We consider the wave equation, g ψ = 0, in fixed flat Friedmann-Lemaître-Robertson-Walker and Kasner spacetimes with topology R + × T 3 . We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface {t = 0}. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate L 2 -sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the L 2 (T 3 ) norms of the solutions blow up towards the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions n + 1 ≥ 3, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow "pushes away" from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. In the second part of this paper we study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the initial soliton metric. We prove a local well-posedness result for the Ricci flow near the singular initial data, which in particular implies that the "opening up" of the singularity persists for the perturbations also.

We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the formx , where a ij (t, x) and p i (x) are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as t → 0 + . These solutions are expected to be highly non-generic, and our construction can be viewed as solving a singular initial value problem with Fuchsian-type analysis where the data are posed on the "singular hypersurface" {t = 0}. This is the first such result without imposing symmetry or analyticity.To carry out the analysis, we study the problem in a synchronized coordinate system. In particular, we introduce a novel way to perform (weighted) energy estimates in such a coordinate system based on estimating the second fundamental forms of the constant-t hypersurfaces.

For (t, x) ∈ (0, ∞) × T D , the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as t ↓ 0, i.e., a singularity along an entire spacelike hypersurface where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents q1, • • • , qD ∈ R, which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at {t = 1}, as long as the exponents are "sub-critical" in the following sense: maxI,J,B=1,••• ,D I

We study the behaviour of smooth solutions to the wave equation, g ψ = 0, in the interior of a fixed Schwarzschild black hole. In particular, we obtain a full asymptotic expansion for all solutions towards r = 0 and show that it is characterised by its first two leading terms, the principal logarithmic term and a bounded second order term. Moreover, we characterise an open set of initial data for which the corresponding solutions blow up logarithmically on the entirety of the singular hypersurface {r = 0}. Our method is based on deriving weighted energy estimates in physical space and requires no symmetries of solutions. However, a key ingredient in our argument uses a precise analysis of the spherically symmetric part of the solution and a monotonicity property of spherically symmetric solutions in the interior.