black hole fall off as t −2ℓ−3 for t → ∞ provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be t −2ℓ−4 . We give a proof of t −2ℓ−2 decay for general data in the form of weighted L 1 to L ∞ bounds for solutions of the Regge-Wheeler equation. For initially static perturbations we obtain t −2ℓ−3 . The proof is based on an integral representation of the solution which follows from self-adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.

We prove sharp pointwise t −3 decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates t −4 , and t −6 , respectively. We proceed by decomposition into angular momentum ℓ and summation of the decay estimates on the Regge-Wheeler equation for fixed ℓ. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in ℓ is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy.1 1 The notation a± stands for a± ε where ε > 0 is arbitrary (the choice determines the constants involved). Also, instead of (/ ∇ 10 , / ∇ 9 ) in (1.3) one needs less, namely (/ ∇ σ+1 , / ∇ σ ) where σ > 8 is arbitrary, see the proof in Section 5 for details.2 From the point of view of the decay estimates in [17], these values need to be excluded as they are precisely the ones that give rise to a zero energy resonance.

Abstract. We consider the critical focusing wave equation (−∂ 2 t + ∆)u + u 5 = 0 in R 1+3 and prove the existence of energy class solutions which are of the form2 is the ground state soliton, µ is an arbitrary prescribed real number (positive or negative) with |µ| ≪ 1, and the error η satisfiesfor all t ≫ 1. Furthermore, the kinetic energy of u outside the cone is small. Consequently, depending on the sign of µ, we obtain two new types of solutions which either concentrate as t → ∞ (with a continuum of rates) or stay bounded but do not scatter. In particular, these solutions contradict a strong version of the soliton resolution conjecture.

We consider the semilinear wave equationThis equation admits an explicit spatially homogeneous blow up solution ψ T given bywhere T > 0 and κp is a p-dependent constant. We prove that the blow up described by ψ T is stable against small perturbations in the energy topology. This complements previous results by Merle and Zaag.