The Wave Equation Near Flat Friedmann–Lemaître–Robertson–Walker and Kasner Big Bang Singularities Artur Alho, Grigorios Fournodavlos, Anne Franzen

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The wave equation near flat Friedmann–Lemaître–Robertson–Walker and Kasner Big Bang singularities Artur Alho, Grigorios Fournodavlos, Anne Franzen

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Artur Alho, Grigorios Fournodavlos, Anne Franzen. The wave equation near flat Friedmann–Lemaître– Robertson–Walker and Kasner Big Bang singularities. Journal of Hyperbolic Differential Equations, World Scientific Publishing, 2019, 16 (02), pp.379-400. 10.1142/S0219891619500140. hal-02967520

HAL Id: hal-02967520 https://hal.sorbonne-universite.fr/hal-02967520 Submitted on 15 Oct 2020

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THE WAVE EQUATION NEAR FLAT FRIEDMANN-LEMAˆITRE-ROBERTSON-WALKER AND KASNER BIG BANG SINGULARITIES

Artur Alho1,?, Grigorios Fournodavlos2,†, and Anne T. Franzen3,?

?Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

† Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom

Abstract. We consider the wave equation, gψ = 0, in fixed flat Friedmann-Lemaˆıtre-Robertson-Walker 3 and Kasner spacetimes with topology R+ × T . 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 L2-sense, up to two spatial derivatives 2 3 of the Dirichlet data. For these initial configurations, the L (T ) 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.

1. Introduction and main theorems In this note, we analyse the behaviour of solutions to the wave equation on cosmological backgrounds towards the initial singularity. Our spacetimes of interest are the spatially homogenous, isotropic ﬂat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW hereafter) backgrounds and the anisotropic vacuum Kas- ner spacetimes. The former plays an important role in physics, since observational evidence suggests that at suﬃciently large scales the universe seems to be spatially homogeneous and isotropic. The Kasner solutions also play an important role in the theory of general relativity, since they form the past attractor of Bianchi type I spacetimes, the Kasner circle, which in turn are the basic building blocks in the “BKL conjecture” [5] concerning spacelike cosmological singularities, see [7] for recent developments on this subject in the setting of spatially homogeneous solutions to the Einstein-vacuum equations. 3 The spacetimes have the topology R+ × T and are endowed with the metrics:

2 4 2 2 2 2 (1.1) g = − dt + t 3γ (dx + dx + dx ), < γ < 2, FLRW 1 2 3 3 3 3 3 2 X 2pj 2 X X 2 (1.2) gKasner = − dt + t dxj , pj = 1, pj = 1, pj < 1, j=1 j=1 j=1 respectively. Both metrics gFLRW, gKasner have a Big Bang singularity at t = 0, where the curvature blows up |Riem| ∼ t−2, as t → 0. Metrics of the form (1.1) are solutions of the Einstein-Euler system for ideal fluids with linear equation of state p = (γ − 1)ρ, where p is the pressure, and ρ the energy density. The case γ = 2 corresponds to stiff fluids, i.e., p = ρ, where incompressibility is expressed by the velocity of sound cs equating the velocity of light c = 1. For the stiff case the dynamics of the Einstein equations towards the 2 singularity are completely understood by the work of [31, 32, 33]. The other endpoint γ = 3 corresponds

1e-mail address: [email protected] 2e-mail address: [email protected] 3e-mail address: [email protected] 1 2 to the coasting universe which does not have a spacelike singularity. On the other hand, the Kasner metric (1.2) is a solution to the Einstein vacuum equations. When one of the pj = 1 equals one, and the other two vanish (flat Kasner), it corresponds to the Taub form of Minkowski space and there is no singularity as the spacetime is flat. Our goal is to understand the behaviour of smooth solutions to the wave equation towards these singularities from the initial value problem point of view and by deriving appropriate energy estimates in physical space, which may also prove useful for dynamical studies. According to the references in the literature, such as [1, 26, 28, 30], these waves are shown to blow up in certain cases. We wish to characterize open sets of initial data at a given time t0 > 0 for which such blow up behaviour occurs at t = 0. Denote the constant t hypersurfaces by Σt. First, we give the general asymptotic profile of all solutions:

Theorem 1.1. Let ψ be a smooth solution to the wave equation, gψ = 0, for either of the metrics gFLRW, gKasner, arising from initial data (ψ0, ∂tψ0) on Σt0 . Then, ψ can be written in the following form: 1− 2 (1.3) ψFLRW(t, x) = AFLRW(x)t γ + uFLRW(t, x),

(1.4) ψKasner(t, x) = AKasner(x) log t + uKasner(t, x),

2 −1 −1 where A(x), u(t, x) are smooth functions and uF LRW t γ , uKasner(log t) tend to zero, as t → 0. We prove the preceding theorem by deriving appropriate stability estimates for renormalized variables, which as a corollary imply the continuous dependence of A(x) on initial data. For instance, solutions coming 1− 2 from initial conﬁgurations close to those of the homogeneous solutions t γ , log t in FLRW and Kasner respectively, will blow up with leading order coeﬃcients A(x) ∼ 1. Hence, the set of all blowing up solutions to the wave equation is open and dense.4 Our next theorem yields a characterization of open sets of initial data for which the corresponding solutions to the wave equation blow up in L2(T3) at the Big Bang hypersurface t = 0.

Theorem 1.2. Let ψ be a smooth solution to the wave equation, gψ = 0, for either of the metrics 2 3 gFLRW, gKasner, arising from initial data (ψ0, ∂tψ0) on Σt0 , t0 > 0. If ∂tψ0 is non-zero in L (T ), t0 is suﬃciently small such that

4 2− 3γ 3 2t0 X 2 2 (1.5) k∂ ∂ ψ k 2 3 < k∂ ψ k 2 3 , (FLRW) 1 − ( 2 )2 t xi 0 L (T ) t 0 L (T ) 3γ i=1

3 2−2pi X 2t0 2 2 (1.6) k∂ ∂ ψ k 2 3 < k∂ ψ k 2 3 , (Kasner) (1 − p )2 t xi 0 L (T ) t 0 L (T ) i=1 i and ψ0, ∂tψ0 satisfy the open conditions 3 4 2 − 3γ X 2 (1.7) (1 − )k∂ ψ k 2 3 > t k∂ ψ k 2 3 t 0 L (T ) 0 xi 0 L (T ) i=3 8 2− 3γ 3 2t0 X (FLRW) + k∂ ∂ ψ k 2 3 , 1 − ( 2 )2 xj xi 0 L (T ) 3γ i,j=1 3 2 X −2pi 2 (1.8) (1 − )k∂ ψ k 2 3 > t k∂ ψ k 2 3 t 0 L (T ) 0 xi 0 L (T ) i=3

3 2−2pi−2pj X 2t0 (Kasner) + k∂ ∂ ψ k 2 3 , (1 − p )2 xj xi 0 L (T ) i,j=1 i

2 3 for some 0 < < 1, then kA(x)kL (T ) > 0.

4 In the sense that if a solution blows up in a compact set at Σ0, i.e., A(x) 6= 0 in that compact set, then this property persists under suﬃciently small perturbations. On the contrary, if A(x) = 0 in an open subset of Σ0, one can always add a small 1− 2 multiple of t γ , log t to produce a new solution with A(x) 6= 0 in that open set. 3

Remark 1.3. Given a blowing up solution to the wave equation in either FLRW or Kasner, having non- vanishing leading order coefficient A(x), it is easy to see, using the expansions (1.3), (1.4), that the solution satisfies the conditions in Theorem 1.2 for t0 > 0 sufficiently small.

We also prove a local version of Theorem 1.2, giving open initial conditions in a neighbourhood of Σt0 ,

Ut0 , whose domain of dependence intersects the singular hypersurface Σ0 at a neighbourhood U0, where the 2 L (U0) norm of the corresponding solutions blows up.

Σ0 U0

Σ Ut t + −∂t − Nl Nl U Σt0 t0

Figure 1. Domain of dependence of an open neighborhood Ut0 of the initial hypersurface Σt0 .

Theorem 1.4. Let Ut0 be an open neighborhood in Σt0 , t0 > 0, whose domain of dependence intersects Σ0 3 in U0 = (0, δ) , and let ψ be a smooth solution to the wave equation, gψ = 0, for either of the metrics gFLRW, gKasner, arising from initial data (ψ0, ∂tψ0) on Ut0 . If ∂tψ0 is non-zero in Ut0 , t0 is suﬃciently small such that

4 2− 3γ 3 2t0 X 2 k∂t∂xi ψ0kL2(U ) 1 − ( 2 )2 t0 3γ i=1 1− 2 3γ 3 4t0 2 X 2 (1.9) + 3k∂tψ0kL2(U ) + k∂t∂xl ψ0kL2(U ) 1 − 2 t0 t0 3γ l=1 2 1− 3γ 2 t0 2 (FLRW) +6 log 1 + k∂tψ0k 2 < k∂tψ0kL2(U ), 2 L (Ut0 ) t0 1 − 3γ δ

3 2−2pi X 2t0 2 k∂t∂xi ψ0kL2(U ) (1 − p )2 t0 i=1 i

3 1−pl X 4t0 2 2 (1.10) + k∂ ψ k 2 + k∂ ∂ ψ k 2 t 0 L (Ut ) t xl 0 L (Ut ) 1 − pl 0 0 l=1

3 1−pl X 2 t0 2 2 (Kasner) + 2 log 1 + k∂ ψ k 2 < k∂ ψ k 2 , t 0 L (Ut ) t 0 L (Ut ) 1 − pl δ 0 0 l=1 and (ψ0, ∂tψ0) satisfy the open conditions: 2 (1 − )k∂tψ0k 2 L (Ut0 ) 3 2− 8 3 4 3γ − 3γ X 2 2t0 X 2 > t0 k∂xi ψ0kL2(U ) + k∂xj ∂xi ψ0kL2(U ) t0 1 − ( 2 )2 t0 i=1 3γ i,j=1 1− 2 3 3 γ 4 4 4t0 − 3γ X 2 − 3γ X 2 (1.11) + 3t0 k∂xi ψ0kL2(U ) + t0 k∂xi ∂xl ψ0kL2(U ) 1 − 2 t0 t0 3γ i=1 i,l=1 2 1− γ 3 − 4 2 t0 3γ X 2 + 6 log 1 + t0 k∂xi ψ0kL2(U ), (FLRW) 1 − 2 δ t0 3γ i=1 2 (1 − )k∂tψ0k 2 L (Ut0 ) 4

3 3 2−2pi−2pj X −2pi 2 X 2t0 2 > t0 k∂xi ψ0kL2(U ) + k∂xj ∂xi ψ0kL2(U ) t0 (1 − p )2 t0 i=1 i,j=1 i

3 1−pl X 4t0 −2pi 2 −2pi 2 (1.12) + t k∂ ψ k 2 + t k∂ ∂ ψ k 2 0 xi 0 L (Ut ) 0 xi xl 0 L (Ut ) 1 − pl 0 0 i,l=1 2 3 1− γ 3 X 2 t0 X −2pi 2 + 2 log 1 + t k∂ ψ k 2 , (Kasner) 0 xi 0 L (Ut ) 1 − pl δ 0 l=1 i=1

2 for some 0 < < 1, then kA(x)kL (U0) > 0. The blow up behaviour of linear waves observed near Big Bang singularities is reminiscent of the behaviour of waves in black hole interiors containing spacelike singularities [8, 14]. Examples are the Schwarzschild singularity or black hole singularities occurring in spherically symmetric solutions to the Einstein-scalar ﬁeld model [10], where a logarithmic blow up behaviour has been observed for spatially homogeneous waves. Such logarithmic blow up behaviour was recently conﬁrmed [16] for generic linear waves in the Schwarzschild black hole interior. The aforementioned blow up behaviours, however, are in contrast to the behaviour of waves observed near null boundaries, where linear and dynamical waves have been shown in general to extend continuously past the relevant null hypersurfaces [11], [17]-[23], see also [25, 27]. Lastly, we should note that although we only deal with spatially homogeneous spacetimes, our method of proof is applicable to cosmological spacetimes with Big Bang singularities exhibiting asymptotically velocity term dominated (AVTD) behaviour [2, 4, 6, 9, 13, 15, 21, 24, 29].

2. Proof of main theorems 2.1. Energy argument and notation. In order to derive the energy estimates required for the proof of Theorem 1.2, we will apply the vector field method and define certain energy currents constructed from the stress-energy tensor 1 (2.1) T [ψ] = ∂ ψ∂ ψ − g ∂cψ∂ ψ , ab a b 2 ab c of the scalar field ψ. The divergence of Tab[ψ] reads

a (2.2) ∇ Tab[ψ] = ∂bψ · gψ, where ∇ stands for the spacetime covariant connection. Hence, if ψ satisﬁes the homogeneous wave equation

(2.3) gψ = 0,

a it follows that we have energy-momentum conservation ∇ Tab[ψ] = 0. Contracting the stress-energy tensor with a vector ﬁeld multiplier X, then deﬁnes the associated current

X b (2.4) Ja [ψ] = X Tab[ψ], whose divergence, according to (2.2), equals

a X a b (2.5) ∇ Ja [ψ] = (∇ X )Tab[ψ] + Xψ · gψ

Applying the divergence theorem to (2.5), over the spacetime domain {Us}s∈[t,t0] (Figure 1), we thus obtain:

Z 3 Z X a X X a (2.6) J [ψ]n vol + J [ψ]n ± vol ± a Ut Ut a N ± Nl l Ut l=1 ∪Nl Z Z t0 Z X a a X = J [ψ]n volU − ∇ J [ψ]volU ds, a Ut0 t0 a s Ut0 t Us 5

where nUt = −∂t, volUt is the intrinsic volume form of Ut and − 2  ± 3γ n = −∂t ± t ∂xl , Nl  4  vol ± = t 3γ dtdxidxj, (FLRW) Nl − 2 ±  t 3γ dt = ±dx , on N  (2.7) l l −p  ± l n = −∂t ± t ∂xl , Nl  1−p  vol ± = t l dtdxidxj, (Kasner) Nl −pl ±  t dt = ±dxl, on Nl  for each l = 1, 2, 3; i < j; i, j 6= l.

Below we will choose the vector ﬁeld X to be a suitable rescaling nUt . Note that, na J X [ψ] = J X [ψ] = 1 [(∂ ψ)2 + |∇ψ|2] and J X [ψ]na ≥ 0, where ∇ is the covariant derivative intrinsic Ut a 0 2 t a ± Nl to the level sets of t. Hence, if we can control the second term on the RHS of (2.6) (the bulk), in terms of X J0 [ψ], we obtain an energy estimate for ψ.

Notice that in the case of the whole torus, Ut0 = Σt0 , due to the absence of causal boundary terms, (2.6) becomes Z Z Z t0 Z X a X a a X (2.8) J [ψ]n volΣ = J [ψ]n volΣ − ∇ J [ψ]volΣ ds. a Σt t a Σt0 t0 a s Σt Σt0 t Σs In the analysis that follows, we will often require higher order energy estimates that we can obtain by commuting the wave equation with spatial derivatives and applying the above energy argument. Note, that we are considering homogeneous spacetimes in which the spatial coordinate derivatives {∂xi } are Killing and α hence [g, ∂xi ] = 0, i = 1, 2, 3. This means that the above identities (2.6), (2.8) are also valid for ∂x ψ, where we use the standard multi-index notation ∂α = ∂α1 ∂α2 ∂α3 for an iterated application of spatial derivatives, x x1 x2 x3 k 3 α = (α1, α2, α3), |α| = α1 +α2 +α3. In this notation, the H (Σt) norm of a smooth function f : (0, +∞)×T equals Z 2 X α 2 (2.9) kfk k = (∂ f) vol , H (Σt) x Euc |α|≤k Σt k 2 where volEuc = dx1dx2dx3. We will often omit Σt from the norms to ease notation and use H ,L for the corresponding time-dependent, non-intrinsic norms. 2.2. Flat FLRW. Let ψ be a smooth solution to the scalar wave equation

(2.10) gFLRW ψ = 0. Consider the orthonormal frame 2 − 3γ (2.11) e0 = −∂t, ei = t ∂xi adapted to the constant t hypersurfaces Σt with the past normal vector ﬁeld e0 pointing towards the singularity. In this frame, the second fundamental form Kij of Σt reads 2 1 (2.12) K := g(∇ e , e ) = − , i = 1, 2, 3. ii ei 0 i 3γ t

Further, the intrinsic volume form on Σt equals 2 γ (2.13) volΣt = t volEuc. Proposition 2.1. The following energy inequality holds:

2 Z 2 Z e α γ e α (2.14) t γ J 0 [∂ ψ]vol ≤ t J 0 [∂ ψ]vol , 0 x Σt 0 0 x Σt0 Σt Σt0 for all t ∈ (0, t0] and any multi-index α. Moreover, ψ satisﬁes the pointwise bound

1 2 2 1− 1− γ X 2 Z 2 t γ − t (2.15) |ψ(t, x)| ≤ C t γ J e0 [∂αψ]vol 0 + |ψ(t , x)|, 0 0 x Σt0 2 0 Σ − 1 |α|≤2 t0 γ where C > 0 is a constant independent of t0, γ. 6

2 γ t e0 Proof. We compute the divergence of Ja [ψ]: 2 γ (2.5) 2 2 2 a t e0 a γ b γ ab γ (2.16) ∇ Ja [ψ] = ∇ (t e0) Tab[ψ] = t K Tab[ψ] − e0t T00[ψ]

2 2 1 2 i 2 1 2 −1 2 2 = t γ K |∇ψ| − t γ K |∇ψ| + t γ [(e ψ) + |∇ψ| ] 11 2 i γ 0

(2.12) 4 1 2 2 = t γ |∇ψ| . 3γ t

2 Hence, utilising (2.8) for X = t γ e0 yields

t0 2 Z 2 Z Z Z 2 e0 γ e0 4 −1 2 (2.17) t γ J [ψ]volΣ = t J [ψ]volΣ − s γ |∇ψ| volΣ ds. 0 t 0 0 t0 3γ s Σt Σt0 t Σs 2 Z (2.18) ≤ t γ J e0 [ψ]vol 0 0 Σt0 Σt0 α The same identity is valid for ∂x ψ, leading to (2.14). In particular, taking into account the volume form (2.13), we have the following bounds for ∂tψ:

4 2 Z γ α 2 γ e0 α (2.19) t k∂ ∂ ψk 2 ≤ 2t J [∂ ψ]vol , t x L 0 0 x Σt0 Σt0 2 3 ∞ 3 for all t ∈ (0, t0] and α. Integrating ∂tψ in [t, t0] and employing the Sobolev embedding H (T ) ,→ L (T ) we derive: Z t

(2.20) |ψ(t, x)| = ∂sψ(s, x)ds + ψ(t0, x) t0 Z t0 ≤ C k∂sψkH2 ds + |ψ(t0, x)| t

2 2 2 Z C 1− 1− γ X γ e α 1 ≤ (t γ − t )( t J 0 [∂ ψ]vol ) 2 + |ψ(t , x)|, 2 0 0 0 x Σt0 0 − 1 Σ γ |α|≤2 t0 for γ < 2. 1− 2 Remark 2.2. The bounds (2.14), (2.15) are saturated by the homogeneous function t γ , which is an exact solution of (2.10).

1− 2 From the previous proposition we understand that t γ is the leading order of ψ at t = 0. To prove this ψ rigorously, we derive analogous energy bounds for the renormalised variable 1− 2 that satisﬁes the wave t γ equation: ψ 2 2 ψ (2.21) = − (1 − )e ( ). 1− 2 0 1− 2 t γ t γ t γ 2 Proposition 2.3. Let ψ be a smooth solution to the wave equation in FLRW backgrounds with 3 < γ < 2. Then, the following bounds hold uniformly in t ∈ (0, t0]:

6 Z 6 Z 4− e α ψ 4− γ e α ψ 4 (2.22) t γ J 0 [∂ ]vol ≤ t J 0 [∂ ]vol , ≤ γ < 2, 0 x 2 Σt 0 0 x 2 Σt0 1− γ 1− γ 3 Σt t Σt0 t

2 Z 2 Z − e α ψ − 3γ e α ψ 2 4 (2.23) t 3γ J 0 [∂ ]vol ≤ t J 0 [∂ ]vol , < γ ≤ , 0 x 2 Σt 0 0 x 2 Σt0 1− γ 1− γ 3 3 Σt t Σt0 t for all t ∈ (0, t0] and any multi-index α. Moreover, the limit ψ (2.24) A(x) := lim 1− 2 t→0 t γ 1− 2 exists, it is a smooth function and the diﬀerence u(t, x) := ψ − A(x)t γ satisﬁes

2 Z γ e0 α (2.25) lim t J0 [∂x u]volΣs = 0. t→0 Σt 7

Proof. Let η > 0. We compute

η ψ (2.5) ψ ψ ψ (2.26) ∇a(J t e0 [ ]) = ∇a(tηe )bT [ ] + tηe ( ) · a 1− 2 0 ab 1− 2 0 1− 2 g 1− 2 t γ t γ t γ t γ ψ ψ 2 2 ψ = tηKabT [ ] − (e tη)T [ ] − (1 − )tη[e ( )]2 ab 1− 2 0 00 1− 2 0 1− 2 t γ t γ t γ t γ (2.12) 1 η ψ η 3 ψ = tη−1 ( + )|∇ |2 + ( + − 2)[e ( )]2 1− 2 0 1− 2 3γ 2 t γ 2 γ t γ

This leads to diﬀerent choices of η depending on the value of γ, given by

6 4 (2.27) η = 4 − , for ≤ γ < 2, γ 3 2 2 4 (2.28) η = − , for < γ ≤ . 3γ 3 3

4 We refer to the former as the stiﬀest region and the latter as the softest region. The case γ = 3 corresponds 1 to radiation, where η = − 2 . For the two cases, (2.26) reads

4− 6 a t γ e ψ 8 3− 6 ψ 2 (2.29) ∇ (J 0 [ ]) = (2 − )t γ |∇ | , a 1− 2 1− 2 t γ 3γ t γ − 2 a t 3γ e ψ 8 −1− 2 ψ 2 (2.30) ∇ (J 0 [ ]) = ( − 2)t 3γ [e ( )] . a 1− 2 0 1− 2 t γ 3γ t γ

ψ 4− 6 4 Then, the energy identity (2.8) for , X = t γ e , in the stiﬀest region { ≤ γ < 2}, implies that 1− 2 0 3 t γ

6 Z 6 Z 4− e ψ 4− γ e ψ t γ J 0 [ ]vol ≤ t J 0 [ ]vol 0 2 Σt 0 0 2 Σt0 1− γ 1− γ Σt t Σt0 t t0 8 Z 6 Z ψ 3− γ 2 − (2 − ) s |∇ 2 | volΣs ds 1− γ 3γ t Σs t 4− 6 Z ψ (2.31) ≤ t γ J e0 [ ]vol . 0 0 2 Σt0 1− γ Σt 0 t0

α Commuting with ∂x yields the bound (2.22). Similarly, we obtain statement (2.23) in the softest region 2 4 { 3 < γ ≤ 3 }. In particular, taking into account the volume form (2.13), we have the bounds:

1 4− 6 Z 2 ψ ψ C X γ e0 α ψ (2.32) |∂ | ≤ Ck∂ k 2 ≤ t J [∂ ]vol , t 2 t 2 H 2 0 0 x 2 Σt0 1− γ 1− γ 2− γ 1− γ t t t Σt |α|≤2 0 t0 4 for ≤ γ < 2, 3

1 − 2 Z 2 ψ ψ C X 3γ e0 α ψ (2.33) |∂t | ≤ Ck∂t k 2 ≤ t J [∂ ]volΣ , 1− 2 1− 2 H 2 0 0 x 1− 2 t0 γ γ 3γ Σ γ t t t |α|≤2 t0 t 2 4 for < γ ≤ , 3 3 which imply that ∂ ψ(t, x) ∈ L1([0, t ]), uniformly in x, for all 2 < γ < 2. Thus, ψ has a limit function t 0 3 1− 2 t γ α ψ A(x), as t → 0. The smoothness of A(x) follows by repeating the preceding argument for ∂x 1− 2 . t γ 8

1− 2 Consider now the energy ﬂux of the diﬀerence ψ − A(x)t γ ,

2 Z 2 2 γ 1− γ 2 1− γ 2 (2.34) t |e0(ψ − A(x)t )| + |∇(ψ − A(x)t )| volΣt Σt 2 Z 2 ψ 2 2 ψ γ 1− γ − γ 2 = t |t e0 2 + (1 − )t ( 2 − A(x))| 1− γ 1− γ Σt t γ t 2− 4 ψ 2 + t γ |∇( − A(x))| vol 1− 2 Σt t γ 2 Z 4 ψ 2 4 ψ 4 ψ γ 2− γ 2 2 − γ 2 2− γ 2 ≤ 2t t |e0 2 | + (1 − ) t | 2 − A(x)| + t |∇ 2 | 1− γ 1− γ 1− γ Σt t γ t t 4 2− γ 2 + t |∇A(x)| volΣt Z Z 2 e0 ψ 2 2 ψ 2 = 4t J0 [ 2 ]volEuc + 2(1 − ) | 2 − A(x)| volEuc 1− γ 1− γ Σt t γ Σt t Z 2 2 + 2t |∇A(x)| volEuc Σt (2.22),(2.23) Z 2 2 ψ 2 ≤ o(1) + 2(1 − ) | 2 − A(x)| volEuc 1− γ γ Σt t 3 4 Z 2− 3γ X 2 + 2t |∂xi A(x)| volEuc, i=1 Σt 4 for all 3 ≤ γ < 2. The third term in the preceding RHS clearly tends to zero, as t → 0, and by the deﬁnition 2 α 1− γ of A(x), so does the second term. Since the above argument also applies to ∂x [ψ − A(x)t ], this proves (2.25). Remark 2.4. The renormalised estimate (2.22) yields an improved control over the spatial gradient of ψ compared to (2.14). Indeed,

4 2 Z 6 Z 2− 2 4− γ e ψ 4 t γ t γ |∇ψ| vol ≤ t J 0 [ ]vol , ≤ γ < 2, Σt 0 0 2 Σt0 1− γ Σt Σt t 3 (2.35) 0 4 2 Z 2 Z −2 2 − 3γ e ψ 2 4 t 3γ t γ |∇ψ| vol ≤ t J 0 [ ]vol , < γ ≤ , Σt 0 0 2 Σt0 1− γ 3 3 Σt Σt0 t 4 4 holds for all t ∈ (0, t0], where in the stiﬀest case 2 − γ < 0, while in the softest 3γ − 2 < 0. The previous proposition validates the asymptotic proﬁle (1.3) of ψ, as stated in Theorem 1.1.

2 Lemma 2.5. The following estimate for the L norm of ∂xi ψ holds:

2 2 1 1− γ 1− √ γ 2 Z 2 (t0 − t ) γ e0 (2.36) k∂x ψkL2(Σ ) ≤ k∂x ψkL2(Σ ) + 2 t J [∂x ψ]volEuc , i t i t0 1 − 2 0 0 i γ Σt0 for all t ∈ (0, t0].

Proof. Diﬀerentiating in e0 we have:

1 2 (2.37) e0k∂x ψk 2 ≤ k∂x ψkL2(Σ )ke0ψkL2(Σ ) 2 i L (Σt) i t t 1 2 Z 2 1 γ e0 (using (2.14)) ≤ 2t J [∂ ψ]vol k∂ ψk 2 2 0 0 xi Σt0 xi L (Σt) γ t Σt0 or

2 1 √ γ Z 2 t0 e0 2 (2.38) e0k∂xi ψkL (Σt) ≤ 2 2 J0 [∂xi ψ]volEuc . γ t Σt0

Integrating the above on [t, t0] gives (2.36) for γ < 2. 9

Remark 2.6. The bounds that we have proven so far, stated in Propositions 2.1, 2.3 and Lemma 2.5, are also valid if we replace the integral domains Σt, Σt0 by Ut,Ut0 . This can be easily seen from the fact that in the corresponding energy identity (2.6), the null boundary terms have a favourable sign for an upper bound and therefore can be dropped.

2 Now we may proceed to derive the blow up criterion given in Theorem 1.2. First, notice that for γ > 3 , e0 the main contribution of the energy ﬂux generated by J [ψ] comes from the e0ψ term. Indeed, by (2.35) it follows that

2 Z 1 Z 4 γ e0 γ 2 η (2.39) t J0 [ψ]volΣt = t (∂tψ) volEuc + O(t ) Σt 2 Σt

4 4 4 2 4 where η = γ − 2 > 0, for γ ∈ [ 3 , 2) and η = 2 − 3γ > 0, for γ ∈ ( 3 , 3 ]. Hence, taking the limit t → 0 in the preceding identity and utilizing (2.25) leads to

2 Z 1 2 Z γ e0 2 2 (2.40) lim t J0 [ψ]volΣt = (1 − ) A (x)volEuc. t→0 Σt 2 γ Σ0

Combining (2.17), (2.40) we derive:

Z 1 2 2 2 (2.41) (1 − ) A (x)volEuc 2 γ Σ0 t0 2 Z Z 2 Z γ e0 4 −1 2 = t J [ψ]volΣ − s γ |∇ψ| volΣ ds 0 0 t0 3γ s Σt0 0 Σs 3 4 4 4 1 γ 2 1 γ − 3γ X 2 (by (2.36)) ≥ t0 k∂tψkL2(Σ ) + t0 k∂xi ψkL2(Σ ) 2 t0 2 t0 i=1

t0 3 8 Z 4 4 γ −1− 3γ X 2 − s ds k∂xi ψkL2(Σ ) 3γ t0 0 i=1 1− 2 2 t0 γ 1− 2 3 Z 4 4 γ 2 Z 16 −1− (t0 − s ) X γ e0 − s γ 3γ ds t J [∂x ψ]volΣ 3γ (1 − 2 )2 0 0 i t0 0 γ i=1 Σt0 3 4 8 1 γ 2 1 3γ X 2 = t0 k∂tψkL2(Σ ) − t0 k∂xi ψkL2(Σ ) 2 t0 2 t0 i=3 2− 4 3 3 3γ 4 8 t0 X γ 2 3γ X 2 − t0 k∂t∂xi ψkL2(Σ ) + t0 k∂xj ∂xi ψkL2(Σ ) . 1 − ( 2 )2 t0 t0 3γ i=1 j=1

2 3 Now is evident now that if the assumptions of Theorem 1.2 for FLRW are satisﬁed, then kA(x)kL (T ) > 0. To prove Theorem 1.4 for FLRW, we use the local energy identity (2.6) and plug in (2.7), (2.13), (2.16):

2 Z γ e0 (2.42) t J0 [ψ]volUt Ut 3 4 4 4 1 γ 2 1 γ − 3γ X 2 = t0 k∂tψkL2(U ) + t0 k∂xi ψkL2(U ) 2 t0 2 t0 i=1

t0 3 4 Z 4 4 γ −1− 3γ X 2 − s k∂x ψk 2 ds 3γ i L (Us) t i=1 3 Z t0 Z X 2 + 4 1 2 2 2 (i < j; i, j 6= l) − s γ 3γ |(e0 ± el)ψ| + |eiψ| + |ejψ| dsdxidxj. ± 2 l=1 t Nl ∩U s 10

2 − 3γ ± Since t dt = ±dxl along Nl , it follows by integrating that the closure U t of the neighbourhood Ut is the 1− 2 1− 2 3 t 3γ t 3γ 5 cube I , where I = [− 2 , δ + 2 ]. We make use of the following 1-dimensional Sobolev inequality 1− 3γ 1− 3γ

2 2 Z 2 1− 3γ −1 2 2 (2.43) f (t, xl) ≤ (δ + 2 t ) f (t, xl)dxl + kf(t, xl)kH1(I), 1 − 3γ I

1 for f ∈ H (I), t ∈ (0, t0]. First, we take the limit t → 0 in (2.42), employing (2.40), and then we apply (2.36) to the integrand in ± the third line of (2.42) and (2.43) to the integral over Nl ∩ U s in the last line of (2.42) to deduce the lower bound: Z 1 2 2 2 (2.44) (1 − ) A (x)volU0 2 γ U0 3 4 4 4 1 γ 2 1 γ − 3γ X 2 ≥ t0 k∂tψkL2(U ) + t0 k∂xi ψkL2(U ) 2 t0 2 t0 i=1

t0 3 8 Z 4 4 γ −1− 3γ X 2 − s ds k∂xi ψkL2(U ) 3γ t0 0 i=1 1− 2 2 t0 γ 1− 2 3 Z 4 4 γ 2 Z 16 −1− (t0 − s ) X γ e0 − s γ 3γ ds t J [∂x ψ]volU 3γ (1 − 2 )2 0 0 i t0 0 γ i=1 Ut0 t0 Z 1 2 Z 4 − 3γ γ e0 − (1 + 2 )s s 12J0 [ψ]volEucds 2 1− 3γ 0 δ + 2 s Us 1− 3γ

t0 3 Z 2 Z 4 − 3γ γ X e0 − s 4s J0 [∂xl ψ]volEuc 0 Us l=1 3 4 8 1 γ 2 1 3γ X 2 ≥ t0 k∂tψkL2(U ) − t0 k∂xi ψkL2(U ) 2 t0 2 t0 i=1 2− 4 3 3 3γ 4 8 t0 X γ 2 3γ X 2 − t0 k∂t∂xi ψkL2(Σ ) + t0 k∂xj ∂xi ψkL2(Σ ) 1 − ( 2 )2 t0 t0 3γ i=1 j=1

t0 Z 1 2 Z 4 − 3γ γ e0 (by (2.14) for {Ut}) − (1 + 2 )s ds 12t0 J0 [ψ]volEuc 2 1− 3γ 0 δ + 2 s Ut0 1− 3γ

t0 3 Z 2 Z 4 − 3γ γ X e0 − s ds 4t0 J0 [∂xl ψ]volEuc 0 U t0 l=1 3 4 8 1 γ 2 1 3γ X 2 = t0 k∂tψkL2(U ) − t0 k∂xi ψkL2(U ) 2 t0 2 t0 i=1 2− 4 3 3 3γ 4 8 t0 X γ 2 3γ X 2 − t0 k∂t∂xi ψkL2(Σ ) + t0 k∂xj ∂xi ψkL2(Σ ) 1 − ( 2 )2 t0 t0 3γ i=1 j=1 1− 2 3 3γ 4 8 t0 γ 2 3γ X 2 − 6t0 k∂tψkL2(U ) + 6t0 k∂xi ψkL2(U ) 1 − 2 t0 t0 3γ i=1 3 3 4 8 γ X 2 3γ X 2 + 2t k∂t∂xl ψk 2 + 2t k∂xi ∂xl ψk 2 0 L (Ut0 ) 0 L (Ut0 ) l=1 i,l=1

5 2 2 R −1 2 Proof by fundamental theorem of calculus: f (t, x ) ≤ min f (t, x ) + 2|f||∂ f|dx ≤ |I| kfk + kfk 1 . l xl∈I l I xl l L2(I) H (I) 11

1− 2 3 3γ 4 8 2 t0 γ 2 3γ X 2 − 3 log 1 + t0 k∂tψkL2(U ) + t0 k∂xi ψkL2(U ) . 1 − 2 δ t0 t0 3γ i=1

2 Thus, if the assumptions of Theorem 1.4 for FLRW are satisﬁed, then the RHS of (2.44) gives kA(x)kL (U0) > 0. This completes the proofs of the main theorems for FLRW.

2.3. Kasner. For Kasner the adapted orthonormal frame to the constant t hypersurfaces reads:

−pi (2.45) e0 = −∂t, ei = t ∂xi .

In this frame, the non-zero components of the second fundamental form K of the Σt hypersurfaces are p (2.46) K := g(∇ e , e ) = − i , i = 1, 2, 3. ii ei 0 i t

Further, the intrinsic volume form volΣt on Σt equals

(2.47) volΣt = tvolEuc.

Proposition 2.7 (Upper bound). Let ψ be a smooth solution to the wave equation, gψ = 0, in Kasner. Then the following energy inequality holds: Z Z (2.48) t J e0 [∂αψ]vol ≤ t J e0 [∂αψ]vol 0 x Σt 0 0 x Σt0 Σt Σt0 β for all t ∈ (0, t0] and any multi-index α. Moreover, ∂x ψ satisﬁes the pointwise bound

e0 Proof. We compute the divergence of the current Ja [ψ]:

(2.5) a te0 a b ab ∇ Ja [ψ] = ∇ (te0) Tab[ψ] = tK Tab[ψ] − (e0t)T00[ψ] 3 X 1 1 (2.50) = K t(e ψ)2 − Ki t|∇ψ|2 + (e ψ)2 + |∇ψ|2 ii i 2 i 2 0 i=1 3 (2.46) X 2 = (1 − pi)(eiψ) . i=1

Hence, by (2.8), for X = te0, we have

3 Z Z Z t0 Z X (2.51) t J e0 [ψ]vol = t J e0 [ψ]vol + (p − 1)(e ψ)2)vol ds 0 Σt 0 0 Σt0 i i Σs Σt Σt0 t Σs i=1 Z (p ≤ 1) ≤ t J e0 [ψ]vol . i 0 0 Σt0 Σt0 β β Note, that the same inequality holds for ∂x ψ by commuting the wave equation with ∂x . In particular, taking into account the volume form, we control Z Z 2 2 e0 2 β 2 (2.52) t (∂ ψ) vol ≤ 2t J [ψ]vol , t k∂ ∂ ψk 2 t Euc 0 0 Σt0 t x H Σt Σt0 X Z ≤ 2t J e0 [∂α∂βψ]vol , 0 0 x x Σt0 Σ |α|≤2 t0 for all t ∈ (0, t0]. 12

2 3 ∞ 3 Using the fundamental theorem of calculus along e0 and Sobolev embedding H (T ) ,→ L (T ), we then derive Z t β β (2.53) |∂x ψ(t, x)| = | ∂s∂x ψds + ψ(t0, x)| t0 Z t0 β β 2 ≤ Ck∂s∂x ψkH (volEuc)ds + |∂x ψ(t0, x)| t 1 Z 2 X 2 e0 α β t0 β (by (2.52)) ≤ C t0 J0 [∂x ∂x ψ]volEuc log + |∂x ψ(t0, x)|, Σ t |α|≤2 t0 for all t ∈ (0, t0]. Instead of deriving renormalised energy estimates, as in the previous subsection for FLRW (see Proposition 2.3), we prove the validity of the expansion (1.4) by using (2.49) to view the wave equation as an inhomoge- neous ODE in t. This procedure is more wasteful in the number of derivatives of ψ that we need to bound from initial data, but it is slightly simpler.

Proof of Theorem 1.1 for Kasner. We express the wave equation for ψ in terms of the (t, x1, x2, x3) coordinate system and treat the spatial derivatives of ψ as error terms: 1 3 3 2 X −2pi 2 X 1−2pi 2 (2.54) −∂ ψ − ∂tψ + t ∂ ψ = 0 ⇒ ∂t(t∂tψ) = t ∂ ψ. t t xi xi i=1 i=1

Integrating in [t, t0] we obtain the formula 3 Z t0 X t∂ ψ = t ∂ ψ − s1−2pi ∂2 ψds t 0 t 0 xi t i=1 3 t Z t0 1 Z t0 X 1−2pi 2 (2.55) ψ(t, x) = ψ(t0, x) + t0∂tψ0 log + s ∂ ψdsds t s xi 0 t s i=1 3 Z t0 t X 1−2pi 2 = ψ(t0, x) + t0∂tψ0 + s ∂ ψds log xi t 0 i=1 0 3 Z t0 1 Z s X + s1−2pi ∂2 ψdsds s xi t 0 i=1 = A(x) log t + u(t, x), where 3 Z t0 X (2.56) A(x) = t ∂ ψ + s1−2pi ∂2 ψds, 0 t 0 xi 0 i=1 3 Z t0 X (2.57) u(t, x) = ψ(t , x) − t ∂ ψ + s1−2pi ∂2 ψds log t 0 0 t 0 xi 0 0 i=1 3 Z t0 1 Z s X + s1−2pi ∂2 ψdsds. s xi t 0 i=1 β Note, that since by the assumption 1 − 2pi > −1 and by (2.49) k∂x ψkL∞ ≤ C| log t|, t ∈ (0, t0], the s 3 functions s1−2pi ∂2 ψ, 1 R P s1−2pi ∂2 ψds are integrable6 in [0, t ] and hence the above formulas make xi s 0 i=1 xi 0 sense. Moreover, it is implied by (2.56), (2.57) that A(x), u(t, x) are smooth functions and u = uKasner and its spatial derivatives are in fact uniformly bounded up to t = 0: α ∞ (2.58) k∂x ukL (Σt) ≤ C, for all t ∈ (0, t0], any multi-index α, where C > 0 is a constant depending on initial data.

6 1 By Proposition 2.7, their L ([0, t0]) norm is bounded by initial data. 13

Next, we prove the blow up result stated in Theorem 1.2 for Kasner. Notice that since pi < 1, the main e0 contribution of the energy ﬂux generated by the current J [ψ], as t → 0, comes from the ∂tψ term: 3 Z 1 Z X (2.59) t J e0 [ψ]vol = t2(∂ ψ)2 + t2−2pi (∂ ψ)2vol Σs 2 t xi Euc Σt Σt i=1 3 1 Z X = t2(∂ ψ)2vol + t2−2pi O(| log t|2). 2 t Euc Σt i=1 Utilising (1.4), (2.58) it follows that Z 1 Z 1 Z e0 2 2 2 (2.60) lim t J [ψ]volΣ = lim t (∂tψ) volEuc = A (x)volEuc. t→0 s t→0 Σt 2 Σt 2 Σ0 Thus, returning to (2.51) and taking the limit t → 0 we obtain the identity:

3 Z Z Z t0 Z X (2.61) A2(x)vol = t J e0 [ψ]vol + (p − 1)(e ψ)2vol ds Euc 0 0 Σt0 i i Σs Σ0 Σt0 0 Σs i=1 3 1 2 2 1 X 2−2pi 2 = t0k∂tψkL2(Σ ) + t0 k∂xi ψkL2(Σ ) 2 t0 2 t0 i=1 3 Z t0 Z X 1−2pi 2 + (pi − 1)s (∂xi ψ) volEucds. 0 i=1 Σs 2 We bound the L norm of ∂xi ψ as follows: 2 Lemma 2.8. The following estimate for the L norm of ∂xi ψ holds:

1 Z 2 e0 t0 (2.62) k∂x ψkL2(Σ ) ≤ k∂x ψkL2(Σ ) + 2t0 J [∂x ψ]volΣ log , i t i t0 0 i t0 t Σt0 for all t ∈ (0, t0]. Proof. We have

C−S 1 2 − ∂tk∂x ψk 2 ≤ k∂x ψkL2(Σ )k∂t∂x ψkL2(Σ ) 2 i L (Σt) i t i t

2 2 − ∂tk∂xi ψkL (Σt) ≤ k∂t∂xi ψkL (Σt) Z t0 k∂ ψk 2 ≤ k∂ ψk 2 + k∂ ∂ ψk 2 ds xi L (Σt) xi L (Σt0 ) s xi L (Σs) t 1 (2.52) Z 2 e0 t0 (2.63) k∂x ψkL2(Σ ) ≤ k∂x ψkL2(Σ ) + 2t0 J [∂x ψ]volΣ log , i t i t0 0 i t0 t Σt0 where in the last inequality we made use of (2.52). Applying (2.63) to (2.61) we derive: Z 2 (2.64) A (x)volEuc Σ0 3 1 2 2 1 X 2−2pi 2 ≥ t0k∂tψkL2(Σ ) + t0 k∂xi ψkL2(Σ ) 2 t0 2 t0 i=1 3 Z t0 X 1−2pi 2 + (pi − 1)s ds(2k∂xi ψk 2 ) L (Σt0 ) i=1 0 3 Z t0 s X 1−2pi 2 2 + (pi − 1)s | log | ds 2t k∂t∂x ψkL2(Σ ) t 0 i t0 i=1 0 0 14

3 X 2−2pi + 2 t k∂ ∂ ψk 2 0 xj xi L (Σt0 ) j=1 3 1 2 2 1 X 2−2pi 2 ≥ t0k∂tψkL2(Σ ) − t0 k∂xi ψkL2(Σ ) 2 t0 2 t0 i=1

3 2−2pi 3 X t0 2 2 X 2−2pi 2 − t0k∂t∂xi ψkL2(Σ ) + t0 k∂xj ∂xi ψkL2(Σ ) . (1 − p )2 t0 t0 i=1 i j=1 If the assumptions of Theorem 1.2 for Kasner are satisﬁed, then it is clear from the preceding lower bound

2 3 that kA(x)kL (T ) > 0. To prove the local version of the blow up criterion in Theorem 1.4, we argue similarly, but also take into account the contribution of the null ﬂux terms in (2.8). Note that the upper bounds (2.48), (2.49), (2.63) are also valid for the integral domains Ut,U0 in place of Σt, Σt0 , since the null ﬂux terms in (2.8) have a favourable sign for an upper bound. Hence, taking the limit t → 0 in (2.6) and employing (2.7), (2.50), (1.4), (2.58) we obtain: Z 2 (2.65) A (x)volEuc U0 3 1 2 2 1 X 2−2pi 2 ≥ t0k∂tψkL2(U ) + t0 k∂xi ψkL2(U ) 2 t0 2 t0 i=1 3 Z t0 Z X 1−2pi 2 + (pi − 1)s (∂xi ψ) volEucds 0 i=1 Us 3 Z t0 Z 1 X 2−pl 2 2 2 (i < j; i, j 6= l) − s |(e0 ± el)ψ| + |eiψ| + |ejψ| dsdxidxj. ± 2 l=1 t Nl ∩U s

−pl ± Since t dt = ±dxl along Nl , it follows by integrating that the closure U t of the neighbourhood of Ut is t1−pi t1−pi the product I1 × I2 × I3, where Ii = [− , δ + ]. The analogous inequality to (2.43) then reads 1−pi 1−pi 2 Z 2 1−pl −1 2 2 (2.66) f (t, x ) ≤ (δ + t ) f (t, x )dx + kf(t, x )k 1 , l l l l H (Il) 1 − pl Il 1 with f ∈ H (Il), t ∈ (0, t0]. ± Applying the latter bound to the integral over Nl ∩ U s on the RHS of (2.65), along with (2.63), we derive Z 2 (2.67) A (x)volEuc U0 3 1 2 2 1 X 2−2pi 2 ≥ t0k∂tψkL2(U ) + t0 k∂xi ψkL2(U ) 2 t0 2 t0 i=1 3 Z t0 X 1−2pi 2 + (pi − 1)s ds(2k∂xi ψk 2 ) L (Ut0 ) i=1 0 3 Z t0 s X 1−2pi 2 2 + (pi − 1)s | log | ds 2t k∂t∂x ψkL2(U ) t 0 i t0 i=1 0 0 3 X 2−2pi + 2 t k∂ ∂ ψk 2 0 xj xi L (Ut0 ) j=1 3 X Z t0 1 Z − (1 + )s−pl 4sJ e0 [ψ]vol ds 2 1−p 0 Us t δ + s l U l=1 1−pl s Z t0 Z −pl e0 + s 4sJ0 [∂xl ψ]volUs ds t Us 15

3 1 2 2 1 X 2−2pi 2 ≥ t0k∂tψkL2(U ) − t0 k∂xi ψkL2(U ) 2 t0 2 t0 i=1

3 2−2pi 3 X t0 2 2 X 2−2pi 2 − t0k∂t∂xi ψkL2(Σ ) + t0 k∂j∂xi ψkL2(Σ ) (1 − p )2 t0 t0 i=1 i j=1

3 1−pl 3 X 2t0 2 2 X 2−2pi 2 − t k∂ ψk 2 + t k∂ ψk 2 0 t L (Ut ) 0 xi L (Ut ) 1 − pl 0 0 l=1 i=1 3 3 2 X 2 X 2−2pi 2 + t k∂t∂xl ψk 2 + t k∂xi ∂xl ψk 2 0 L (Ut0 ) 0 L (Ut0 ) l=1 i=1

3 1−pl 3 X 2 t0 2 2 2−2pl X 2 − log 1 + t k∂ ψk 2 + t k∂ ψk 2 . 0 t L (Ut ) 0 xi L (Ut ) 1 − pl δ 0 0 l=1 i=1

2 Thus, given the assumptions in Theorem 1.4 for Kasner, it follows that kA(x)kL (U0) > 0, as required.

Acknowledgements The authors would like to thank Mihalis Dafermos for useful interactions. Further, A.A. and A.F. ben- efited from discussions with JoséNatário. This work was partially supported by FCT/Portugal through UID/MAT/04459/2013, grant (GPSEinstein) PTDC/MAT-ANA/1275/2014 and through the FCT fellowships SFRH/BPD/115959/2016 (A.F.) and SFRH/BPD/85194/2012 (A.A.). G.F. was supported by the EPSRC grant EP/K00865X/1 on “Singularities of Geometric Partial Differential Equations”.

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