A class of dust-like self-similar solutions of the massless Einstein-Vlasov system
aa r X i v : . [ g r- q c ] S e p A CLASS OF DUST-LIKE SELF-SIMILAR SOLUTIONS OF THEMASSLESS EINSTEIN-VLASOV SYSTEM.
Alan D. Rendall , Juan J. L. Vel´azquez Abstract
In this paper the existence of a class of self-similar solutions of theEinstein-Vlasov system is proved. The initial data for these solutions arenot smooth, with their particle density being supported in a submani-fold of codimension one. They can be thought of as intermediate betweensmooth solutions of the Einstein-Vlasov system and dust. The motivationfor studying them is to obtain insights into possible violation of weak cos-mic censorship by solutions of the Einstein-Vlasov system. By assuming asuitable form of the unknowns it is shown that the existence question canbe reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on twoparameters. This solution starts at a particular point P and converges toa stationary solution P as the independent variable tends to infinity. Theexistence proof is based on a shooting argument and involves relating thedynamics of solutions of the four-dimensional system to that of solutionsof certain two- and three-dimensional systems obtained from it by limitingprocesses. It is well known that solutions of the Einstein equations coupled with suitablemodels of matter can yield singularities in finite time. The unknowns in theseequations are the spacetime metric and some matter fields. The exact natureof the latter depends on the physical situation being considered. The usualterminology in general relativity is that there is said to be a singularity if themetric fails to be causally geodesically complete, i.e. if there are timelike ornull geodesics which in at least one direction are inextendible and of finite affinelength. The singularity is said to be in the future or the past according to theincomplete direction of the geodesics. It is expected on the basis of physicalintuition, and known to be true in some simple cases, that the geodesic in-completeness is associated with the energy density or some curvature invariantsblowing up. For background on this subject see textbooks such as [13], [28] and Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am M¨uhlenberg1, 14476 Potsdam, Germany ICMAT (CSIC-UAM-UC3M-UCM), Universidad Complutense, Madrid 28035, Spain. h ab , a symmetric tensor k ab and some matter fieldswhich for the moment will be denoted generically by F , all defined on a three-dimensional manifold S . Solving the Cauchy problem for the Einstein-matterequations means embedding the manifold S into a four-dimensional manifold M on which are defined a Lorentzian metric g αβ and matter fields F such that h ab and k ab are the pull-backs to S of the induced metric and second funda-mental form of the image of the embedding of S while F is the pullback ofthe matter fields. The metric g αβ and the matter fields F are required to sat-isfy the Einstein-matter equations. A comprehensive treatment of the Cauchyproblem for the Einstein equations can be found in [26]. Initial data on R arecalled asymptotically flat if the metric h ab tends to the flat metric at infinityin a suitable sense while k ab and F tend to zero. Physically this correspondsto concentrating attention on a particular physical system while ignoring theinfluence of the rest of the universe.A solution of the Einstein-matter equations evolving from initial data is saidto be a development of that data if each inextendible causal curve intersectsthe initial hypersurface precisely once. When this property holds the initialhypersurface is said to be a Cauchy hypersurface for that solution. In general,a solution is called globally hyperbolic if it admits a Cauchy hypersurface. Forprescribed data there is a development which is maximal in the sense that anyother development can be embedded into it. It is unique up to a diffeomorphismwhich preserves the initial hypersurface.In a spacetime evolving from asymptotically flat data it is often possibleto define future null infinity I + as a set of ideal endpoints of complete future-directed null geodesics. We can say that any singularity occurring does notinfluence events near infinity if there is no inextendible causal curve to the fu-ture of the initial hypersurface which is incomplete in the past while intersectinga future-complete null geodesic. The first of these properties means intuitivelythat this curve represents a signal which comes out of a singularity while thesecond property means that it reaches a region which can communicate withinfinity. If a curve of this type does exist it is said that a globally naked singular-2ty exists. The past of null infinity, J − ( I + ), is the set of points for which thereis a future-directed causal curve starting there and going to null infinity. Thecomplement of J − ( I + ) is called the black hole region. Its boundary is calledthe event horizon and is a null hypersurface in M .There is a notion of completeness of null infinity. A precise definition willnot be given here but roughly speaking it corresponds to the situation wherethere are timelike curves contained in J − ( I + ) which exist for a infinite timetowards the future. Physically this means that there are observers which canremain outside the black hole for an unlimited amount of time. If the maximalglobally hyperbolic development of asymptotically flat initial data always has acomplete null infinity then this ensures the absence of globally naked singulari-ties. For any inextendible causal curve to the future of the initial surface whichgoes to null infinity must intersect the initial hypersurface. Hence it cannot beincomplete in the past. The completeness of I + ensures that the solution islarge enough to represent the whole future of a system evolving from the initialdata under consideration. The intuitive content of the weak cosmic censorshiphypothesis is that in the time evolution corresponding to initial data for the Ein-stein equations coupled to reasonable (non-pathological) matter the existenceof a singularity implies that of an event horizon which covers the singularityand hides it from distant observers. Often this is weakened to the requirementthat a horizon exists in the case of generic initial data. Up to now this intuitivepicture has only been developed into a precise mathematical formulation underspecial circumstances. In general finding the correct formulation is part of theproblem to be solved.Due to the mathematical complexity of the Einstein equations many of thestudies related to singularity formation for these equations have been carriedout for spherically symmetric solutions. In spherical symmetry the Einsteinvacuum equations are non-dynamical due to Birkhoff’s theorem, which saysthat any spherically symmetric vacuum solution is locally isometric to theSchwarzschild solution and, in particular, static. Thus it is essential to in-clude matter of some kind. A matter model which has proved very useful forthis task is the scalar field. This is a real-valued function φ which satisfies thewave equation ∇ α ∇ α φ = 0. In this case the Einstein equations take the form R αβ = 8 π ∇ α φ ∇ β φ , where R αβ is the Ricci curvature of g αβ . The sphericallysymmetric Einstein-scalar field equations were studied in great detail in a seriesof papers by Demetrios Christodoulou. This culminated in [7] and [8]. In [7]it was shown that in this system naked singularities can evolve from regularasymptotically flat initial data. This represents a problem for the weak cosmiccensorship hypothesis but the conjecture can be saved by a genericity assump-tion since it was shown in [8] that generic initial data do not lead to nakedsingularities.For the spherically symmetric Einstein-scalar field equations it is knownfrom the work of Christodoulou [6] that small asymptotically flat initial datalead to a solution which is geodesically complete and hence free of singularities.(In fact this small data result has recently been extended to the case withoutsymmetry [15].) On the other hand there are certain large initial data for which3t is known that a black hole is formed. The threshold between these two typesof behaviour was studied in influential work by Choptuik [3] and many otherpapers since. This area of research is known as critical collapse and is surveyedin [12]. It is entirely numerical and heuristic and unfortunately mathematicallyrigorous results are not yet available.The scalar field provides a simple and well-behaved matter model. At thesame time no such field has been experimentally observed and the matter fieldsof importance for applications to astrophysics are of other kinds. One astro-physically relevant matter field which has good mathematical properties is col-lisionless matter described by the Vlasov equation. The necessary definitionsare given in the next section. For the moment let it just be noted that the un-known in the Vlasov equation is a non-negative real-valued function f ( t, x a , v b )depending on local coordinates ( t, x a ) on M and velocity variables v b . Ana-logues of a number of the results proved for the scalar field have been provedfor the Einstein-Vlasov system. For small initial data the solutions are geodesi-cally complete [22]. There are certain large initial data for which a black hole isformed [1]. The threshold between these two types of behaviour has been inves-tigated numerically in [23] and [18]. A closely related matter model which hasbeen very popular in theoretical general relativity is dust, a fluid with vanishingpressure. It is equivalent to consider distributional solutions of the Vlasov equa-tion of the form f ( t, x a , v b ) = ρ ( t, x a ) δ ( v b − u b ( t, x a )) where the δ is a Diracdistribution. From many points of view dust is relatively simple to analyse.Unfortunately it has a strong tendency to form singularities where the energydensity blows up, even in the absence of gravity. For this reason it must beregarded as pathological and of limited appropriateness for the investigation ofcosmic censorship. A detailed mathematical study of formation of singularitiesin the Einstein equations coupled to dust was given in [5]. In spherical sym-metry dust particles move as spherical shells. It can easily happen that shellsincluding a strictly positive total mass come together at one radius and thiscauses the density to blow up. This effect is known as shell-crossing.The motivation for this paper is the wish to understand cosmic censorshipbetter for spherically symmetric solutions of the Einstein-Vlasov system. Is ittrue that in asymptotically flat spherically symmetric solutions of the Einstein-Vlasov system there are no naked singularities for generic data so that colli-sionless matter is as well-behaved as the scalar field? Could it even be that theVlasov equation is better-behaved and that there are no naked singularities atall? No answers to these questions, positive or negative, are available althoughconsiderable effort has been invested into obtaining a positive answer. In whatfollows we try to obtain new insights by approaching a negative result throughan interpolation between dust and smooth solutions of the Vlasov equation andlooking for self-similar solutions. There are some results on related equationswhich give some hints. In the case of the Vlasov-Poisson system, the non-relativistic analogue of the Einstein-Vlasov system, global existence for generaldata, not necessarily symmetric, was proved by Pfaffelmoser [20] and Lions andPerthame [16]. The relativistic Vlasov-Poisson system, which is in some senseintermediate between the Vlasov-Poisson and Einstein-Vlasov systems, (but not4n all ways) has been shown to have solutions which develop singularities in fi-nite time. Rather precise information is available about the nature of thesesingularities [14].As a side remark, we mention a paper [27] where it was suggested thatnaked singularities are formed in solutions of the Einstein-Vlasov system. Thesolutions concerned were axially symmetric but not spherically symmetric. Thework is purely numerical but trying to understand what it means for the an-alytical problem leads to the conclusion that the solutions computed in [27]were dust solutions rather than smooth solutions of the Einstein-Vlasov system.This is discussed in [24]. There are also reasons for doubting that the numericalresults really show the formation of a naked singularity [29].A class of distributional solutions of the Einstein-Vlasov system intermedi-ate between smooth solutions and dust is given by the Einstein clusters [11].These are spherically symmetric and static, i.e. there exists a timelike Killingvector field which is orthogonal to spacelike hypersurfaces. It is supposed thatthe support of f consists of v a such that the geodesics with these initial dataare tangent to the spheres of constant distance from the centre of symmetryon these spacelike hypersurfaces. This means that the radial velocity and itstime derivative in the geodesic equation are zero. These are in general two inde-pendent conditions on the data at a given time. A wider class, the generalizedEinstein clusters [9], [2], is obtained as follows. In the case of the Einstein clus-ters taking the union of the spheres at a fixed distance from the centre definesa foliation of the spacetime by timelike hypersurfaces and the condition on thesupport means that the four-velocity of a particle with the given initial datais everywhere tangent to these timelike hypersurfaces. The generalized Ein-stein clusters are obtained by dropping the condition of staticity and replacingthe family of timelike hypersurfaces invariant under the timelike Killing vectorfield by another foliation by timelike hypersurfaces which intersect any Cauchysurface in spheres and whose equation of motion follows from the Vlasov equa-tion. Once again the four-velocity of a particle in the support of f is tangentto these hypersurfaces at all times. An analytical formulation of this definitionwill be given in the next section. It should be noted that the generalized Ein-stein clusters exhibit shell-crossing singularities and thus can still be thoughtof as pathological. We are interested in them as an intermediate step towardsbetter-behaved matter models.There are two major differences between the generalized Einstein clustersand the solutions studied in this paper. In the case of Einstein clusters thevalue of the angular momentum of the particles F is uniquely determined bythe distance r to the centre of symmetry. By contrast, in the solutions studiedin this paper the angular momentum takes a continuous range of values for eachvalue of r. The second difference is that in the case of Einstein clusters at eachspacetime point the component of the velocity vector v a of a particle in thedirection of the vector ∂ r takes on only one value. In the case of the solutionsobtained in this paper the component along the direction ∂ r of the velocityvector takes on two different values at most spacetime points. This only failsat some exceptional values of r at a given time. This difference in the structure5f the generalized Einstein clusters and the solutions considered in this paper iswhat gives some plausibility to the idea that the solutions described here couldbe a big step towards better-behaved matter models. From the physical pointof view, in the case of the generalized Einstein clusters the material particlewith the smallest value of r would not experience any gravitational field, andtherefore could not approach the centre r = 0 unless its angular momentumvanished. In the solutions studied in this paper, since two radial velocities areallowed at each spacetime point, the material particle with the smallest valueof r changes in time. This allows the occurrence of a collective collapse of thewhole distribution of particles towards the origin with some of them comingcloser and closer to the center as the value of some suitable time coordinate t increases.Self-similar solutions of the massless Einstein-Vlasov system have also beenconsidered in the paper [17]. There are several differences between the approachin [17] and the one considered in this paper. The first one is the choice of therescaling group under which the solutions are invariant. The massless Einstein-Vlasov system is invariant under a two-dimensional group of rescalings. Thechoice of a particular one-dimensional rescaling group has been made in thispaper by imposing that the distribution function f for the particles remainsalways of order one (see Sections 2, 3). This condition is natural, because thefunction f is invariant along characteristic curves. On the contrary, the choice ofone-dimensional rescaling group for the solutions in [17] imposes that f becomesunbounded near the singularity for the particles within the self-similar region,something that can be achieved assuming that the distribution of matter issingular near the light-cone. The second difference between the solutions in [17]and those in this paper is that the solutions in [17] can be thought of as self-similar perturbations of the flat Minkowski space. As a matter of fact they havebeen computed by means of a perturbative iteration procedure that takes flatspace as a starting point and where the terms in the resulting series have beencomputed numerically. By contrast, the solutions of this paper are obtained bymeans of a shooting procedure in which a parameter that measures the amountof energy in the self-similar region is of order one. The approach in this paperuses purely analytical methods and does not rely on numerical computations.On the other hand, in order to simplify the arguments, we have restricted theanalysis in this paper to the study of dust-like solutions, an assumption thatwas not made in [17].The plan of the paper is as follows. We will first reduce the problem offinding self-similar solutions of the Einstein-Vlasov system to an ODE problemthat can be transformed into a four-dimensional system using suitable changesof variables. Using these transformations it will be seen that the constructionof the desired self-similar solutions reduces to finding a particular orbit in thecorresponding four-dimensional space connecting a certain point with a steadystate that has a three-dimensional stable manifold. The existence of such anorbit will be shown by adjusting a parameter that measures the density of par-ticles in a particular perturbative limit. The precise limit under consideration,which has the goal of making the problem feasible using analytical methods, cor-6esponds to assuming that the radius of the region empty of particles, measuredin the natural self-similar variables, is small. We do not use exactly the classical Schwarzschild coordinates, but a slight mod-ification of them that normalizes the time to be the proper time at the center r = 0. The metric is given by (cf. [21]): ds = − e µ ( t,r ) dt + e λ ( t,r ) dr + r (cid:0) dθ + sin θdϕ (cid:1) . (2.1)If we restrict our attention to spherically symmetric solutions it is convenientto use the quantities (cf. [21]): r = | x | , w = x · vr , F = | x ∧ v | to parametrize the velocity variables. In particular F is constant along charac-teristics. Writing the particle density as f = f ( r, w, F, t )the Einstein-Vlasov system for spherically symmetric solutions in these coordi-nates becomes: ∂ t f + e µ − λ wE ∂ r f − (cid:18) λ t w + e µ − λ µ r E − e µ − λ Fr E (cid:19) ∂ w f = 0 (2.2)where: E = r w + Fr (2.3)and the functions λ, µ that characterize the gravitational field satisfy: e − λ (2 rλ r −
1) + 1 = 8 πr ρ, (2.4) e − λ (2 rµ r + 1) − πr p (2.5)with boundary conditions: µ (0) = 0 , λ (0) = 0 , (2.6) λ ( ∞ ) = 0 . (2.7)On the other hand ρ and p are given by: ρ = ρ ( r, t ) = πr Z ∞−∞ (cid:20)Z ∞ Ef dF (cid:21) dw, (2.8) p = p ( r, t ) = πr Z ∞−∞ (cid:20)Z ∞ w E f dF (cid:21) dw. (2.9)7ith these basic equations in hand it is possible to give some details con-cerning generalized Einstein clusters, as promised in the introduction. Theseare not required to understand the main results of the paper but help to putthose results into a wider context. A distributional solution of the Vlasov equa-tion whose support is a smooth submanifold Σ has the property that Σ is aunion of characteristics of the equation. A simple example is that of dust wherethe support is the graph of a function u a ( t, x ) of the form W ( t, r ) x a r . Whenexpressed in terms of polar coordinates this becomes the graph of a function W ( t, r ) augmented by the condition F = 0. Here the function W solves theequations dRdt = e µ ( t,R ) − λ ( t,R ) WE , (2.10) dWdt = − ( ˙ λ ( t, R ) W + e µ ( t,R ) − λ ( t,R ) µ ′ ( t, R ) E ) (2.11)where E = √ W .Now consider the generalized Einstein clusters. They are only defined underthe condition of spherical symmetry. They can be thought of as defining amatter model which can be used in the spherically symmetric Einstein-matterequations. Here they will be described in terms of Schwarzschild coordinates.The basic unknown is a function R ( t, r ) which satisfies R (0 , r ) = r . It is thearea radius at time t of the shell which had area radius r at time 0. As inputwe require a function F ( r ) which is the angular momentum of the particles onthe shell which was at radius r at time zero and N ( r ) which is the density ofparticles per shell evaluated on the shell which had area radius r at t = 0. Forsome purposes it is more convenient to use R as a radial coordinate instead of r and this is what was done in the original papers [9] and [2]. For a given shell ata given time the angular momentum and radial velocity of the particles are fixedand so the intersection of the support of the solution with the fibre of the massshell over the point with coordinates ( t, r ) has codimension two. The followingequations should be satisfied: dRdt = e µ − λ WE , (2.12) dWdt = − (cid:18) ˙ λW + e µ − λ µ ′ E − e µ − λ FR E (cid:19) (2.13)where E = q W + FR and the functions λ and µ are to be evaluated at thepoint ( t, R ). These are the full characteristic equations for the Vlasov equation.The difference in the coupled system comes from the fact that the expressionsfor the components of the energy-momentum tensor are different in the twocases. In the case of Einstein clusters the characteristics of interest have W = 0and dWdt = 0. It follows immediately that dRdt = 0. In that case the angularmomentum is related to the geometry by the relation F = r µ ′ − rµ ′ .The equations which have been written up to now describe particles of unit8ass. We are interested in the construction of solutions of (2.2)-(2.9) supportedin a region where (cid:0) w + Fr (cid:1) takes large values near the formation of the singu-larity. This suggests replacing (2.3) by: E = r w + Fr . (2.14)The system (2.2), (2.4)-(2.9), (2.14) is invariant under the rescaling: r → θr , t → θt for t < , w → √ θ w , F → θF (2.15)for any θ > . It is then natural to look for solutions of (2.2), (2.4)-(2.14)invariant under the rescaling (2.15). They will be the self-similar solutions inwhich we will be interested in this paper.The system obtained when (2.3) is replaced by (2.14) can be interpreted asdescribing particles of zero rest mass. The rationale for this assumption is thatnear the singularity the derived solution will satisfy w + Fr >> , and thereforeit could be expected that it is possible to treat the whole Einstein-Vlasov systemwith massive particles as a perturbation of the massless problem.In what follows we will consider solutions of (2.2), (2.4)-(2.14) where f isnot a bounded function, but a measure concentrated on some hypersurfaces thatwill be described in detail later. As was mentioned in the introduction thereis a class of distributional solutions of the Einstein-Vlasov system which areequivalent to what is usually known in the literature as dust. From this point ofview the solutions considered in this paper are intermediate between dust andsmooth solutions and hence will be called dust-like solutions. Note, however,that in contrast to dust they do have some velocity dispersion. The dimensionof the support of f in the tangent space at a given spacetime point is zero fordust, one for generalized Einstein clusters, two for the solutions in this paperand three for smooth solutions. For the solutions here it will be possible todescribe the distribution of velocities for the particles at a given point using afunction depending on one coordinate, while a general distribution of velocitiescompatible with the assumption of spherical symmetry would depend on twocoordinates. In this section we formulate the system of equations satisfied by the solutionsof (2.2), (2.4)-(2.9), (2.14) that are invariant under the transformation (2.15).We will call these self-similar solutions in what follows. It is convenient, as afirst step, in order to transform (2.2), (2.4)-(2.14) to a more convenient form todefine a new variable: v = w √ F . (3.1)9e will assume in the rest of the paper that f = 0 for ( r, v, F, t ) = ( r, v, , t )in order to avoid singularities in (3.1). Moreover, we can even assume a morestringent condition on f, namely f = 0 for 0 ≤ F ≤ δ for some δ > . Concerning the support in the r coordinate, the solutions constructed in thispaper will vanish for r ≤ y ( − t ) for some y > . Making the change of variables ( r, w, F, t ) → ( r, v, F, t ) and denoting the newdistribution function by f with a slight abuse of notation we can transform thesystem (2.2), (2.8)-(2.14) into: ∂ t f + e µ − λ v ˜ E ∂ r f − (cid:18) λ t v + e µ − λ µ r ˜ E − e µ − λ r ˜ E (cid:19) ∂ v f = 0 , (3.2)˜ E = r v + 1 r , (3.3) ρ = πr Z ∞−∞ ˜ E (cid:20)Z ∞ f F dF (cid:21) dv, (3.4) p = πr Z ∞−∞ v ˜ E (cid:20)Z ∞ f F dF (cid:21) dv. (3.5)Notice that the change of variables (3.1) eliminates the dependence on the vari-able F for the characteristic curves associated to the Vlasov equation (cf. (3.2)).Moreover, the functions ρ and p and therefore the functions λ, µ characteriz-ing the gravitational fields depend on f only through the reduced distributionfunction: ζ ( r, v, t ) ≡ Z ∞ f F dF. (3.6)In particular, it is possible to write a closed problem for the reduced distribu-tion function that can be obtained multiplying (3.2) by F and integrating withrespect to this variable: ∂ t ζ + e µ − λ v ˜ E ∂ r ζ − (cid:18) λ t v + e µ − λ µ r ˜ E − e µ − λ r ˜ E (cid:19) ∂ v ζ = 0 , (3.7)˜ E = r v + 1 r , (3.8) ρ = πr Z ∞−∞ ˜ Eζdv, (3.9) p = πr Z ∞−∞ v ˜ E ζdv. (3.10)The system (3.7)-(3.10) complemented with (2.4), (2.5) is a closed system ofequations.We will now study the class of self-similar solutions of the system (2.4), (2.5),103.2)-(3.5). These are the functions having the functional dependence: f ( r, v, F, t ) = G ( y, V, Φ) , µ ( r, t ) = U ( y ) , λ ( r, t ) = Λ ( y ) , (3.11) y = r ( − t ) , V = ( − t ) v , Φ = F ( − t ) . (3.12)The solutions of (2.4), (2.5), (3.2)-(3.5) with this functional dependence satisfy: yG y − V G V + Φ G Φ + e U − Λ V ˆ E G y − (cid:18) y Λ y V + e U − Λ U y ˆ E − e U − Λ y ˆ E (cid:19) G V = 0 (3.13)where: ˆ E = r V + 1 y (3.14)and e − (2 y Λ y −
1) + 1 = 8 πy ˜ ρ, (3.15) e − (2 yU y + 1) − πy ˜ p (3.16)with boundary conditions: U = 0 , Λ = 0 at y = 0 . (3.17)Here: ˜ ρ = πy Z ∞−∞ ˆ E (cid:20)Z ∞ G Φ d Φ (cid:21) dV, (3.18)˜ p = πy Z ∞−∞ V ˆ E (cid:20)Z ∞ G Φ d Φ (cid:21) dV. (3.19)The function G which is a solution of (3.13)-(3.19) is constant along thecharacteristic curves of (3.13) which are given by: dydσ = y + e U − Λ V q V + y = y + e U − Λ V y p V y + 1 , (3.20) dVdσ = − V − y Λ y V + e U − Λ U y y p V y + 1 − e U − Λ y p V y + 1 ! , (3.21) d Φ dσ = Φ . (3.22)In these equations σ is just a parameter that is used to parametrize the charac-teristic curves. Its precise definition will be given later in some specific cases.11he equations (3.20)-(3.22) can be integrated explicitly for any pair of func-tions U = U ( y ) , Λ = Λ ( y ). Indeed, the first two equations can be rewrittenas: dydσ = e − Λ ∂H∂V , (3.23) dVdσ = − e − Λ ∂H∂y (3.24)where: H ≡ e U y p V y + 1 + yV e Λ . (3.25)The trajectories in the ( y, V )-plane associated to the solutions of (3.20), (3.21)are contained in the level sets: H = h. (3.26)We will also need the self-similar formulation of the integrated form of theequation (3.7). In this case the function ζ in (3.6) has the functional dependence: ζ ( r, v, t ) = ( − t ) Θ ( y, V ) . Notice that: Θ ( y, V ) = Z ∞ G Φ d Φ . (3.27)The function Θ satisfies: y Θ y − V Θ V −
2Θ + e U − Λ V ˆ E Θ y − (cid:18) y Λ y V + e U − Λ U y ˆ E − e U − Λ y ˆ E (cid:19) Θ V = 0 (3.28)and: ˜ ρ = πy Z ∞−∞ ˆ E Θ dV, (3.29)˜ p = πy Z ∞−∞ V ˆ E Θ dV. (3.30)The characteristic curves associated to (3.28) are (3.20), (3.21) and: d Θ dσ = 2Θ . (3.31)12 SINGULAR SELF-SIMILAR SOLUTIONS:GENERAL PROPERTIES.
The main goal of this paper is to construct a family of distributional solutions of(3.13)-(3.19) for which G = G ( y, V, Φ) is a measure supported on some surfacesin the three-dimensional space with coordinates ( y, V, Φ) . In this section wewill describe in a heuristic manner the argument yielding the construction ofsuch solutions. The arguments will be made rigorous in the rest of the paper.The key idea behind the argument is that the problem can be transformed intoa system of ordinary differential equations for the particular class of solutionsdescribed in this section.Taking into account that the singularities of the distribution G might beexpected to be propagated by characteristics it is natural to look for solutionsof (3.13)-(3.19) of the form: G ( y, V, Φ) = A ( y, V, Φ) δ ( H ( y, V ) − h ) (4.1)satisfying (3.13) in the sense of distributions. Let us assume that A, H have thedifferentiability properties required for all the following formal computations.Plugging (4.1) into (3.13) we obtain:( a ( y, V ) A y + b ( y, V ) A V + Φ A Φ ) δ ( H − h )+ A ( a ( y, V ) H y + b ( y, V ) H V ) δ ′ ( H − h )= 0where: a ( y, V ) ≡ y + e U − Λ V ˆ E = e Λ ∂H∂V , (4.2) b ( y, V ) ≡ − V − (cid:18) y Λ y V + e U − Λ U y ˆ E − e U − Λ y ˆ E (cid:19) = − e Λ ∂H∂y . (4.3)Notice that a ( y, V ) H y + b ( y, V ) H V = 0 . Then:( a ( y, V ) A y + b ( y, V ) A V + Φ A Φ ) δ ( H − h ) = 0 . This equation is satisfied if: a ( y, V ) A y + b ( y, V ) A V + Φ A Φ = 0 (4.4)on the surface { H = h } × R + . Let us assume that the curve { H = h } can beparametrized, at least locally, using a parameter σ satisfying: y = y ( σ ) , V = V ( σ ) ,dy ( σ ) dσ = a ( y ( σ ) , V ( σ )) , dV ( σ ) dσ = b ( y ( σ ) , V ( σ )) . (4.5)13hen the function A can be written on the surface { H = h } × R + as a functionof the variables ( σ, Φ) . We can write: A ( y ( σ ) , V ( σ ) , Φ) = ¯ A ( σ, Φ) for ( y, V, σ ) ∈ { H = h } × R + (4.6)and using (4.5) we can rewrite (4.4) as:¯ A σ + Φ ¯ A Φ = 0 . (4.7)Since the curves { H = h } can be determined in terms of Θ alone it is con-venient to compute this distribution explicitly. If G has the form (4.1) thedistribution Θ defined in (3.27) is given by:Θ ( y, V ) = βδ ( H − h ) (4.8)where: β = Z ∞ A Φ d Φ . Since A is given by (4.6) it follows that: β = β ( σ ) = Z ∞ ¯ A ( σ, Φ) Φ d Φ for ( y, V ) ∈ { H = h } . (4.9)We can compute β ( σ ) along the curve { H = h } . To this end we multiply (4.7)by Φ and integrate in the Φ variable in the interval [0 , ∞ ) . Then: β σ = 2 β. The function β then takes the form: β ( σ ) = β e σ (4.10)for some β ≥ . In the rest of the paper we prove that there exist functions ¯ A ( σ, Φ) as in(4.6) and curves { H = h } with Λ , U solving (3.15)-(3.17) and ˜ ρ, ˜ p as in (3.18),(3.19) such that (4.1) solves (3.13) in the sense of distributions. In this section we describe in a precise manner the form of the curved surfacecontaining the support of the distribution G for the self-similar solutions con-structed in this paper. Such a surface is contained in the surface S = γ × R + , where γ ⊂ { ( y, V ) : y > , V ∈ R } is an unbounded curve, at a strictly positivedistance from the line { y = 0 } ≡ { ( y, V ) : y = 0 , V ∈ R } with a discontinuity14n its curvature at the point ( y , V ) ∈ γ placed at the minimum distance fromthe line { y = 0 } . In order to avoid such irregular curves it is more convenientto assume that the curve γ is the union of two analytic curves γ and γ thatcan be parametrized in the form: γ i = { ( y, V ) : y < y < ∞ , V = V i ( y ) } , i = 1 , V i ( y ) are analytic and satisfy:lim y → y +0 V ( y ) = lim y → y +0 V ( y ) = V = − p − y , (5.2) V ( y ) < V ( y ) for y < y < ∞ (5.3)for some y ∈ (0 , . Since the curves γ i are contained in the curve { H = h } itfollows that the functions V i ( y ) are the two roots of the equation: e U y p V y + 1 + yV e Λ = h (5.4)assuming that such roots exist. Then: V ( y ) = 1( e U − y e ) " − ye Λ h − s ( ye Λ h ) − ( e U − y e ) (cid:18) e U y − h (cid:19) , (5.5) V ( y ) = 1( e U − y e ) " − ye Λ h + s ( ye Λ h ) − ( e U − y e ) (cid:18) e U y − h (cid:19) . (5.6)Notice that for such solutions the support of G in (4.1) is contained in thehalf-plane { y ≥ y } . Therefore, ρ ( y ) = p ( y ) = 0 for y < y . Then (3.15)-(3.17)imply U ( y ) = Λ ( y ) = 0 for y < y . Under suitable regularity assumptions for the curves γ i near the point ( y , V )that will be made precise below the functions U and Λ are continuous at thepoint y = y . In such a case (5.4) implies: h = p V y + 1 y + y V = p − y y . (5.7)We will prove later that it is possible to construct the desired curves γ i , i = 1 , y → y +0 V i ( y ) − V √ y − y = K i , K i ∈ R , i = 1 , , K < K . (5.8)15oreover, the quotients of the functions Λ and U by √ y − y also tend to finitelimits. Let: lim y → y +0 Λ ( y ) √ y − y = θ ∈ R , (5.9)lim y → y +0 U ( y ) √ y − y = θ ∈ R . (5.10)We parametrize the curve γ = { H = h } as in the previous section using aparameter σ. We will denote a parameter of this kind on the curves γ , γ by σ , σ respectively. Due to (4.2), (4.5), (5.1) it follows that: dσ i dy = 1 a ( y, V i ( y )) = 1 y + e U − Λ V i ( y ) y √ ( V i ( y )) y +1 , i = 1 , . (5.11)We will normalize the parameters σ i = σ i ( y ) by means of the condition: σ i ( y ) = 0 , i = 1 , . (5.12)Finally we remark that in order to obtain the functions U and Λ we needto prescribe the distribution Θ defined by (3.27). Using (4.8), (4.10) it thenfollows that: Θ ( y, V ) = β χ { y>y } e σ ( y ) (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) δ ( V − V ( y ))+ β χ { y>y } e σ ( y ) (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) δ ( V − V ( y )) (5.13)where χ { y>y } is the characteristic function of the half-plane { y > y } . Using(3.29), (3.30) it follows that:˜ ρ ( y ) = πβ χ { y>y } y " e σ ( y ) (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) q ( V ( y )) y + 1+ e σ ( y ) (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) q ( V ( y )) y + 1 , (5.14)˜ p ( y ) = πβ χ { y>y } y e σ ( y ) (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) ( V ( y )) q ( V ( y )) y + 1+ e σ ( y ) (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) ( V ( y )) q ( V ( y )) y + 1 (5.15)and the functions U and Λ can then be obtained using the equations (3.15),(3.16). 16ue to the dust-like character of the solutions considered in this paper, theyexhibit a singular behaviour for ˜ ρ and ˜ p at the radius y = y . This singularityis due to the fact that at this point the radial velocity of the particles, in self-similar variables, vanishes. However, since the motion of the trajectories afterthey reach the singularity continues in a smooth way, and since ˜ ρ and ˜ p areintegrable near this radius, this singularity can be expected to disappear if thedust-like assumption is relaxed and some thickness is given to the support ofthe distribution function in the phase space.The main result of this paper is the following: Theorem 1
There exists ε > small such that, for any y ∈ (0 , ε ) there exista value of β > and two curves γ , γ that can be parametrized as in (5.1)with the functions V ( y ) , V ( y ) as in (5.5), (5.6) satisfying (5.2), (5.3), (5.8),the functions U, Λ satisfying (3.15), (3.16) and (5.9), (5.10) with ˜ ρ, ˜ p as in(5.14), (5.15) and σ , σ solving (5.11), (5.12). Using Theorem 1 it is possible to obtain distributional solutions of the prob-lem (3.13)-(3.19). In order to make the definition of the distribution G in(4.1) precise we use (4.6), (4.7). Let us prescribe a smooth function ¯ A (Φ) inΦ ∈ (0 , ∞ ) . Taking into account (4.7) we can then define:¯ A ( σ, Φ) = ¯ A (cid:0) e − σ Φ (cid:1) . Using the structure of the curves γ , γ it would then follow that the distribution G in (4.1) would be given by: G ( y, V, Φ) = ¯ A (cid:0) e − σ ( y ) Φ (cid:1) χ { y>y } (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) δ ( V − V ( y ))+ ¯ A (cid:0) e − σ ( y ) Φ (cid:1) χ { y>y } (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) δ ( V − V ( y )) . (5.16)We then have the following result: Theorem 2
Suppose that the function ¯ A ( · ) ∈ C (0 , ∞ ) satisfies Z ∞ ¯ A (Φ) Φ d Φ = β . (5.17) Let us define a Radon measure G ∈ M ( R + × R × R + ) by means of (5.16) withthe functions V ( · ) , V ( · ) , σ ( · ) , σ ( · ) as in Theorem 1. Then the functions ˜ ρ, ˜ p defined (3.18), (3.19) belong to the spaces L ploc (0 , ∞ ) for ≤ p < . The functions Λ , U defined by means of (3.15)-(3.17) belong to W ,ploc (0 , ∞ ) for ≤ p < . The measure G satisfies (3.13) in the sense of distributions. emark 3 The space C (0 , ∞ ) is the space of compactly supported continu-ously differentiable functions and the space M ( R + × R × R + ) is the space ofRadon measures on R + × R × R + . It is not necessary to require A ( · ) to becompactly supported. Actually this condition could be replaced by assumptionsof fast enough decay near the origin and infinity. Remark 4
It is worth noticing that the functions ˜ ρ, ˜ p associated to the distri-bution G have an integrable singularity as y → y +0 . In the rest of this section we will prove Theorem 2. Theorem 1 will be provedin the remaining sections of the paper using a shooting argument and refinedasymptotics of the solutions for y small. The following auxiliary result will beused in the proof of Theorem 2 and it will be proved in Section 6. We remarkthat Theorem 2 will not be used in either the proof of Theorem 1 or that ofProposition 5 below. Proposition 5
The curves γ , γ whose existence has been proved in Theorem1 satisfy the following conditions: lim y → y +0 ∂H∂V ( y, V ( y )) √ y − y = L , lim y → y +0 ∂H∂V ( y, V ( y )) √ y − y = L (5.18) for some constants L < L . Proof of Theorem 2.
Using (3.27), (5.16) and (5.17) we obtain:Θ ( y, V ) = Z ∞ G Φ d Φ = β e σ ( y ) χ { y>y } (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) δ ( V − V ( y ))+ β e σ ( y ) χ { y>y } (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) δ ( V − V ( y )) . (5.19)We can then compute ˜ ρ, ˜ p using (3.29), (3.30):˜ ρ ( y ) = πβ χ { y>y } y e σ ( y ) q y ( V ( y )) (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) + e σ ( y ) q y ( V ( y )) (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) , (5.20)˜ p ( y ) = πβ χ { y>y } y e σ ( y ) ( V ( y )) (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) q y ( V ( y )) + e σ ( y ) ( V ( y )) (cid:12)(cid:12) ∂H∂V ( y, V ( y )) (cid:12)(cid:12) q y ( V ( y )) . (5.21)18sing (5.8)-(5.10), (5.18), we obtain: | ˜ ρ ( y ) | + | ˜ p ( y ) | ≤ Cχ { y>y } √ y − y (5.22)whence the estimate ˜ ρ, ˜ p ∈ L ploc (0 , ∞ ) , ≤ p <
2, in the theorem follows. Onthe other hand (3.15)-(3.17) imply:Λ = −
12 log (cid:18) − πy Z yy ξ ˜ ρ ( ξ ) dξ (cid:19) , (5.23) U = Z yy (cid:2)(cid:0) πξ ˜ p ( ξ ) + 1 (cid:1) e ξ ) − (cid:3) ξ dξ. (5.24)Due to Theorem 1 the functions Λ , U are bounded for any finite value y > . On the other hand, (5.23), (5.24) imply Λ , U ∈ W ,ploc (0 , ∞ ) , ≤ p < . In order to conclude the proof of Theorem 2 it only remains to prove that G solves (3.13) in the sense of distributions. This is equivalent to showing that: Z R + × R × R + " − ( yϕ ) y + ( V ϕ ) V − (Φ ϕ ) Φ − (cid:18) e U − Λ V ˆ E ϕ (cid:19) y + (cid:18)(cid:18) y Λ y V + e U − Λ U y ˆ E − e U − Λ y ˆ E (cid:19) ϕ (cid:19) V (cid:21) GdydV d
Φ= 0 (5.25)for any ϕ = ϕ ( y, V, Φ) ∈ C ∞ ( R + × R × R + ) . Using (5.16) we can rewrite (5.25)as: J ≡ X i =1 Z ∞ y Z ∞ " − ( yϕ ) y + ( V ϕ ) V − (Φ ϕ ) Φ − (cid:18) e U − Λ V ˆ E ϕ (cid:19) y + (5.26) (cid:18)(cid:18) y Λ y V + e U − Λ U y ˆ E − e U − Λ y ˆ E (cid:19) ϕ (cid:19) V (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ( y,V i ( y ) , Φ) ¯ A (cid:0) e − σ i ( y ) Φ (cid:1)(cid:12)(cid:12) ∂H∂V ( y, V i ( y )) (cid:12)(cid:12) d Φ dy = 0and making the change of variables e − σ i ( y ) Φ → Φ we can transform J into: J ≡ X i =1 Z ∞ y Z ∞ " − ( yϕ ) y + ( V ϕ ) V − (Φ ϕ ) Φ − (cid:18) e U − Λ V ˆ E ϕ (cid:19) y + (cid:18)(cid:18) y Λ y V + e U − Λ U y ˆ E − e U − Λ y ˆ E (cid:19) ϕ (cid:19) V (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ( y,V i ( y ) , Φ e σi ( y ) ) ¯ A (Φ) e σ i ( y ) (cid:12)(cid:12) ∂H∂V ( y, V i ( y )) (cid:12)(cid:12) dyd Φ(5.27)19otice that we can write: F ≡ − ( yϕ ) y + ( V ϕ ) V − (Φ ϕ ) Φ − (cid:18) e U − Λ V ˆ E ϕ (cid:19) y + (cid:18)(cid:18) y Λ y V + e U − Λ U y ˆ E − e U − Λ y ˆ E (cid:19) ϕ (cid:19) V = − yϕ y + V ϕ V − (Φ ϕ ) Φ − e U − Λ V y p V y ϕ ! y + y Λ y V + e U − Λ U y y p V y − e U − Λ y p V y ! ϕ ! V and, using Leibniz’s rule: F = − y Λ y ϕ − ye Λ (cid:0) e − Λ ϕ (cid:1) y + V ϕ V − Φ ϕ Φ − ϕ − U y e U − Λ V y p V y ϕ − e U − Λ V p V y ϕ + e U − Λ V y (1 + V y ) ϕ − e U V y p V y (cid:0) e − Λ ϕ (cid:1) y + y Λ y + U y e U − Λ V y p V y + e U − Λ V (1 + V y ) ! ϕ + y Λ y V + e U − Λ U y y p V y − e U − Λ y p V y ! ϕ V . After some cancellations: F ( y, V, Φ) = − ye Λ + e U V y p V y ! (cid:0) e − Λ ϕ (cid:1) y − Φ e Λ (cid:0) e − Λ ϕ (cid:1) Φ − ϕ + y Λ y V e Λ + V e Λ + e U U y y p V y − e U y p V y ! (cid:0) e − Λ ϕ (cid:1) V . (5.28)Then (5.27) can be rewritten as: X i =1 Z ∞ y Z ∞ F (cid:16) y, V i ( y ) , Φ e σ i ( y ) (cid:17) ¯ A (Φ) e σ i ( y ) (cid:12)(cid:12) ∂H∂V ( y, V i ( y )) (cid:12)(cid:12) d Φ dy = 0 . Due to Proposition 5 as well as the fact that the curves γ , γ are globallydefined it follows that: (cid:12)(cid:12)(cid:12)(cid:12) ∂H∂V ( y, V i ( y )) (cid:12)(cid:12)(cid:12)(cid:12) = ( − i − ∂H∂V ( y, V i ( y )) . Then: J = X i =1 ( − i − Z ∞ y Z ∞ F (cid:16) y, V i ( y ) , Φ e σ i ( y ) (cid:17) ¯ A (Φ) e σ i ( y ) ∂H∂V ( y, V i ( y )) d Φ dy. (5.29)20sing (3.25) and (5.28) we obtain: F ( y, V, Φ) ye Λ + e U V y √ V y +1 = − (cid:0) e − Λ ϕ (cid:1) y − Φ y + e U − Λ V y √ V y +1 (cid:0) e − Λ ϕ (cid:1) Φ − y + e U − Λ V y √ V y +1 (cid:0) e − Λ ϕ (cid:1) + 1 y + e U − Λ V y √ V y +1 (cid:18) y Λ y V + V + e U − Λ U y y p V y (cid:19) (cid:0) e − Λ ϕ (cid:1) V − e U − Λ y + e U − Λ V y √ V y +1 y p V y (cid:0) e − Λ ϕ (cid:1) V . Equations (4.3), (4.4) and (5.11) give: dV i dy ( y ) = − y + e U − Λ V y √ V y +1 (cid:18) y Λ y V + V + e U − Λ U y y p V y − e U − Λ y p V y ! . (5.30)Therefore F (cid:16) y, V i ( y ) , Φ e σ i ( y ) (cid:17) e σ i ( y ) ∂H∂V ( y, V i ( y ))= e σ i ( y ) (cid:20) − (cid:0) e − Λ ϕ (cid:1) y − Φ dσ i dy (cid:0) e − Λ ϕ (cid:1) Φ − dσ i dy (cid:0) e − Λ ϕ (cid:1) − dV i dy ( y ) (cid:0) e − Λ ϕ (cid:1)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) V ( y,V i ( y ) , Φ e σi ( y ) ) . It then follows, using the chain rule that: ddy (cid:16) e σ i ( y ) e − Λ( y ) ϕ (cid:16) y, V i ( y ) , Φ e σ i ( y ) (cid:17)(cid:17) = − F (cid:16) y, V i ( y ) , Φ e σ i ( y ) (cid:17) e σ i ( y ) ∂H∂V ( y, V i ( y )) . Formula (5.29) then becomes: J = X i =1 ( − i − Z ∞ y Z ∞ ¯ A (Φ) ddy (cid:16) e σ i ( y ) e − Λ( y ) ϕ (cid:16) y, V i ( y ) , Φ e σ i ( y ) (cid:17)(cid:17) d Φ dy or, equivalently: J = X i =1 ( − i − Z ∞ ¯ A (Φ) (cid:16) e σ i ( y ) e − Λ( y ) ϕ (cid:16) y , V i (cid:0) y +0 (cid:1) , Φ e σ i ( y ) (cid:17)(cid:17) d Φ21nd using (5.9), (5.10), (5.12): J ≡ X i =1 ( − i − Z ∞ ¯ A (Φ) ϕ ( y , V i ( y ) , Φ) d Φ . Due to the fact that ϕ (cid:0) y , V i (cid:0) y +0 (cid:1) , Φ (cid:1) = ϕ ( y , V , Φ) for i = 1 , J = 0and (5.26) follows. This concludes the proof of the theorem.We now remark that it is possible to derive some detailed information aboutthe behaviour of the curves γ , γ as y → ∞ . Theorem 6
Suppose that the curves γ , γ are as in Theorem 1. Then, thefollowing asymptotic formulas hold: U = log (cid:18) yy (cid:19) + log (cid:18)q − y (cid:19) + o (1) as y → ∞ , Λ → log (cid:16) √ (cid:17) as y → ∞ ,V = − y p − y )(1 − y ) y (1 + o (1)) as y → ∞ ,V = − p − y √ y C y (cid:18) y y (cid:19) (1 + o (1)) as y → ∞ for a suitable constant C ∈ R . Notice that the asymptotic behaviour of the solutions in Theorem 6 showsthat the support of these solutions approaches the line { V = 0 } away from theself-similar region (i.e. for y → ∞ ). This is the one of the main differencesbetween the solutions described in this paper and the ones in [17].It is relevant to notice that the spacetime described by the solutions inTheorem 6 exhibits curvature singularities and not just coordinate singularities.To this end we use Kretschmann scalar (cf. [25]): R αβγδ R αβγδ = 4 K + 16 m r + 12 r − ∇ a ∇ b r ∇ a ∇ b r where K is the gaussian curvature of the quotient of the spacetime by thesymmetry group and m is the Hawking mass that can be computed by meansof: m = r − ∂ a r∂ a r ) . Combining (2.1), (3.11), (3.12) we obtain the following self-similar form for theHawking mass: m = r (cid:16) − e − ( r ( − t ) ) (cid:17) m ∼ r r ( − t ) sufficiently large . On the other hand, the last term in the Kretschmann scalar can be written as(cf. [10], Appendix A):24 r − (cid:18) r ( k − ∇ b r ∇ c r ) + 2 πr tr T (cid:19) +96 π (cid:18) T ab − tr T g ab (cid:19) (cid:18) T ab − tr T g ab (cid:19) . The last term turns out to be positive for any matter model satisfying the dom-inant energy condition, which includes in particular the case of Vlasov matter.Therefore R αβγδ R αβγδ ≥ m r and so the curvature becomes singular as r → r ( − t ) . We remark that the solutions which have been derived do not provide anexample of violation of the cosmic censorship hypothesis for Vlasov matter, be-cause the spacetimes concerned are not asymptotically flat as r → ∞ . Moreover,it turns out that the region contained inside the light cone reaching the singularpoint at r = 0 , t = 0 − in the spacetime described by Theorem 6 is dependent onthe data on the whole region with 0 ≤ r < ∞ . This implies that a gluing of thisspacetime with another one causally disconnected from the singular point isnot possible, because this would require doing some gluing along regions where r = ∞ . In order to check these statements it is convenient to rewrite the metric(2.1) in double null coordinates. Notice that (2.1), (3.11) and (3.12) yield thefollowing self-similar structure for the metric: ds = − e U ( r ( − t ) ) dt + e ( r ( − t ) ) dr + r (cid:0) dθ + sin θdϕ (cid:1) . The double null coordinates are then just the constants of integration associatedto the pair of differential equations: − e U ( r ( − t ) ) dt + e Λ ( r ( − t ) ) dr = 0 ,e U ( r ( − t ) ) dt + e Λ ( r ( − t ) ) dr = 0 . The solutions of these equations can be written in terms of two integrationconstants u and v that will define the double null coordinates. The particularchoice of coordinates has been made in order to obtain u and v taking values incompact sets: arctanh ( u ) = log ( − t ) + Z y e Λ( ξ ) − U ( ξ ) ξe Λ( ξ ) − U ( ξ ) dξ arctanh ( v ) = log ( − t ) − Z y e Λ( ξ ) − U ( ξ ) − ξe Λ( ξ ) − U ( ξ ) dξ In the region close to the centre (i.e. y <<
1) the structure of the metricis similar to Minkowski. On the other hand, Theorem 6 yields the following23symptotics for r >> ( − t ) :arctanh ( v ) ∼ log ( − t ) + √ y p − y log (cid:18) r ( − t ) (cid:19) , arctanh ( u ) ∼ − log ( − t ) + √ y p − y log (cid:18) r ( − t ) (cid:19) . The light cone approaching the singular point is described in these coordinatesby the line u = 1 . Notice along such a line, for v of order one we would have r = ∞ , whence the assertion above follows.For these reasons a spacetime behaving asymptotically as Minkowski can-not be obtained gluing the self-similar solution obtained in this paper with aspacetime causally disconnected from the singular point. This kind of gluingmight be possible for non self-similar solutions of the Einstein equations behav-ing asymptotically near the singular point like those described in this paper.However, such an analysis is beyond the scope of this paper. The strategy used to prove Theorem 1 is the following. We first transformthe original problem (3.15), (3.16), (5.2), (5.3), (5.5), (5.6), (5.8)-(5.12), (5.14),(5.15) into a family of four-dimensional autonomous systems depending on theparameter β by means of a change of variables. It will be shown that provingTheorem 1 is equivalent to finding an orbit for this system connecting twospecific points P , P of the four-dimensional phase space. The point P isa unstable saddle point with an associated three-dimensional stable manifold M = M ( β ) that can be described in detail in the limit y →
0. A shootingargument will show that for a suitable choice of the parameter β the manifold M ( β ) contains the point P . In the rest of this section we give the details ofthis argument.
Instead of the set of variables ( y, U, Λ , V i , σ i ) it is more convenient to use theset of variables ( s, u, Λ , ζ i , Q i ) where: s = log (cid:18) yy (cid:19) , U = log (cid:18) yy (cid:19) + u , ζ i = yV i , Q i = y y e σ i , i = 1 , . (6.1)24hen, the evolution equations (3.15), (3.16), (5.4), (5.11) become: e u q ζ i + 1 + y ζ i e Λ = q − y , i = 1 , , ζ < ζ , (6.2) dQ i ds = − e u Q i ζ i (cid:20) y e Λ q ( ζ i ) + 1 + ζ i e u (cid:21) , i = 1 , , (6.3) e − (2Λ s −
1) + 1 = θ Q (cid:2) ζ + 1 (cid:3)(cid:12)(cid:12)(cid:12) ζ e u + y e Λ p ζ + 1 (cid:12)(cid:12)(cid:12) + Q (cid:2) ζ + 1 (cid:3)(cid:12)(cid:12)(cid:12) ζ e u + y e Λ p ζ + 1 (cid:12)(cid:12)(cid:12) , (6.4) e − (2 u s + 3) − θ Q ( ζ ) (cid:12)(cid:12)(cid:12) ζ e u + y e Λ p ζ + 1 (cid:12)(cid:12)(cid:12) + Q ( ζ ) (cid:12)(cid:12)(cid:12) ζ e u + y e Λ p ζ + 1 (cid:12)(cid:12)(cid:12) (6.5)where θ = 16 π β y . (6.6)The initial conditions (3.17), (5.12) imply: u = 0 , Λ = 0 , Q i = 1 , i = 1 , s = 0 . (6.7)Notice that the system (6.3)-(6.5) with ζ i as in (6.2) is a four-dimensional au-tonomous system of equations for the unknown functions ( Q , Q , Λ , u ) . Noticehowever that the system seems to becomes singular if the variables ( Q , Q , Λ , u )approach the values in (6.7) due to the vanishing of the denominators in (6.4),(6.5). To treat these singularities we rewrite the terms (cid:20) y e Λ q ( ζ i ) + 1 + ζ i e u (cid:21) . Notice that (6.2) implies: ζ i = 1 (cid:0) − y e − u ) (cid:1) (cid:20) − y q − y e Λ − u ∓ Z (cid:21) , i = 1 , , (6.8) Z = q ( e − u (1 − y ) − (cid:0) − y e − u ) (cid:1) + y (1 − y ) e − u ) . (6.9)Then: y e Λ q ( ζ i ) + 1 + ζ i e u = ∓ e u Z , i = 1 , dQ ds = Q ζ Z , (6.10) dQ ds = − Q ζ Z , (6.11) e − (2Λ s −
1) + 1 = θe − u (cid:20) Q Z (cid:2) ζ + 1 (cid:3) + Q Z (cid:2) ζ + 1 (cid:3)(cid:21) , (6.12) e − (2 u s + 3) − θe − u (cid:20) Q ζ Z + Q ζ Z (cid:21) . (6.13)We now eliminate the variables Λ , u in (6.3)-(6.5) and replace them by thefunctions Z and G where Z is as in (6.9) and G is defined by means of: G = e − . (6.14)Then (6.12) becomes: G s = 1 − G − θe − u (cid:20) Q Z (cid:2) ζ + 1 (cid:3) + Q Z (cid:2) ζ + 1 (cid:3)(cid:21) . (6.15)On the other hand (6.9) implies: e − u = Z + 1[(1 − y ) + y e ] = (cid:0) Z + 1 (cid:1) G [ G + y (1 − G )] (6.16)whence: u = −
12 log (cid:0) Z + 1 (cid:1) G [ G + y (1 − G )] ! . Differentiating this formula we obtain: u s = − ZZ s ( Z + 1) − y G s [ G + y (1 − G )] G .
Eliminating u s from this formula using (6.13), (6.15) we obtain: ZZ s = (cid:18) − G − ∆ (cid:19) (cid:0) Z + 1 (cid:1) (6.17)where: ∆ ≡ y G s [ G + y (1 − G )] G + θe − u G " Q ( ζ ) Z + Q ( ζ ) Z . Using (6.15) it then follows, after some computations, that:4 GZ (cid:2) G + y (1 − G ) (cid:3) ∆ = 2 (1 − G ) y Z (6.18)+ θe − u (cid:2) − y (cid:2) Q (cid:2) ζ + 1 (cid:3) + Q (cid:2) ζ + 1 (cid:3)(cid:3) + h Q ( ζ ) + Q ( ζ ) i (cid:2) G + y (1 − G ) (cid:3)i . h − y (cid:2) Q (cid:2) ζ + 1 (cid:3) + Q (cid:2) ζ + 1 (cid:3)(cid:3) + h Q ( ζ ) + Q ( ζ ) i (cid:2) G + y (1 − G ) (cid:3)i = Q (cid:2) ζ − y (cid:0) ζ + 1 (cid:1)(cid:3) + Q (cid:2) ζ − y (cid:0) ζ + 1 (cid:1)(cid:3) + (cid:0) − y (cid:1) h Q ( ζ ) + Q ( ζ ) i ( G − . (6.19)Using (6.8) we obtain: (cid:2) ζ i − y (cid:0) ζ i + 1 (cid:1)(cid:3) = (cid:0) − y (cid:1) Z (cid:0) − y e − u ) (cid:1) ± y (cid:0) − y (cid:1) e Λ − u (cid:0) − y e − u ) (cid:1) Z + y " (cid:0) − y (cid:1) e − u ) (cid:0) − y e − u ) (cid:1) − , i = 1 , . (6.20)Plugging (6.20) into (6.19) it then follows that: h − y (cid:2) Q (cid:2) ζ + 1 (cid:3) + Q (cid:2) ζ + 1 (cid:3)(cid:3) + h Q ( ζ ) + Q ( ζ ) i (cid:2) G + y (1 − G ) (cid:3)i = (cid:0) − y (cid:1) Z (cid:0) − y e − u ) (cid:1) (cid:0) Q + Q (cid:1) + 2 y (cid:0) − y (cid:1) e Λ − u (cid:0) − y e − u ) (cid:1) (cid:0) Q − Q (cid:1) Z + y " (cid:0) − y (cid:1) e − u ) (cid:0) − y e − u ) (cid:1) − Q + Q (cid:1) + (cid:0) − y (cid:1) h Q ( ζ ) + Q ( ζ ) i ( G − − G ) y G [ G + y (1 − G )]+ θe − u G [ G + y (1 − G )] " (cid:0) − y (cid:1) Z (cid:0) − y e − u ) (cid:1) (cid:0) Q + Q (cid:1) + 2 y (cid:0) − y (cid:1) e Λ − u (cid:0) − y e − u ) (cid:1) (cid:0) Q − Q (cid:1) + 1 Z Φ (6.21)where: Φ = y " (cid:0) − y (cid:1) e − u ) (cid:0) − y e − u ) (cid:1) − Q + Q (cid:1) + (cid:0) − y (cid:1) h Q ( ζ ) + Q ( ζ ) i ( G − . (6.22)In order to obtain analytic solutions it is convenient to introduce the change ofvariables: ds = 2 GZdχ , χ = 0 at s = 0 . (6.23)27hen the system (6.10), (6.11), (6.15), (6.17) becomes: dQ dχ = 2 GQ ζ , (6.24) dQ dχ = − GQ ζ , (6.25) dGdχ = 2 G (cid:20) Z (1 − G ) − θe − u (cid:2) Q (cid:2) ζ + 1 (cid:3) + Q (cid:2) ζ + 1 (cid:3)(cid:3)(cid:21) , (6.26) dZdχ = (3 G − − G ∆) (cid:0) Z + 1 (cid:1) (6.27)with the initial conditions: Q = Q = 1 , G = 1 , Z = 0 , at χ = 0 . (6.28)We can further simplify Φ in (6.22) using (6.8):Φ = y " (cid:0) − y (cid:1) e − u ) (cid:0) − y e − u ) (cid:1) − Q + Q (cid:1) (6.29)+ (cid:0) − y (cid:1) ( G − (cid:0) − y e − u ) (cid:1) hh y (cid:0) − y (cid:1) e − u ) + Z i (cid:0) Q + Q (cid:1)i +2 y q − y Ze Λ − u (cid:0) Q − Q (cid:1)(cid:21) . In order to identify the behaviour of Φ as Z → " (cid:0) − y (cid:1) e − u ) (cid:0) − y e − u ) (cid:1) − = 1 (cid:0) − y e − u ) (cid:1) h(cid:0) − y (cid:1) (cid:16) e − u ) − (cid:17) + 2 (cid:0) − y (cid:1) y (cid:16) e − u ) − (cid:17) − y (cid:16) e − u ) − (cid:17) (cid:21) . Then (6.29) becomes:Φ = y (cid:0) − y (cid:1) (cid:0) Q + Q (cid:1)(cid:0) − y e − u ) (cid:1) h(cid:0) − y (cid:1) (cid:16) e − u ) − (cid:17) + 2 y (cid:16) e − u ) − (cid:17) + (cid:0) − y (cid:1) ( G − e − u ) i + (cid:0) − y (cid:1) ( G − (cid:0) − y e − u ) (cid:1) (cid:20) Z (cid:0) Q + Q (cid:1) + 2 y q − y Ze Λ − u (cid:0) Q − Q (cid:1)(cid:21) − y (cid:0) Q + Q (cid:1)(cid:0) − y e − u ) (cid:1) (cid:16) e − u ) − (cid:17) . (6.30)28n order to simplify this formula we write, using (6.14), (6.16): h(cid:0) − y (cid:1) (cid:16) e − u ) − (cid:17) + 2 y (cid:16) e − u ) − (cid:17) + (cid:0) − y (cid:1) ( G − e − u ) i = − (cid:0) − y (cid:1) (cid:0) − e − u (cid:1) + 2 y (cid:16) e − u ) − (cid:17) = − (cid:0) − y (cid:1) − (cid:18) G [ G + y (1 − G )] (cid:19) ! + 2 y (cid:18) G + y (1 − G )] − (cid:19) + (cid:0) − y (cid:1) (cid:0) Z + Z (cid:1) (cid:18) G [ G + y (1 − G )] (cid:19) + 2 y (cid:18) Z [ G + y (1 − G )] (cid:19) , (cid:16) e − u ) − (cid:17) = Z + (1 − G ) (cid:0) − y (cid:1) [ G + y (1 − G )] . Plugging these formulas into (6.30) we obtain, after some computations:Φ Z = y (cid:0) − y (cid:1) (cid:0) Q + Q (cid:1)(cid:0) − y e − u ) (cid:1) "(cid:0) − y (cid:1) (cid:0) Z + Z (cid:1) (cid:18) G [ G + y (1 − G )] (cid:19) +2 y (cid:18) Z [ G + y (1 − G )] (cid:19)(cid:21) + (cid:0) − y (cid:1) ( G − (cid:0) − y e − u ) (cid:1) (cid:20) Z (cid:0) Q + Q (cid:1) + 2 y q − y e Λ − u (cid:0) Q − Q (cid:1)(cid:21) − y (cid:0) Q + Q (cid:1)(cid:0) − y e − u ) (cid:1) " (cid:0) − y (cid:1) Z (1 − G ) + Z [ G + y (1 − G )] . (6.31)Summarizing, we have transformed the original problem (3.15), (3.16), (5.4),(5.11) into the system of equations (6.24)-(6.27) with ∆ as in (6.21), Φ Z as in(6.31), ζ i as in (6.8) and Λ , u given by (6.14), (6.16). The initial data for( Q , Q , G, Z ) are as in (6.28).Some of the forms that we have derived for the ODE problems above are moreconvenient for describing the solutions in different regions of the phase space.We will change freely between the different groups of equivalent variables in thefollowing. γ , γ . With the reformulation of the problem obtained in the previous subsection theexistence of the curves γ , γ in a neighbourhood of the point ( y , V ) can beobtained using standard ODE theory. 29 roposition 7 For any y ∈ (0 , and any β > there exist δ > and twocurves γ , γ that can be parametrized as γ i = { ( y, V ) : y < y < y + δ , V = V i ( y ) } , i = 1 , with the functions V ( y ) , V ( y ) as in (5.5), (5.6) satisfying (5.2), (5.3), (5.8)the functions U, Λ satisfying (3.15), (3.16) and (5.9), (5.10) with ˜ ρ, ˜ p as in(5.14), (5.15) and σ , σ solving (5.11), (5.12). Proof.
The arguments in Subsection 6.1 show that the proposition followsfrom proving local existence and uniqueness for (6.24)-(6.27) with initial data(6.28). Since the right-hand side of (6.24)-(6.27) is analytic in a neighbourhoodof ( Q , Q , G, Z ) = (1 , , ,
0) it follows that there exists a unique solution of(6.28), (6.24)-(6.27) on an interval of the form 0 < χ < δ for some δ > . Moreover, for such a solution ∆ → χ → + , whence Z ∼ χ as χ → + . Therefore (6.23) yields: s ∼ χ as χ → + , χ ∼ r s s → + ,Z ∼ √ s as s → + . (6.33)Using (6.1) it follows that: s ∼ y − y y as y → y +0 . (6.34)Combining then (6.1) and (6.8) we obtain (5.8). The asymptotics (5.9), (5.10)follows from the asymptotics for G, Z in an analogous way.Moreover, we can prove Proposition 5 in a similar way.
Proof of Proposition 5.
It follows from (3.25), (6.1), (6.33), (6.34).We notice for further reference that we have also proved the following result:
Proposition 8
There exists a unique solution of the system (6.3)-(6.5) with ζ i as in (6.2) and initial data ( Q , Q , Λ , u ) = (1 , , , as s → + . In order to study the steady states of (6.24)-(6.27) it is more convenient to usethe form of the equations in (6.2)-(6.5). Then the steady states are characterized30y: Q i ζ i = 0 i = 1 , , (6.35) − e − + 1 = θ Q (cid:12)(cid:12)(cid:12) ζ e u + y e Λ p ζ + 1 (cid:12)(cid:12)(cid:12) (cid:2) ζ + 1 (cid:3) + Q (cid:12)(cid:12)(cid:12) ζ e u + y e Λ p ζ + 1 (cid:12)(cid:12)(cid:12) h ( ζ ) + 1 i , (6.36)3 e − − θ Q ( ζ ) (cid:12)(cid:12)(cid:12) ζ e u + y e Λ p ζ + 1 (cid:12)(cid:12)(cid:12) + Q ( ζ ) (cid:12)(cid:12)(cid:12) ζ e u + y e Λ p ζ + 1 (cid:12)(cid:12)(cid:12) . (6.37)The first and third equations imply:3 e − − . (6.38)Then, the second equation reduces to:23 = θ Q (cid:12)(cid:12)(cid:12) ζ e u + y e Λ p ζ + 1 (cid:12)(cid:12)(cid:12) + Q (cid:12)(cid:12)(cid:12) ζ e u + y e Λ p ζ + 1 (cid:12)(cid:12)(cid:12) . (6.39)Notice that (6.39) implies that at least one of the variables Q , Q is differentfrom zero at the steady state. Suppose that both of them are different fromzero. Then ζ = ζ = 0 , whence, using e u q ζ i + 1 + y ζ i e Λ = q − y , i = 1 , , ζ ≤ ζ it follows that: e u = q − y (6.40)and (6.39) reduces to: (cid:0) Q + Q (cid:1) = 4 y e Λ θ = 4 y √ θ . This defines a family of steady states. Local analysis near these solutionsindicates that they are reached for finite values of y. Since we are interested insolutions defined for arbitrarily large values of y > y a more detailed analysisof these solutions will not be pursued here. We will then restrict our analysisto the solutions for which Q Q = 0.Suppose that Q = 0. Then ζ = 0. (6.40) implies: q ζ + 1 + y ζ e Λ p − y = 1 ,ζ = p − y y e Λ (cid:20) − q ζ + 1 (cid:21) < . ζ ≤ ζ . Therefore for solutions with Q Q = 0 we must have Q = 0 whence ζ = 0 . Then (6.40) is satisfied and (6.39) yields: Q = s √ θ y = 2 √ y √ θ . We remark that for this solution: ζ = − he Λ ∞ ( h − e ∞ ) = − y p − y )1 − y . In order to have ζ < ζ = 0 we need y ∈ (cid:0) , (cid:1) . Summarizing, for each y ∈ (cid:0) , (cid:1) the system (6.24)-(6.27) has the followingsteady state: Q = Q , ∞ = 0 , (6.41) Q = Q , ∞ = 2 √ y √ θ, (6.42)Λ = Λ ∞ = log (3)2 , (6.43) u = u ∞ = log (cid:18)q − y (cid:19) . (6.44)We also introduce the following notation for further reference: ζ , ∞ = − he Λ ∞ ( h − e ∞ ) = − y p − y )1 − y , (6.45) ζ , ∞ = 0 . (6.46) The main result that we prove in this subsection is the following:
Theorem 9
For each y ∈ (cid:0) , (cid:1) the point P = ( Q , ∞ , Q , ∞ , Λ ∞ , u ∞ ) definedby (6.41)-(6.44) is an unstable hyperbolic point of the system (6.2)-(6.5). Thecorresponding stable manifold of the point ( Q , ∞ , Q , ∞ , Λ ∞ , u ∞ ) that will bedenoted by M θ is three-dimensional and it is tangent at this point to the subspacegenerated by the vectors , − ( − y ) y √ θ − , − √ − y y √ θ − √ − y y . (6.47)32 roof. The key ingredient in the proof of this theorem is the linearizationof the system (6.2)-(6.5) around the point ( Q , ∞ , Q , ∞ , Λ ∞ , u ∞ ) . Let us write:Λ = Λ ∞ + L,u = u ∞ + ν,Q = Q , ∞ + q = q ,Q = Q , ∞ + q . Neglecting terms quadratic in | L | + | ν | + | q | + | q | we obtain, after some tedious,but mechanical computations, the following linearized problem: dq ds = − h ( h − q = − (cid:0) − y (cid:1) (1 − y ) q , (6.48) dq ds = 2 (cid:0) − y (cid:1) √ θy ν, (6.49) L s = 3 √ θ √ y q + (cid:0) − y (cid:1) y ν − L, (6.50) ν s = 3 L. (6.51)Looking for solutions of the linearized problem with the form: e γs A A A A we obtain the following possible values of γ with their corresponding eigenvec-tors: γ = − (cid:0) − y (cid:1) (1 − y ) ↔ A A A A = ,γ = − ↔ A A A A = − ( − y ) y √ θ − ,γ = − p (1 − y ) y ↔ A A A A = − √ − y y √ θ − √ − y y , = p (1 − y ) y ↔ √ − y y √ θ √ − y y . The theorem then follows from standard results for stable manifolds (cf. forinstance [4], [19]).
Our goal now is to obtain a trajectory connecting the point ( Q , Q , Λ , u ) =(1 , , ,
0) at s = 0 with the point P at s = ∞ for a suitable value of θ (orequivalently β ). Let us remark that such a trajectory would satisfy the re-quirements in Theorem 1. Indeed, notice that such a trajectory behaves nearthe point ( y , V ) as stated in Theorem 1 due to Proposition 7. On the otherhand, such a trajectory would belong to the stable manifold of the point P andtherefore its asymptotic behaviour as s → ∞ would be given by: Q Q Λ u ∼ Q , ∞ Q , ∞ Λ ∞ u ∞ + C e − s − y ) √ θy − + C e − ( − y ) ( − y ) s + ... for sufficiently small y (cf. [4]). Notice that the smallness of y guarantees thatthe last term yields a contribution larger for s → ∞ than the first quadraticcorrections if C = 0 . Using (6.1) we obtain the following asymptotics for the original set of vari-ables U, Λ , σ i , V i , i = 1 , U = log (cid:18) yy (cid:19) + u ∼ log (cid:18) yy (cid:19) + log (cid:18)q − y (cid:19) + o (1) as y → ∞ , Λ → log (cid:16) √ (cid:17) as y → ∞ ,e σ ∼ C (cid:18) yy (cid:19) − y ( − y ) as y → ∞ ,e σ ∼ Q , ∞ (cid:18) yy (cid:19) as y → ∞ ,V ∼ ζ , ∞ y = − y p − y )(1 − y ) y as y → ∞ ,V ∼ − p − y √ y C y (cid:18) y y (cid:19) as y → ∞ .
34n particular these formulas prove Theorem 6. M θ for small y . Since the stable manifold M θ is three-dimensional we cannot expect the point( Q , Q , Λ , u ) = (1 , , ,
0) to belong to M θ for generic values of θ. The intuitiveidea of the proof which follows is to show that the manifold M θ divides the set { < G < , Z > , Q i > , i = 1 , } into two different regions. If the point(1 , , ,
0) lies on different sides of M θ for different values of θ then by continuitythere must exist a value θ ∗ of θ such that (1 , , , ∈ M θ . In the rest of thepaper we will obtain approximations to the manifold M θ for y small that willshow that the point (1 , , ,
0) lies on different sides of M θ for large positivevalues of θ and small positive values of θ. More precisely, the main result of thissubsection is the following:
Theorem 10
There exists ¯ y small enough such that, for any y in the interval [0 , ¯ y ] there exists θ ∗ = θ ∗ ( y ) > such that (1 , , , ∈ M θ ∗ . Proof.
In order to prove Theorem 10 it is convenient to use the coordinates( Q , Q , G, Z ) (cf. (6.9), (6.14)). These variables satisfy the system of equations(6.24)-(6.27). The steady state P = P ( y ) is given in these coordinates by: P = ( Q , ∞ , Q , ∞ , G ∞ , Z ∞ ) = , √ y √ θ , , s y (1 − y ) ! . (6.52)The point P depends continuously on y if y ∈ (cid:2) , (cid:3) . If y = 0 the system(6.24)-(6.27) becomes: dQ dζ = − GZQ , (6.53) dQ dζ = − GZQ , (6.54) dGdζ = 2 G Z (1 − G ) − θ (cid:2) Z + 1 (cid:3) (cid:0) Q + Q (cid:1) , (6.55) dZdζ = (cid:18) G − − θe − u Z (cid:0) Q + Q (cid:1)(cid:19) (cid:0) Z + 1 (cid:1) . (6.56)Theorem 9 shows that the point P ( y ) is hyperbolic for y ∈ (cid:0) , (cid:3) with athree-dimensional stable manifold M θ = M θ ( y ) . On the other hand two of35he eigenvalues associated to the linearization around P of the system (6.24)-(6.27) degenerate for y = 0 . More precisely, let us write G = + g. Since P (0) = (cid:0) , , , (cid:1) we obtain the following linearization of (6.53)-(6.56) near P (0): dQ dζ = 0 , dQ dζ = 0 , dGdζ = 4 Z , dZdζ = 3 g. The corresponding eigenvalues are n , , − √ , √ o and the correspondingeigenvectors are , , − √ , √ . Standard results(cf. [4]) show the existence of a centre-stable manifold that will be denoted by M θ (0) that is invariant under the flow defined by the system (6.53)-(6.56) andis tangent at P (0) to the plane spanned by , , − √ . Classical results (cf. [4]) then show that it is possible to obtain a con-tinuously differentiable four-dimensional manifold M θ, ext ⊂ (cid:2) , (cid:3) × R , with( y , Q , Q , G, Z ) ∈ M ext such that: M θ, ext ∩ { y = b } = M θ ( b ) (6.57)for any b ∈ (cid:0) , (cid:1) . Indeed, the manifold M θ, ext is any centre-stable manifoldat the point ( y , Q , Q , G, Z ) = (cid:0) , , , , (cid:1) associated to the system (6.24)-(6.27) complemented with the additional equation dy dζ = 0 . (6.58)More precisely, we make use of the fact that the dynamical system of interesthas a smooth extension to an open neighbourhood of the stationary point un-der consideration. The manifold M θ, ext is the intersection of a centre-stablemanifold for the extended system with the subset defined by the inequality y ≥
0. The manifold M ext contains all the points of the form ( y , P ( y ))with y ∈ (cid:2) , (cid:3) since they remain in a neighbourhood of (cid:0) , , , , (cid:1) for ar-bitrary times. Moreover, the manifolds M θ, ext ∩ { y = b } are invariant underthe flow (6.24)-(6.27) and since they are formed by points that remain in aneighbourhood of (cid:0) , , , , (cid:1) for arbitrarily long times, it follows from (6.58)that the points in M θ, ext ∩ { y = b } are contained in the stable manifold asso-ciated to the point P ( y ) . The uniqueness of the stable manifold then implies M θ ( b ) ⊂ M θ, ext ∩ { y = b } . Moreover, the form of the tangent space to M θ, ext at the point (cid:0) , , , , (cid:1) implies that the dimension of M θ, ext ∩ { y = b } isthree for small b. Since this is also the dimension of M θ ( b ) the relation (6.57)follows. The continuity of M ext then implies that the centre-stable manifold36 θ (0) can be uniquely obtained as limit of the manifolds M θ ( y ) as y → + . In particular the manifold M θ (0) is unique.The properties of the manifold M θ (0) can be analysed in more detail. Weremark that the curve: p ( Z + 1) √ G (1 − G ) = 23 , Q = Q = 0 (6.59)belongs to M θ (0) since the hyperplane { Q = Q = 0 } is invariant under thedynamics induced by (6.53)-(6.56). On the other hand, the invariance of (6.53)-(6.56) under rotations in the ( Q , Q )-plane allows the problem to be reducedto one with smaller dimensionality. More precisely, defining Q = q ( Q + Q )leads to the system: dQdζ = − GZQ, (6.60) dGdζ = 2 G h Z (1 − G ) − θ (cid:2) Z + 1 (cid:3) Q i , (6.61) dZdζ = (cid:16) G − − θZQ p ( Z + 1) (cid:17) (cid:0) Z + 1 (cid:1) . (6.62)We will denote by N θ the (two-dimensional) invariant manifold associated tothe system (6.60)-(6.62) that is obtained from M θ by taking the quotient byrotations in the Q i and which contains the curve (6.59).Our goal is to show the existence for any y sufficiently small of a value θ ∗ = θ ∗ ( y ) of θ such that the manifold M θ ∗ ( y ) contains the point Q = Q = 1 , G = 1 , Z = 0 . This will be done by showing that the correspondingstatement holds in the case y = 0 and then doing a perturbation argument.The statement about the manifold M θ ∗ (0) is equivalent to the statement that N θ ∗ contains the point (1 , , θ is varied through the value θ ∗ the manifold N θ moves through (1 , ,
0) with non-zero velocity. It then follows that M θ (0)moves through (1 , , ,
0) with non-zero velocity. Note that the coefficients of thesystem extend smoothly to an open neighbourhood of the manifold M θ ∗ (0). Asa consequence the manifold M θ, ext extends smoothly to small negative values of y . The desired statement concerning M θ ( y ) is a consequence of the implicitfunction theorem. In more detail, the statement that M θ depends on θ and y in a way which is continuously differentiable means that there is a C mappingΨ from the product of a neighbourhood of (0 , θ ∗ ) in R with M θ (0) into aneighbourhood of (1 , , ,
0) with the properties that its restriction to y = 0and θ = θ ∗ is the identity and that the image of { ( y , θ ) } × M θ ∗ (0) underΨ is M θ ( y ). The condition that the manifold moves with non-zero velocityimplies that if x denotes the point of M θ ∗ (0) with coordinates (1 , , ,
0) thelinearization of Ψ at the point (0 , θ ∗ , x ) with respect to the last four variablesis an isomorphism. This allows the implicit function theorem to be applied.In order to check the existence of θ ∗ it is enough to study the behaviour ofthe manifolds N θ for θ → + and θ → ∞ . These manifolds are two-dimensional37anifolds in the three-dimensional space (
Q, G, Z ) . Notice that the structureof the manifolds N θ can be easily understood using the fact that the parameter θ can be rescaled out of the system (6.60)-(6.62) using the change of variables: Q = 1 √ θ q. (6.63)Then (6.60)-(6.62) becomes: dqdζ = − GZq, (6.64) dGdζ = 2 G h Z (1 − G ) − (cid:2) Z + 1 (cid:3) q i , (6.65) dZdζ = (cid:16) G − − Zq p ( Z + 1) (cid:17) (cid:0) Z + 1 (cid:1) . (6.66)Let us denote by e N the centre-stable manifold at the point ( q, G, Z ) = (cid:0) , , (cid:1) for the dynamics (6.64)-(6.66). The manifold e N contains the curve n(cid:0) Z + 1 (cid:1) G (1 − G ) = , q = 0 o . Notice that:(
Q, G, Z ) ∈ N θ ⇐⇒ (cid:16) √ θQ, G, Z (cid:17) ∈ e N . Therefore the family of manifolds N θ can be obtained from the manifold e N by means of the rescaling (6.63) while keeping the same value of the variables G, Z.
In order to check if (
Q, G, Z ) = (1 , , ∈ N θ we just need to describein detail the intersection of the manifold e N with the line { G = 1 , Z = 0 } . Oncethe existence of a value θ ∗ of θ for which the manifold N θ ∗ contains the point(1 , ,
0) has been shown the statement that the manifold N θ moves through thispoint with non-zero velocity follows immediately from the rescaling property.Notice that the plane { q = 0 } is invariant for the system of equations (6.64)-(6.66). The analysis of the trajectories of (6.64)-(6.66) in this plane can be doneusing phase portrait arguments. There is a unique equilibrium point at ( G, Z ) = (cid:0) , (cid:1) with stable manifold n(cid:0) Z + 1 (cid:1) G (1 − G ) = o . This manifold splitsthe plane (
G, Z ) in two connected regions. The trajectories starting their motionin the region that contains the point (
G, Z ) = (0 ,
0) reach the line Z = 0 fora finite value of ζ if Z > Z approaches −∞ at a finite value of ζ. On the other hand, the trajectoriesstarting their motion in the region containing the point (
G, Z ) = (1 ,
0) move inthe direction of increasing Z towards Z = ∞ , G = 1 , a value that is achievedfor a finite value of ζ. Notice that the solutions of (6.64)-(6.66) starting their dynamics in the set { ≤ G ≤ , Z ≥ } can only evolve in two different ways. Either the trajectoryremains in the region where Z ≥ ζ or thetrajectory enters the region { Z < } . In the second case this can only happenthrough the set G ≤ . Since G is decreasing it remains in the set { Z < } for38arger values of ζ and eventually it approaches Z = −∞ for some finite value of ζ. Suppose otherwise that the trajectory remains in the region where Z ≥ ζ. Then q decreases to zero and the behaviour of the trajec-tories is then similar to the ones in the plane { q = 0 } . We now claim that eitherthis trajectory belongs to the stable manifold e N or it satisfies lim ζ → ζ ∗ Z ( ζ ) = ∞ for some ζ ∗ ≤ ∞ . In order to avoid breaking the continuity of the argument wewill prove this result in Lemma 11 in Section 7.We will show that there exists a point of the line { G = 1 , Z = 0 } in themanifold e N . The points of this line enter the region { < G < , Z > } dueto the form of the vector field associated to (6.64)-(6.66). If q (0) > Z approaches Z = ∞ for a finite value of ζ. Suppose nowthat q (0) > { Z < } for a finite value of ζ as the following argument shows. A solution which startsat ( q , ,
0) with q large immediately enters the region Z > G <
1. Theinequality Z ≤ since dZdζ ≤ Z ≤ q sufficiently large Z will become negative withinthe interval [0 , ]. From now on only that interval is considered. Integratingthe equation for q gives the inequality q ( ζ ) ≥ e − q . The equation for G thenshows that G ( ζ ) ≤ e − α ( q ) ζ where α ( q ) = q e − −
1. Choose q large enough sothat e − α ( q ) ≤ . When ζ = the inequality Z ≤ still holds. Under thegiven circumstances G is decreasing on the whole interval [0 , ]. The equationfor Z shows that by the time ζ = at the latest Z has reached zero.Let U be the set of positive real numbers q for which the solution startingat ( q , ,
0) is such that Z → −∞ as ζ → ζ ∗ , where ζ ∗ denotes the maximaltime of existence, and let U be the set of positive real numbers q for whichthe solution starting at ( q , ,
0) is such that Z → + ∞ as ζ → ζ ∗ . It followsfrom Lemma 13 that U is open. We also know that U is open. Moreover, ithas been proved that both U and U are non-empty. By connectedness of theinterval (0 , ∞ ) it follows that there must be a value of q for which the solutionstarting at ( q , ,
0) is neither in U or U . For that solution Z is non-negativeand does not tend to infinity and thus, by Lemma 11, it is the desired solutionwhich lies on ˜ N .The equivalence between the existence of the self-similar solution describedin Section 5 and the existence of a trajectory connecting the points ( Q , Q , G, Z ) =(1 , , ,
0) and ( Q , ∞ , Q , ∞ , Λ ∞ , u ∞ ) proved in Subsection 6.5 concludes theproof of Theorem 10. Theorem 1 is just a Corollary of Theorem 10. Lemma 11
Suppose that a solution of (6.64)-(6.66) is defined for ζ ∈ [ ζ ∗ , ζ ∗ ) , where ζ ∗ is the maximal time of existence. Suppose that Z ( ζ ) > for any ∈ [ ζ ∗ , ζ ∗ ) and also that G ( ζ ∗ ) ∈ (0 , , q ( ζ ∗ ) > . Then, either the curve { ( q ( ζ ) , G ( ζ ) , Z ( ζ )) : ζ ∈ ( ζ ∗ , ζ ∗ ) } is contained in the stable manifold e N or lim ζ → ζ ∗ Z ( ζ ) = ∞ . Proof.
The plane { G = 0 } is invariant under the flow associated to (6.64)-(6.66). On the other hand, the vector field on the right-hand side of (6.64)-(6.66)points into the region { G < } if q = 0 . Therefore the region { < G < , q > } is invariant for the flow defined by (6.64)-(6.66) and we can assume that theinequalities 0 < G ( ζ ) < , q ( ζ ) > ζ ∈ [ ζ ∗ , ζ ∗ ) . We now have twopossibilities: lim sup ζ → ζ ∗ Z ( ζ ) < ∞ , (7.1)lim sup ζ → ζ ∗ Z ( ζ ) = ∞ . (7.2)Suppose first that (7.1) holds. Then, there exists M > Z ( ζ ) ≤ M for any ζ ∈ [ ζ ∗ , ζ ∗ ) . (7.3)We claim that in this case the trajectory { ( q ( ζ ) , G ( ζ ) , Z ( ζ )) : ζ ∈ ( ζ ∗ , ζ ∗ ) } iscontained in e N . Notice that in this case, the boundedness of | ( q, G, Z ) | impliesthat ζ ∗ = ∞ . Since (
GZq ) ( ζ ) > ζ ∈ [ ζ ∗ , ∞ ) it follows from (6.64) that q ( ζ ) is decreasing. Therefore q ∞ = lim ζ →∞ q ( ζ ) exists and is non-negative.Suppose that 0 < q ∞ . Then 0 < q ∞ < q ( ζ ) for any ζ ∈ [ ζ ∗ , ∞ ) . Integrating(6.64) we obtain R ∞ ζ ∗ ( GZq ) ( ζ ) dζ < ∞ , whence Z ∞ ζ ∗ ( GZ ) ( ζ ) dζ < ∞ . (7.4)Since dGdζ , dZdζ are bounded, (7.4) implies lim ζ →∞ ( GZ ) ( ζ ) = 0 . Then (6.65)implies: dGdζ ≤ − q ∞ G for ζ ≥ ζ sufficiently large. Therefore lim ζ →∞ G ( ζ ) = 0 . Equation (6.66) thenyields: dZdζ ≤ − ζ ≥ ζ large enough. Then Z ( ζ ) < ζ, but this contradicts thehypothesis of the lemma. It then follows that q ∞ = 0 . Due to (7.3) and since lim ζ →∞ q ( ζ ) = 0 we can approximate the trajec-tories associated to (6.64)-(6.66) for large values of ζ using the correspondingtrajectories associated to (6.64)-(6.66) for q = 0 . The study of the trajecto-ries associated to (6.64)-(6.66) that are contained in { q = 0 } ∩ { < G < } reduces to a two-dimensional phase portrait. These trajectories can have onlythree different behaviours. Either they are contained in e N ∩ { q = 0 } , or they40each the plane { Z = 0 } , with G < , entering { Z < } , or they become un-bounded. The continuous dependence of the trajectories with respect to theinitial values as well as the fact that lim ζ →∞ q ( ζ ) = 0 implies then that eitherlim ζ →∞ dist (cid:16) ( q ( ζ ) , G ( ζ ) , Z ( ζ )) , e N ∩ { q = 0 } (cid:17) = 0 , or Z ( ζ ) < ζ < ∞ , or Z ( ζ ) ≥ M + 1 for some ζ < ∞ . The second alternative con-tradicts the hypothesis of the lemma. The third alternative contradicts (7.3)and therefore only the first alternative is left. However, in that case lim ζ →∞ ( q ( ζ ) , G ( ζ ) , Z ( ζ )) = (cid:0) , , (cid:1) and the trajectory is contained in e N as claimed.Suppose then that (7.2) holds. We claim that in this case lim ζ → ζ ∗ Z ( ζ ) = ∞ . Notice that the monotonicity of q ( ζ ) implies that lim ζ → ζ ∗ q ( ζ ) = q ∞ exists.We will first prove that q ∞ = 0 . Suppose that, on the contrary, q ∞ > . Then q ( ζ ) > q ∞ > ζ ∈ [ ζ ∗ , ζ ∗ ) . Equation (6.66) as well as
G < dZdζ < (cid:16) − Zq ∞ p Z + 1 (cid:17) (cid:0) Z + 1 (cid:1) for any ζ ∈ [ ζ ∗ , ζ ∗ ) . This inequality implies dZdζ < Z > Z ∞ = Z ∞ ( q ∞ ) . Therefore Z ( ζ ) < Z ∞ for ζ ∈ [ ζ ∗ , ζ ∗ ) and this contradicts (7.2). From now ontake q ∞ = 0 . We can then assume (7.2) andlim ζ → ζ ∗ q ( ζ ) = 0 . (7.5)Suppose also that lim inf ζ → ζ ∗ Z ( ζ ) < ∞ . This is equivalent to the existence of0 < M < ∞ and a subsequence { ζ n } with ζ n → ζ ∗ as n → ∞ such that: Z ( ζ n ) ≤ M. (7.6)We now claim that: lim ζ → ζ ∗ [ Z ( ζ ) q ( ζ )] = 0 . (7.7)To prove (7.7) we argue as follows. Combining (6.64), (6.66) we obtain: ddζ ( Zq ) = qZ ( G −
1) + q (3 G − − Zq q ( Z + 1) . (7.8)We now use the inequality Z q ( Z + 1) ≥ Z for Z > . Then, using also theinequality
G < ddζ ( Zq ) ≤ q − h (3 G − q − ( Zq ) i . (7.9)It follows from this inequality, as well as (7.5) that for any ε > , every tra-jectory satisfying the hypothesis of Lemma 11 and entering any of the regions { ( q, G, Z ) : Zq < ε } for ζ sufficiently close to ζ ∗ remains in such a region forlater times. If ζ ∗ = ∞ , the meaning of sufficiently close is large enough. Dueto (7.5) and (7.6), for any ε > , there exist ζ n arbitrarily close to ζ ∗ such that41 Zq ) ( ζ n ) < ε. Then ( Zq ) ( ζ ) < ε for any ζ ∈ ( ζ n , ζ ∗ ) . Since ε is arbitrary weobtain (7.7).Combining (7.5) and (7.7) it follows that:lim ζ → ζ ∗ δ ( ζ ) = lim ζ → ζ ∗ δ ( ζ ) = 0 (7.10)where: δ ( ζ ) = Zq p Z + 1 , δ ( ζ ) = (cid:0) Z + 1 (cid:1) q Z + 1 . We can then rewrite (6.64), (6.66) as: dqdζ = − GZq, (7.11) dGdζ = 2 G [ Z (1 − G ) − ( Z + 1) δ ( ζ )] , (7.12) dZdζ = (3 G − − δ ( ζ )) (cid:0) Z + 1 (cid:1) . (7.13)We now claim the following. Given any ε belonging to the interval (cid:0) , (cid:1) suppose that the trajectory under consideration enters the set:Ω ε = (cid:26) G ≥
13 + ε , Z ≥ (cid:27) for some ζ < ζ ∗ sufficiently close to ζ ∗ . Then lim ζ → ζ ∗ Z ( ζ ) = ∞ and ζ ∗ < ∞ . The proof as the follows. Due to (7.10) the set Ω ε is invariant for (7.11)-(7.13)if ζ is close to ζ ∗ . Then, for ζ close to ζ ∗ we have: dZdζ ≥ ε (cid:0) Z + 1 (cid:1) and this implies lim ζ → ζ ∗ Z ( ζ ) = ∞ and ζ ∗ < ∞ . Therefore, to complete the proof of Lemma 11 it only remains to prove thatthe trajectory enters Ω ε for values of ζ sufficiently close to ζ ∗ . Due to (7.2) and(7.6) there exists a sequence (cid:8) ¯ ζ n (cid:9) with ζ n < ¯ ζ n < ζ ∗ such that: Z (cid:0) ¯ ζ n (cid:1) = 2 M and dZdζ (cid:0) ¯ ζ n (cid:1) ≥ . Due to (7.13) this implies: lim sup n →∞ G (cid:0) ¯ ζ n (cid:1) ≥ . (7.14)On the other hand, a Gronwall type of argument applied to (7.13) implies theexistence of α M >
0, depending only on M such that:0 < M ≤ Z ( ζ ) ≤ M for ζ ∈ (cid:2) ¯ ζ n , ¯ ζ n + α M (cid:3) . (7.15)42omparing the solution of the equation (7.12) with the solution of the equation dGdζ = 2 GZ (1 − G ) with the same initial datum at ζ = ¯ ζ n and taking into ac-count (7.14), (7.15) it then follows that, for n large enough (cid:0) q (cid:0) ¯ ζ n (cid:1) , G (cid:0) ¯ ζ n (cid:1) , Z (cid:0) ¯ ζ n (cid:1)(cid:1) ∈ Ω ε . Therefore lim ζ → ζ ∗ Z ( ζ ) = ∞ . This contradicts (7.6) and the lemma follows.
Lemma 12
There exists δ > sufficiently small such that, the solution of(6.64)-(6.66) with initial value ( q (0) , G (0) , Z (0)) = ( q , , and < q < δ satisfies lim ζ → ζ ∗ Z ( ζ ) = ∞ , where ζ ∗ denotes the maximal time of existence ofthe trajectory. Proof.
The trajectory enters the region { Z > } and as long as it remainsthere, the function q ( ζ ) is decreasing. The inequality ∂Z∂ζ ≤ Z ≤
1. It follows that Z ≤ (cid:2) , (cid:3) . On that interval theinequality ∂ (log G ) ∂ζ ≥ − q holds and hence G ≥ e − q . Furthermore ∂Z∂ζ ≥ e − q − − √ q = β ( q ) . (7.16)Choose δ sufficiently small that β ( δ ) > e − δ > . Then Z (cid:0) (cid:1) > and G > on (cid:2) , (cid:3) . Choose ǫ > δ < ǫ . Then it follows from(7.9) that the set defined by the inequality Zq ≤ ǫ is invariant. Thus the solutionremains in that region on its whole interval of existence. Now δ ( ζ ) ≤ ǫ √ ǫ + δ and δ ( ζ ) ≤ ( ǫ + δ ). Let [0 , ζ ) be the longest interval on which G ≥ . Fromwhat has been shown already ζ ≥ . Reduce the size of ǫ if necessary so that ǫ √ ǫ + δ < . Then it follows from (7.13) that Z is increasing on [0 , ζ ) andhence is greater than for ζ ≥ ζ . Putting this information into (7.12) showsthat provided ǫ + q < then G cannot decrease. For δ sufficiently smallthis gives a contradiction unless ζ = ζ ∗ . In particular there is a positive lowerbound for Z at late times. Furthermore (7.13) implies that lim ζ → ζ ∗ Z ( ζ ) = ∞ and the lemma follows. Lemma 13
Suppose that a solution satisfying the hypotheses of Lemma 11 with ζ ∗ = 0 has the property that lim ζ → ζ ∗ Z ( ζ ) = ∞ . Then any solution startingsufficiently close to the given solution for ζ = 0 also has the property that Z tends to infinity on its maximal interval of existence. Proof.
To start with a number of further consequences of the hypotheses ofLemma 11 will be derived. The assumption on the initial condition only playsa role towards the end of the proof. It has been shown in the proof of Lemma11 that lim ζ → ζ ∗ q ( ζ ) = 0. We now claim that (7.7) holds. Suppose that it isnot true. Then we claim that the limit lim ζ → ζ ∗ ( Zq ) ( ζ ) = L exists and that L >
0. Indeed, notice first that lim inf ζ → ζ ∗ ( Zq ) ( ζ ) > . Otherwise there wouldexist a sequence { ζ n } such that lim n →∞ ζ n = ζ ∗ with lim n →∞ ( Zq ) ( ζ n ) = 0 . Combining this with the fact that q → ζ → ζ ∗ ( Zq ) ( ζ ) > . Using again the fact that q → Zq ) is monotone decreasing for ζ close to ζ ∗ , whencethe limit lim ζ → ζ ∗ ( Zq ) ( ζ ) = L exists. Moreover we have obtained also in thiscase that ( Zq ) ( ζ ) > L for ζ close to ζ ∗ . It follows from the proof of Lemma 11 that ζ ∗ < ∞ . By the boundedness ofthe right hand side of (6.64) it follows by integrating this equation between ζ and ζ ∗ that q ( ζ ) ≤ a − ( ζ ∗ − ζ ) for a positive constant a . Hence q − ( ζ ) ≥ a ( ζ ∗ − ζ ) − .This can be used together with the limiting behaviour of Zq to estimate theright hand side of (7.8) from above. The first term is negative and can bediscarded. The second term tends to zero as ζ → ζ ∗ . The third term can bewritten in a suggestive form as − q − [( Zq ) p (( Zq ) + q ) ]. The expression insquare brackets tends to a positive limit as ζ → ζ ∗ . Thus the right hand sideof (7.8) fails to be integrable, contradicting the fact that Zq is positive. Thiscontradiction completes the proof that lim ζ → ζ ∗ ( Zq ) ( ζ ) = 0.We now use some arguments analogous to the ones used in the proof ofLemmas 11 and 12. As a next step we prove that G ( ζ ) tends to a limit as ζ → ζ ∗ and that this limit is greater than . We first claim that: S = lim sup ζ → ζ ∗ G ( ζ ) ≥ . (7.17)Indeed, suppose first that S = lim sup ζ → ζ ∗ G ( ζ ) < . Since lim ζ → ζ ∗ ( Zq ) ( ζ ) =0 we can approximate (6.60)-(6.62) by the system (7.11)-(7.13). Using (7.13) itfollows that Z ( ζ ) is decreasing for ζ close to ζ ∗ . This contradicts (7.2) and then(7.17) follows. On the other hand (7.12) implies that G is increasing if G > for ζ close to ζ ∗ . Using (7.17) it then follows that G increases for ζ close to ζ ∗ . Therefore the limit lim ζ → ζ ∗ G ( ζ ) exists and:lim ζ → ζ ∗ G ( ζ ) ≥ . Since G is monotonically increasing we can parametrize Z as a function of G. Let us denote the corresponding function by Z = ˜ Z ( G ) . Then by (7.12) and(7.13): d (log ˜ Z ) dG = (3 G − − δ ( ζ ))(1 + ˜ Z − )2 G [(1 − G ) − (1 + ˜ Z − ) δ ( ζ )] . (7.18)If the limit of G were less than one the right hand side of this expression wouldbe bounded and it would follow that Z was bounded, a contradiction. Hencelim ζ → ζ ∗ G ( ζ ) = 1.To complete the proof the condition on the initial data in the hypothe-ses of the lemma will be used. Since lim ζ → ζ ∗ Z ( ζ ) = ∞ , lim ζ → ζ ∗ q ( ζ ) = 0 , lim ζ → ζ ∗ ( Zq ) ( ζ ) = 0 and lim ζ → ζ ∗ G ( ζ ) > it follows that for any sufficientlysmall δ > (cid:0) ¯ q, ¯ G, ¯ Z (cid:1) that is sufficiently close to ( q, G, Z )at ζ = 0 we have for some ζ < ζ ∗ :¯ q ( ζ ) ≤ δ , ¯ G ( ζ ) ≥
13 + δ , (cid:0) ¯ Z ¯ q (cid:1) ( ζ ) ≤ δ , ¯ Z ( ζ ) ≥ δ . (7.19)44t will now be shown that for δ sufficiently small the region defined by thesefour inequalities is invariant. On the part of the boundary of the region where¯ q = δ we have d ¯ qdζ <
0. On the part of the boundary where ¯ Z ¯ q = δ assumingthat δ < − suffices to show, using (7.9), that the derivative of ¯ Z ¯ q is negative.On the part with ¯ G = + δ the following inequality holds: ∂ ¯ G∂ζ ≥ (cid:20) δ − − ( δ + δ ) − ( δ + δ ) (cid:21) . (7.20)Choosing δ sufficiently small implies that the right hand side of this inequalityis positive. On the whole region d ¯ Zdζ ≥ δ (3 − p δ + δ ) . (7.21)If δ is small enough then this quantity is positive. Putting these facts togethershows that the solution starts in the region of interest when ζ = ζ and staysthere. In particular ¯ G ( ζ ) ≥ + δ for ζ ≥ ζ . Therefore ¯ Z blows up in finitetime due to (7.13) and Lemma 13 follows. Acknowledgements : JJLV is grateful to J. M. Mart´ın-Garc´ıa for interestingdiscussions concerning the analogies and differences between the solutions ofthis paper and those in [17]. JJLV acknowledges support of the HumboldtFoundation, the Max Planck Institute for Gravitational Physics (Golm), theMax Planck Institute for Mathematics in the Sciences (Leipzig), the HumboldtUniversity in Berlin and DGES Grant MTM2007-61755 and Universidad Com-plutense. Both authors are grateful to the Erwin Schr¨odinger Institute in Vi-enna, where part of this research was carried out, for support.
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