Egg IIIIHIWIIHIWINIWIIIIIWIHliHNlWlllHlWll W388 llllllllllllllllllnllllllllllllIllIlllllllllllnllllll 3 1293 01688 748 This is to certify that the dissertation entitled Global Existence of Solutions to Nonlinear Nave Equations presented by Mariusz Adam Kepka has been accepted towards fulfillment of the requirements for Ph . D . degree in mm 73588891983 56‘?“ \IMajor professor Date July 1, 1993 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Mlchlgan State Unlverslty PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE ”A!“ 32201 miss 9 is 20m mg, g (912“? AUGUB 4 pBZRn 8-8: .45 l I 1M www.mu Global Existence of Solutions to Nonlinear Wave Equations By Mariusz Adam K epka DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1998 ABSTRACT Global Existence of Solutions to Nonlinear Wave Equations By Mariusz Adam K epka Global in time solutions of the equations Du = H (u, u’ , u” ) on Minkowski space- time are considered. Results avaiable so far, involve complicated decay and energy estimates, and also careful choice of Banach spaces and associated ordinary differential inequalities. This work tries to simplify some of the existing arguments, and to develop a new technique for other nonlinear evolution equations. The method is motivated by the work of Christodoulou, and Baez, Segal and Zhou, on nonlinear wave equations. The key idea is to use the Penrose transform for transforming the equations from Minkowski space to the Einstein universe, in order to change the global existence question to the local one. In Chapter 4, the existence of global I’—1 on Minkowski space is solutions to the semilinear wave equations Bu 2 lull, —u|u proved assuming initial data are small and the power I is big enough. In Chapter 5, corresponding results for the quasilinear equations Du = l'uroll, lurll, are shown. To my mother, Irena Kepka. iii ACKNOWLEDGMENTS I would like to express my gratitude and thanks to my advisor, Professor Zhengfang Zhou for his constant help, encouragement, and excellent advice. I would also like to thank Professors Dennis Dunninger, John McCarthy, Milan Miklavcic, William Sledd, and Clifford Weil for their time and for their valuable suggestions. My warm thanks go to Professor Pawel VValczak, of the University of Lodz, Poland, for his continual inspiration, support, and advice. I also thank my friend, Eugene Belchev, for many interesting and important dis- cussions. Finally, I would like to thank my wife, Ilona, for her help and encouragement in all things. iv TABLE OF CONTENTS Introduction 1 1 Conformal Transform 9 1.1 Definition and Properties ......................... 9 1.2 Conformality of the Penrose Transform ................. 14 2 Transformed Equation 16 2.1 Conformal Laplacian ........................... 16 2.2 Transformed Equations .......................... 22 3 Nonlinear Evolution Equations 24 3.1 ODE’s on Banach Spaces ......................... 24 3.2 Semigroup of Operators for the Wave Equation ............ 26 4 Global Existence for Semilinear Equations 33 4.1 Solutions on the Einstein universe .................... 33 4.2 Solutions on Minkowski space ...................... 47 5 Global Existence for Quasilinear Equations 56 BIBLIOGRAPHY 61 Introduction In this dissertation we study global existence of the Cauchy problem u“ — Au = H(u, Du, D2u), ”(8733 O) = “113), ”1001330) = 907:)» (1) where 1:0 6 R, :1: 6 IR". The space M = IR x IR" is usually called Minkowski space. This equation describes many familiar and important physical processes of interaction. The movement of vibrating strings and drum heads, sound waves and electromagnetic waves are some famous examples. The initial value, or Cauchy, problem has long played a central role in the theory of evolution differential equations like (1), and has been studied extensively with considerable success. In spite of a great deal of recent activity on the nonlinear theory, many physically and mathematically important and difficult problems remain unsolved. Among the most interesting ones are those of global existence, regularity, stability and temporal asymptotics of solutions. When H (u, Du, D221) 2 lull with l > 1, equation (1) becomes the semilinear wave equation u“ — Au = |u|l, u(:L',O) = f(I) “180(3330) : 9(I)a (2) which was first studied by John [12]. He proved, that if I > 1 + \/2, then there are always global, classical solutions of (2), provided (f,g) E C8(R3) x 03(R3) is sufficiently small. John also showed that for 1 < l < 1 + \/2 a global solution does not exist for any smooth non-trivial data with compact support. The strange number I = 1+\/2 appeared also in the work of Strauss [37] concerning low energy scattering of semilinear Schrodinger equations in three dimensions. In fact, he conjectured in [38] that there should always be global solutions of (2) if I > (0(a), where 10(7),.) is the positive root of the quadratic equation ('n.—1)l2 — (71+ 1)l - = 0, that is, n+1+\/n2+10n—7 2(n—1) ' 10(7):) =- This conjecture was verified for n = 2 in 1981 by Glassey [10], who showed the global existence for (2) and I > §+2—‘/fi, provided the initial data are sufficiently small. Lindblad and Sogge [23] proved that for n. g 8 one has global solutions if I > 10(7),). They also showed one has global solutions for any dimension n if I > 10(7).) and the initial data are spherically symmetric. Let H(u, Du, D2u) = —u|u|"1, l > 1. Then Equation (1) becomes u“ — Au 2 —u[u|l‘1, u(:r:,0) = f(;1:), 11.1,0(:17,0) = 9(1). (3) For n = 3,1 < l < (1(a), where 11(7).) 2 :3 = 5, the global existence of the - classical solution (3) was established by Jdrgens [17] in 1961. The case I = 5 is due to Grillakis [11] after earlier work of Ranch [28]. The global existence question for l > 5 so far has eluded all research attempts. Let H (u, Du, D2u) = luxoll, ['uroll, l > 1. Then (1) becomes a quasilinear equation utt _ Au = [urollv lu'rolla u(:r,0) = f(:r), uxo(:r,0) = g(;r). (4) Klainerman and Ponce [21] proved that (4) has small global solutions if the initial data are sufficiently small and (n — 1)(l — l)2 — 21 > 0. However, it was conjectured that thecritical value for the quasilinearcase is 12(n) = "+1. Sideris [35] proved global existence of small radially symmetric solutions for n = 3. and l > 2. Schaeffer [29] established global existence of small radially symmetric solutions in five space dimensions for l > %. Coming back to the dimension n = 3, John [13] showed that every non-trivial C3 solution of _ 2 Du—umo with compactly supported Cauchy data blows-up in finite time. On the other hand, the seemingly similar equation Du = “:0 — 2: u: (5) always has global C°° solutions if the data belong to 08° and are small. The reason for this is the following observation made by Nirenberg. Let u = 1 — e’“. Then the Cauchy problem (5) is changed to the linear Cauchy problem C11) 2 0, u(0,:r) = 1 — EMU“), 221.0(0, T) = umoe'“(0‘x), which has a global smooth solution 2), with [[UHoo < 1, when the initial data are 18° and very small. In fact, Klainerman [20] showed global existence for equation (1) if n = 3 and F satisfies the so called null condition, which implies that F = F0+0(|u|3+ WP), and F0 = Z bjkajuDku, with the property Zikzo bjijCk = 0 whenever (3 — ([2 — (.3 — (if = 0. It should be pointed out that all these results are not easy to obtain. They involve very complicated decay and energy estimates, and careful choice of Banach spaces and associated ordinary differential inequalities. Our work is trying to simplify some existing arguments and to develop a new technique for other nonlinear evolution equations. The method we are going to use is a conformal campactification which is described in more details later. Our approach is motivated by the work of Christodoulou [5], Baez, Segal and Zhou [2] on nonlinear wave equations, and recent developments in the rigorous theory of nonlinear quantum fields [25, 26, 32, 33], which made the first substantial progress in the physically interesting and mathematically quite singular case of four or higher dimensional space-time. In part, this progress results from the extension of the quantum field on the Minkowski space to the Einstein universe. The key idea is to map the Minkowski space to a compact part of the Einstein universe E = 1R x S". To be more precise, let (51:0, 11:1, . . . ,x,,) be the usual coordinates in M = IR >< IR", and let 9 = data — drrf — . . . — (1:103, be Minkowski metric on M. Let 5‘] = dt2 — (132 be the metric on Einstein universe E, where ds is the standard metric on the n-dimensional unit sphere S". We will use the coordinates (t, y1,y2, . . . ,y,,+1) for E, where t E R is the Einstein time and y? + . . . + yf, +1 = 1. These two spaces, and wave operators on them, are very closely related. M is conformally embedded in E. The embedding map 0 is known as the Penrose transform [27], which is briefly described in the following. The conformal map 0 (which preserves Lorentz angles) from M into E is given by (T(f1§(),f171, . . . ,IIIn) : (t1 21/1, ° ° ' ayn-i-l): sint = p(;r):r0, —7r < t < 7r, yj = p(1;)1:j, j = 1,... ,n, 21m = p(rr)(1+ 132/4), 18(11?) = [(1 - 1152/4)‘2 + Iii—”2, ,2_,,2 ,2_ __ 2_,,2_2 .1: — .10 — (Ill . . . 1:71 ‘— .10 T o In terms of spherical coordinates on S "; i.e., y,,+1 = cos p, (yl, . . . ,y,,) = sin ,0 ed, with w E 5"”, p E [0, WU c can be written as sint = p(:r);r0, sinp = p(:r)7‘, cosp + cost = 2p(:2:), where w = (51:1, . . . ,asn)/r. When x0 = 0, the transformation is exactly stereographic transformation from IR" to S". It is easy to see that the image of M under this transformation is C(IW) = {cos‘l‘ + cosp = 2p(:1:) > O} = {Itl +p < 7r,t E (—7r,7r),p E [0,7r)}. (6) The inverse of c, c‘1 : C(M) —8 M, is given by 2 sint 21/8 .170 = - , ij = A] . (7) cost + cos p cost + cos p It is important to note that CU”) is bounded in E since [t] < 7r. The boundary of C(M) consists of two parts C- and C1,. More precisely, C_={p—t=7r}, C+={p+t=7r}. On this boundary, p(a:) 2 (cost + cos p)/2 = 0. The map c is conformal since g = p2(:r)g. We will see that CI = 830 — AIR" on M, and LC = ('92 — Asn + " 2 on E are conformally covariant o erators. In fact, t P —1 2 Due)=p<'8+“>/2[L.v](cm), where um=p>, (8) for any smooth function u on E. From this simple transformation we see that if u is a solution of the homogeneous linear equation Du = 0, then Lav = 0. The conformal factor p(:r) can easily give us many decay estimates for linear homogeneous solutions. Such estimates are vital to study the nonlinear problems [3, 11, 19]. For example, |u(.r)| S c|.1:0[‘("‘1)/2 if the initial data is in 08°. Furthermore, |u(:r)| S c|$0|‘(""l) when :1: is away from the light cone 2:2 = O. This fact is not used fully in the study of the wave equation, yet, should be useful. Another important feature for the free wave equation is the conservation of Minkowski energy E0(:r0) =/ [(onu)2 + IVu|2] (1:171 dz", which played a crucial role in studying this equation. From the conformal transforma- tion c, we can have another conserved quantity: E (t) E C, where E (t) is the Einstein energy dS. 2 EU) =/ [(71, 2 1) 222 + (@102 + [Vul‘2 S71 The Einstein energy is readily expressed in terms of Minkowski space, being the usual relativistic energy plus that of the transform of the solution under inversion .1: ——8 113/1132. At$0=t=0, 2 E(()) =/ {<1+%) [(6tmu)2 + [Vu|2] — $23} (1:1)] dIL‘n 10:0 Its finiteness imposes only a weak condition on the decay of the Cauchy data near spatial infinity. For any nonlinear wave equation Du = H (u, Du, D2u), this equation can be changed to an equation on C(Al) Lc'u(t, y) = [$(cost + cos p)]'("+3)/2H(t,y,v, D’U, D212), (t,y) E C(A/I). In general, this will be a singular PDE on E since %(cost + cos p) = p = 0 on the boundary of C(M). If the right hand side, p(t,y)‘("+3)/2H(t,y,v, Dv,D2v), can be extended to a nice function on E in some sense, then we can solve the equation on E. The global existence question in M becomes a local existence question in E (Einstein time from —7r to 7r). This is especially effective for problems with small initial data, since the time interval of the existence usually can be estimated in terms of the size of the initial data by the general perturbative argument. Christodoulou [5] proved that the “null condition” on IR x R3 introduced by Klainerman [20] means that the transformed equation on C(A/I) can be extended to E smoothly and implies the global existence in .M with small initial data. We will study the global existence based on this simple idea. For example, the nonlinear equation Du + lull‘lu = 0 on M can be changed to Lev + |v|l‘lv[cost + cos ,0]l("_”/l‘Z’MHV2 = 0. When 1 = (n + 3) / (n — 1), which is the conformally invariant case, the equation can be extended to E with smooth coefficients. In general, the question is the following. What kind of singularity on the boundary of C(Il/I) still guarantees the existence of a nice solution on E '? If this question is answered, we will solve many interesting problems regarding the global existence, asymptotic behavior of solutions, and blow- up of solutions [15, 16]. Recently, the nonlinear wave equation Du + lullt‘lu = 0 with the critical power lat: : n+2 n_2 has been studied extensively, and the global existence and regularity of the solution were established [11]. A more difficult question is what happens if I > 1*. Using the compactification, one can see that the corresponding equation on E has continuous coefficient on the boundary of (:(M). It is our hope that this technique will be helpful in attacking the problem with supercritical exponent. At least, the compactification of M can be used to simulate solutions numerically by the finite element method. In this case, the spatial cutoff is no longer necessary due to the compactness of S". The numerical results should give us some indication about the behavior of the solutions involving supercritical exponents. However, we will not deal with numerical solutions in this thesis. I Our work is organized as follows. In Chapter 1, we introduce the Penrose trans- form c and prove some of its properties. In particular, we show that c is a conformal transformation from Minkowski space M to the Einstein universe E. In Chapter 2, we analyze the behaviour of the so called conformal Laplacian subject to any conformal change of a metric. We use results of this analysis, to transform our equations from M to E. A theory of solving nonlinear evolution equations is reviewed in Chapter 3. We rewrite a PDE as an ODE on a suitable Banach space, and solve the ODE using the so called Duhamel’s principle. In Chapter 4, we combine ideas from the previous chapters and prove the main result of the thesis in the case of semilinear equations. Namely, we construct a global solution of an equation on the Minkowski space from a. local solution of the corresponding equation on the Enistein universe. Finally, in Chapter 5, we are able to extend our global existence results to the case of quasilinear equations. CHAPTER 1 Conformal Transform Penrose [27] gave an interesting representation of Minkowski space-time to study the structure of infinity in this space. The transformation 0 : R x IR" —> IR x S ", which we call here the conformal or the Penrose transform, is the Lorentzian metric analog of the usual stereographic transformation from R" to S ". The main property, which we prove in Section 1.2, shows that the Penrose transform is a conformal transformation. 1.1 Definition and Properties In this section, we give a definition of the Penrose transform and state several of its properties. In order to describe the conformal transformation in M, let us fix some notation in IR". Let (r,61, ...,6n_1) to be the polar coordinates; that is, for (.131, ...,:1:,,) 6 IR", we have r 1:1: 7" sin 0,,_1... sin 61, 1172 = 7“ sin 9,,_1... sin 62 cos 61, t . (1.1) 12,, = 7" cos 0,,_1, t 10 with r E'(0,oo), 91 E (0,27r), 6;, E (O,7r),k = 2,...,n — 1. We will denote by g the usual metric in the Minkowski space M given next. Definition 1 .1 n— l g = (1135 — (17‘2 — E 9,, (16,2, (1.2) 121 r23i112 6,,_1...si112 6,11, 2': 1, ..., n — 2, git = r2, 2': n — 1. 011 S", we will use the notation (4)1“, ...,¢n_1, p) for the spherical coordinates; namely, for any point (y1,...,y.,,+1) E S’: with yf + + girl 2 1, we write yl = sinpsin ¢n—1---Sin¢1’ 312 = sinpsin q3,,_1... sin 3252 COS (P1, 313 = sinpsin q5n_1... sin ([53 COS CA2, {:14 = sin p sin ¢n_1 sin €134 cos qfig, [um = COS p, with (251 E (0,27r), gbk E (0,7r),k = 2,...,n — 1, p E (0,7r). Now we will define the metric 57 on the Einstein universe IR X S”. Definition 1 .2 11—1 57 = (1132 — €192 — 2917' (1953, (1.4) i=1 where gjj = sin) psin2 gb,,,_1...sin2 Q6311, j = 1,2, ...,n - 2; g,,_1,,,_1 = sinz p. In the following definition, we describe the Penrose transform. 11 Definition 1.3 Let (1:0,r,01, ...,6n._1) 6 IR X IR". Put C(IOIT, 61, ”wen—1) :: (tap: $1, "°i¢n-l) 6 IR X Sn) where 0 1:2 P230 = smt, P(1— 4) = cost, t6 (—7r, 7T), Claim 1.1 The map c is well defined. PROOF. In fact, a) t is well defined since 192.13% + 192(1 — “’72? = 1, and b) p is well defined since 122(1 + 32,—2)? s 102(1 + £25 + g + £1,453) = 1. [:1 Claim 1.2 y,- = p23,- and yn+1 = p(1 + %), where y1,...,y,,+1 are the associated Carte- sian coordinates in an+1 D S". PROOF. Since . -1 . . y,- = smpcosf),~_1 (112:, sindk), z = 1,... ,n — 1, ya = sinpcos 6n_1, and since 12 we have y,- = pr cosd,8_1(]‘[]:i1 sindk) = p35,, i = 1, ...,n — 1, yn 2 pr cos 6,,_1 = pl‘n. On the other hand, CI :1:2 yn+l = COSp :1) (1+ 71-) ' Claim 1.3 1 p = 5(cost + cosp). PROOF. It is obvious since cost + cosp =1)(1 — I? + 1 + g) = 2]). C] For later use we need partial derivatives of p, t, p, d) with respect to the Minkowski coordinates. These derivatives are denoted here by pm, tmo, pm, etc Proposition 1 .1 1 1 i) 192:0 = ——psintcosp, u) p, = —§pcostsinp, l 1 1 /ii) t$0 = §(1+ costcos ,0), Ah) t, 2 —§ sintsin p, ix) p1,, = ——p2 cos t:1:,-. 1 1 Jiii) p3.0 2 —§ sintsin p, /vii)p §=(1+ costcos p), viii) (97.6,- : 0, 31)) (91-061 2 0 . —% PROOF. Recall that p = [(1 — 1.32/4)2 + 1:3] . Therefore i) p,0 = —%p3[2(1-— ’37)? + 2.130]: —-§p3(1 + §)$0 = —%psintcosp. ii) Using i) and the fact that sint = 111:0, we have 1 . . 1 1 . t,r0 cost = atopro + p = —5 sm2 t cos p + §(cost + cos p) = 5 cost + cosz tcos p). 13 Thus tr 0 l = -2-(1+ costcosp). iii) Using cosp 2 p(1 + §), we have , {132 ([0 1 , I132 (130 —pI0 smp =19Jro 1+ 4 +p—é— = —.—,—psmtcosp 1+ 1 +1)? __1 ' 2 l ‘. —l . j. 2 — 2smtcos p+2s1nt— 2smtsm ,0. Thus p10 2 -—% sin t sin p. iv) Since the qb,’s do not depend on 1:0, we have (310d,- = 0. v) First observe that sin p 2 pr. Indeed, r2 = 22:32: 10—2 :31? = p‘2(1 - .7131“) = p‘2(1 - 0082 p) = 10—2 sinzp. j=1 j=l Since p E (0, 7r), sinp 2 pr. Therefore, 1 3 11:2 r 1 r=—— 21—— —2—— costsin. 19 2p ( 4 ) 2 218 0 vi) From sint = p.130 we have . 1 . . t, cost = 12,330 = —§p:1:0 costsmp 2 ——§ smtcostsm ,0. vii) From sin p 2 pr and Part v), 1 prcosp = prT+p = —§pcostsinpr+p 1 .2 1 1 = ——§costsm p+§(cost+c08p) = 5c08p(1+COStCOSP)8 14 Hence, p, = %(1 + costcos ,0). viii) 8rd),- = 0 since the dis do not depend on 'r. ix) This follows by direct computation. 1 . ,2 1 pr, 2 ——p3 (1— L) :13,- = —§p2 cos t;r,-. CI 1.2 Conformality of the Penrose Transform To check the conformality of c and the covariance of operators C] = 630 — AIR" on A4, and LC 2 8,2 — A371 + ("—3—1)2 on E, we will need the following assertion. Proposition 1 .2 . _ 1+costcos _ sintsin , , 1) (can, _ ———2, a, ——I-’., 0,, 2) (3* (9r _ _ sm tzsm 38‘ + 1+cos2t cos gap; 3) (2.09, 20.15,, i=1,2,...,n—1. PROOF. This follows directly from the formulae ii)—viii) in Proposition 1.1 and the following relations. ,- uv/ (' i") / (3:63:01: treat + pilfoap + 8I0((tbn—1)a¢n—l + + a$0(gb1)a‘f’l; (liar : trat 'I' [1GC + ar((f)n—1)ad>n—1 'I' + at‘((f)1)0¢1i may, 2 8(1)]. D Now we are ready to establish the conformal property of c. Proposition 1.3 The Penrose transform (3 is a conformal transformation with a con- formal factor p2, i.e., 6*? = 1229- 15 PROOF. Note that {I ‘3 I mg 2 mam, c.aB)dA <29 dB, A, B 6 {$0, r, 91, ...,9,,_,}. 9 Using Proposition 1.2, we have 1 4 1) g(c*a1‘030*al‘o)= 2) §(c*3mo,c..(9.,.) =0; [(1+ cos t cos p)2 — sin2 t sin2 p] 2 122; 3) §(c.810,c.89,) = 0; 4) §(c.8,., as.) = p2(by calculations similar to 1); 5) §(c.6,,c.89,) =0; n—l 6) §(c..80,, (2.89,) = §(6¢,, 8%.) = sin2p H sin2 45,, k=i+l 71—1 2 T2132 H Sill? 05k = PQQIao,a 30,)- k=i+1 0;: Therefore, c*§ = p29. CHAPTER 2 Transformed Equation In this chapter we would like to transfer our equations to the Einstein universe, using the Penrose transform. In general, we could have ended up with quite complicated formulae. Fortunately, since c is a conformal transformation, it is useful to consider the so called conformal Laplacian which is preserved under conformal transformations. 2. 1 Conformal Laplacian Theorem 2.1 Let N be an n-dimensional {n 2 2) manifold and let 9,57 be {pseudo-) Riemannian metrics on N with f) = [229. Then _ — n — 2 __n—2 _m~_2 77/ - 2 IA ‘ Rm] (P “I ‘8" IA " le “8 where R, R, and A, A are the scalar curvatures and the Laplacians on N with. respect to the metrics g, g, respectively. PROOF. This theorem is known for Riemannian metrics [1]. To show this theorem for Lorentzian metrics, as in our case, we need several lemmas. Here and throughout this chapter, we adopt the summation convention whereby a. repeated index implies summation over all values of that index. The indices used in this theorem have their 16 17 values in the set {1, 2, . . . ,n}. The definitions and properties which we use in this chapter, can be found in any book on Riemannian geometry; for example in [6]. In the first lemma, we show how the Christoffel symbols change under the confor- mal change of the metric. Lemma 2.1 —l n 1“ ._ = 1“,]. + rife,- + (5595,1— gijgl Wm, 1] and Fl, J are C'hristofiel symbols for g and g, respectively, p = e‘f’, and the .. z where F ,j notation 4),,- := 8,-gb is introduced to simplify future writing. PROOF. Using the definition of the Christoffel symbols and the fact that, by the assumption, 57k, = eipd’gkl, whereas g“ = EQf’gk’, we have P ij = ifiklfajgik + (92931: — 518.925) = %e-2¢gkll2e2¢(¢.jgik + ¢,igjk - ¢,kgij) +€2¢(ajgik + 8,9,1. - (91:90)] = 6:453 + ($45.1 — gklgiflk + Flij' The last equality follows from the fact that gklgik = (5] and gk’gjk = 6].. Cl In the next lemma, we show how the curvature coefficients transform under the conformal change of a metric. Lemma 2.2 Rhijk : Rhijk ‘ (SI-Ida)“ + 6f¢ik + ghl(_gijd)lk + gikdhj) + (—6f:gij + (Sj'lgiklAlfib- Here Alqb := g’jdflbj is the ”first Beltrami operator”; 3b,,- := (by, — (bydj, and 65.0 :2 83¢ — Fruity denotes the second covariant derivative of (j). 18 PROOF. Using the definition of the curvature and Lemma 2.1, we have — ——h , —h —h —m —h —m Rhijk : _ajF M + akr ij _ F jmr ki + F mkF ij = —a (1‘3...- + 6 m. — gas/”WM + 6MP" + 6&8. + W.— 9,98%,» _(P’j1n + 6;1¢.m + 621$]. _ gjfnghrgb,‘r)(I-‘m ik + 6"]ng + 6:" ¢.k — gikaSTs) +(Fhkm+ 53.9% +5},I (bm— gkmghrd),r)(F";j +5I"¢,j +5§",i- gagmvfis ) Now, we need to perform all the calculations indicated above. They produce number of terms, but fortunately many of them cancel out. To be more specific, we get that Rh _ ijk —a,r'8,,, — 6303,95 — 6?an + 8,-(g,,,gh’"¢,,..) + ahr’g, +5’la ,,.¢ + 6’83 ,45 —a.(g..-g’""¢.m) — 1“,... "I. — I‘",,..¢,. — PM)... + g.kg'"-8F",....¢,. — «new m _6;l¢.i¢,k — 6j'¢.i r —g.-tgjmg’"g "”48 a5 s +P’l mkIvn'j + W My + Pity-1,705.1 — gijgmsrhmkébs + Fitz-Mi + 5il¢,j¢,k +5jl¢s¢1c - gijfl’ls + 2aAcb)u + 4(Lgiju,i¢,j “90(2GQAz’C/JJ + tax/{j + 20¢,ic’). j“ + 18.1%) + 27109“"¢.z¢.nzu + "gm {Arum}. When we use the the relation a = —l;—2 and again, the formula for A145, we have 2 \/ — n—' -n 7?. '— 2 n — 2 . " . A(6_—2_2¢u) = 6%,, AU + (( 2 ) A1¢ — 2 A65) u — (n — 2)g”u,,~(,f),,- . I ,8 ‘/ .. .' A n n — 2 ‘ —2g”u,,¢),j + 72g’mujcbm + (n — 2)g” N is a smooth map. and (3‘57 = p2g, then —- - 71,— 2 __1_a;z 1 n 2 71—2 — ,— 2 ., _," = 5 '2 A- — . ,> . [A R4(nl_1)]((p u)oc ) p ( R4(n_1))u, n _2 PROOF. Since (2 : (N, 6*9) —* (N, {1) is an isometry, the result follows from Theorem 2.1 for metrics g and CY} considered on the manifold N. D 2.2 Transformed Equations Now we are ready to transform our equations to the Einstein universe. Since the signature of the metrics used in Theorem 2.2 was not significant for our considerations, we can use the result of this theorem when we work with Lorentzian metrics on the Minkowski space and the Einstein universe. We will denote by D the d’Alambertian with respect to metric g on M, and by If], the d’Alabertian with respect to metric g on E. Theorem 2.3 If (1%, g) and (E, g) are the Minkowski space and the Einstein. universe respectively, then (C) + 7’2)(p_7u) o c"1 = p“23fl[:lu, n 2 1, 11—1 2 . where ’7 = PROOF. Observe, that Theorem 2.1 with n + 1 in place of n gives us 4n since the curvature of the Minkowski space is R = (l. 23 Note also, that "—1 R = gal?“ "‘ gppRpp — ZéiiRfi = —RSn = —7),(Tl — 1). \ C] In the following theorem, we denote by u a function defined on the l\‘lillkOWSl\'i space, and by v, a function on the Einstein universe given by v = (p‘Iu) 0 0‘1. Theorem 2.4 The equation Bu 2 H (u u’ , u” ) cmresponds t0 the equation (If! + 72M 2 H(t,y, U, 'U’, u"), with pk|v|’, if H(zu u' ,u”)— |u|’; k l—l - 11 1-1 _ —p vlvl , if H(u,u’,u )= —u|u| ; H(t, y,v, v', v") = (2.2) pk|T(v)[’, if H(u, u’ 2,1”) = luxollg P"|Q(v)|', Z"f H(u, UCU”) = lurl’; where we have used the notation k = W and T(2))— — %(— 7 sin t cos pv — sin t sin pvp+ +(1 + cos t cos p)vt,); (2.3) (2(1)) 2 §(— 7 costs1npv(1+ costcos p)i),, — sintsin pvt). (2.4) PROOF. W In the first case, using Theorem 2.3, we see that 1:1(t,y,v,v’,v”) = p"'_2fl|p"7u|’ = pkl’ul‘, since 7 = "S1 The second case is analogous to the first one. To check the next case, we differentiate the composition (p71)) 0 c with respect to 3:0 and use formulas i) - iv) in Proposition 1.1. Similarly, to prove the fourth case, we differentiate the composition (1)712) o c with respect to r and use formulas v) - viii) from Proposition 1.1. D CHAPTER 3 Nonlinear Evolution Equations In this chapter we state several facts from the theory of nonlinear evolution equations which we use to solve our equations. Any nonlinear evolution partial differential equation can be written as an evolution equation on a Banach space in the form \I!’ = All + R(\II), where \Il(t) is an element in a Banach space X, and A is a linear transformation defined on X. .A is an unbounded operator since it is a differential operator. The delicate part is to find a suitable Banach space in which to look for a solution. This space must be big enough to guarantee the existence of a solution, at least for an associated linear equation, and small enough so that the nonlinear term makes sense and has some kind of continuity or differentiability. We will restrict our attention only to the problem we are trying to solve. 3.1 ODE’s on Banach Spaces After transforming equation (1) to the Einstein universe, we obtain a nonlinear hy- perbolic equation which we can rewrite as an evolution equation \11’ = All + R(\II), W0) = (F10), 24 25 on the Hilbert space X = H "‘(S") GB H ’"‘1(S"‘), for some fixed integer m. Here A: ,\II= , A is the Laplacian on S", and the transformation R : X —> X is to be specified later. To solve equation (3.1), it is enough to find a solution of a certain integral equation. More specifically, we have the following well-know theorem, called the Duhamel’s principle. Theorem 3.1 Let K = C((0,T),X), ||\Il||K :2 sup||\IJ(t)||X. (097') Assume A : D(A) —> X is a generator of the CO-semigroup of linear bounded oper- ators S(t) : X —> X. Then every solution of t \Il(t) = S(t)‘I’0 +/ S(t — r)R(\I/(r)) dr (3.2) 0 is a weak solution of the Cauchy problem qr = Aw + R(\II), (3 3) ‘I/(O) = ‘110 E 19(14)- The simplest existence result for equation (3.2) is the following theorem, which is a generalization of the familiar Contraction Mapping Principle. Theorem 3.2 Let X be a Banach space and S(t), a CO-semigroup on X. Let us assume that R : X ——> X satisfies the following conditions: 1) R is locally Lipschitz, i.e., ||R(‘1’1) - R(‘I’2)llx S L(ll‘I’illx,ll‘1’2||x)||‘1’1 - ‘1’2llx; 26 2) R(O) = O; L, 3) ¢($, y) is small when a: and y are small enough. Then for any r > 0, the equation t \Il(t) = S(t)\IIO +/ S(t — s)R(\II(s)) ds 0 has a solution \I’ E C([0, r), X), provided ||\I'0||X is sufficiently small. The proof of Theorem 3.2 is standard and uses the Contraction Mapping Principle. The details are omitted here to save space. 3.2 Semigroup of Operators for the Wave Equa- tion In order to solve our problem using Theorem 3.1 and Theorem 3.2, we need to show that operator A in (3.1) generates a CO-semigroup and that R is Lipschitz continuous. In this section we establish the first part. The latter one will be shown in Chapter 4. We will proceed in several steps. First, we need to define the square root of the operator ’72 — A. Let A], (15,- be respectively the eigenvalues and the normalized eigenfunctions of the 11—1 2 operator 72 — A on S". In fact, /\j = (j + )2 and dj is a spherical harmonic of degree j and ||¢j||L2 = 1, j 2 0,1,2... (cf.[24]). It is well-known, that for every v E L2(S""), we have ”U = Zaj¢ja (Lj =/ 21¢)de E R. (3.4) 1:1 " 27 Definition 3.1 The operator B = \/ '72 — A is defined by Bu 2: Zl/Ajajqfij’ where v = ZaJ-(bj. j=l j=1 Now we are going to state several properties of B which will be used later. To simplify our calculations, we introduce on H "’(S ") a. norm which is equivalent to the usual Sobolev norm. Definition 3.2 Let U E H’"(S”), m = O, 1, 2, ..., l 00 2 Ilvllm 2: (Z Ayn?) .. (3.5) i=1 When we use this norm, we immediately have the following assertion. Proposition 3.1 For any m > 0, Elm" : H"‘(S") —> Hm‘1(S") is norm preserving; i.e., ”32)”qu = “21””... for any v E H’". PROOF. In the following we will write B instead of Blnm whenever it is clear from the context which operator we are working with. For v E Hm we have 1 00 2 5 IIB‘va-l = (2%?“ Ma.) ] = llvllm. a j=1 Analogously to Definition 3.1 we define some functions of B as follows. From the spectral theory this is the only reasonable definition anyway. 28 Definition 3.3 For any 2) = Z]. ajqu, 1) costh 2: ZCOS(t‘/)\j)aj¢j; J 2) Bsinth :2 Z‘/)\j8111(t\/)\j)aj¢ji j 3) $112sz 3: Zsin(t,/Aj) 1 x/E From this definition and the norm we defined earlier, we have the next assertion. (ljdb'. Proposition 3.2 The following are bounded operators. costB: Hm —+ Hm, m = 1,2, ...; BsintB: H’" —> Hm’l, m:1,2,...; sintB. B Hm"1——>Hm, m=1,2,.... PROOF. It is enough to observe that by Definition 3.2 and Definition 3.3. \flz (‘"‘ k0 , .1. CO 2 || cos(tB)v||Hm = ZAgneosz(t\/Aj)a§] 3 Ho] _j=1 Hm) NIH .00 2 ||Bsin(tB)v||Hm-1 = Zign-1(,/A,sin(t,//\,)) a3] gnvuflm; -j=1 whereas the inequality l s'ntB 0° sin t,/)\- . 2 1 v = [2 Ag." (——(——’—)> aj] g ||v||Hm—1 proves the third property. C! Using the above properties, we have the following assertion. 29 Proposition 3.3 Let sin tB cos tB —B sin tB cos tB Then S defines a Co-seTmIgToup on X = H’"(S") {D H’"“1(S"). PROOF. First, we need to show that St E B(X), for any t _>_ 0. Let (u, v) E X. Then Proposition (3.2) gives us that HSt(U,U)Hx sintB/ Sllcos tBullm + B U m +II — Bsin(tB)v||m_1 + H cos(tB)w||,,,_1 52(H'Ullm +llwllm—1)= 2ll(u,v)llx- Next we prove that the family of operators {SJQO is a semigroup. Obviously, SO is an identity on X. It is also a direct computation that cos tB sin TB — sin tB cos TB cos tB cos TB — sin tB sin TB St 0 Sr 2 B —B(sin tB cos TB + cos B sin TB) — sin tB sin TB + cos tB cos TB t = B = SH-T- —Bsin(t +T)B —cos(t +T)B Finally, to see that {SJQO is a CO—semigroup, we need to prove that. liming ||5t(u,v) — (u,v)||x = 0, for all (do) 6 X. This follows from the defini- tion of St and the norm estimate above. [:1 Before we state the next result, let us recall a definition from the group theory. 30 Definition 3.4 We say that an opeTatOT A : D(.A) ——> X is a generatOT of a semi- gToup {SJQO z'ffOT any (1), w) E D(A) S , 1’ _ , . lim t(v u) (v w) t—»o t = A('U, w). (3.7) With this definition we can state the following fact. Proposition 3.4 The 0])67‘at07‘ \ '. U5 \ a) ’ i Q i 0 1 ( Q 0 A = _ i w A — ’72 0 geneTates the semz'gToup sin tB cos tB St = B ? —BsintB cos tB with ’D(.A) = Hm+1(S") G} Hm(S") C X. PROOF. It is obvious that D(A) C Hm+1(S‘") GB H"’(S"). To show the other inclusion, let (v,w) E H’"+1(S") GB Hm(S"), m = 1,2, . It follows from (3.6), that S nw —— o, w costh—v sintB sintB costh—w t(' )t ( ) (—-t-—+—t—B——w,— t v+ t ). (3.8) Hence, the proposition follows from the following lemmas, which estimate the indi- vidual terms in (38.) Lemma 3.1 For any 2) E H’"+1(S"), we have cos(tB)v — v t ——>0. __,+ mtO 31 PROOF. Let 1) = :00 aj o- as in (3.4) Then using Definition 3.2 we get j=1 1 I- w 2 5 os(tB)1.1 — o _ [\m cost,/)\j — 1 2 t — 2 j t aj m j=1 1 F a : oo/\m+l (COS:\/—_ 1)2 (1.]2- . j=1t\/— cos .17— 1 Observe also, that since is bounded by 1 uniformly on [0, 00), taking 231- = t /\J-, we have t 5 m cos MA] ZAj +1 < t\//\_:' _12) a? < ll"”m+l- (39) j=1 It is also easy to see that liming 30:59: O, for all j. Therefore (3.9) and the Dominated Convergence Theorem imply that C03 ”3”” —+ 0, as t —) 0, [:1 Lemma 3.2 FOT any w E H’"(S”),m = 1,2, ..., we have sin tB 0 tB w — w —>t_.0 . m PROOF. Let w = Zj_1 J-gbj. Then we have 1 ' tB 0° sint,/)\- 2 2 8mg w — w = 2A;" (-—J — I) b32- . (3.10) t m j=1 tV Aj As in the proof of the previous lemma, we want to use the Dominated Convergence . A Theorem. In order to do th1s we observe that % — 1 is bounded uniformly in t 1 and j, so the series on the right. hand side of (3.10) is bounded by C llwllm. Also, 32 s' t A- . . . . m \/_’ > 1, for all j. The des1red result follows from these estimates. t\/z\—,- t—.0 Finally, we prove the next lemma. Lemma 3.3 FOT any 11 E Hm‘+l(S"),m = 1, 2, ..., sin tB B o — B22) —+ 0. m_, t—»0+ PROOF. By Proposition 3.1, it follows that . sintB s1n tB l B2( ’0 - v) _ v - v tB m. 1 tB m+1 Also, by Lemma 3.2, we have sin tB v — v —> 0 t8 m+1 tdo which completes the proof of the lemma. CHAPTER 4 Global Existence for Semilinear Equations In this chapter we prove the existence of global solutions to Equations (2) and (3). Although we will work only with the first equation, our method applies to both of them. As we mentioned earlier, in order to find a global solution u on Minkowski space, it is enough to find the corresponding local solution, 2) on the Einstein universe. 4.1 Solutions on the Einstein universe In this section we find a solution \II to the abstract ODE, and next, a solution 2) to the Cauchy problem on the Einstein universe. First, we prove one technical result which will be used later. Proposition 4.1 Let N be a compact manifold, q 2 1, and m > :—;. If f is a Teal function, f E Cm(R), and f (0) = 0, then theTe exists a function hmq, such that Hf(U)||w'v-n.q(N) S hm.q(Hullli’"'vq(N))a ’U. E [’V1n’q(N). 33 34 (4.1) fl<0, n l_ The function hm” satisfies the following conditions. #OSTSM (hll hm,q(37) S hmqu) if m g m, q (’12) hm,q($) S Chm,q(Cl‘) if .T, —> 0. (h3) lime) —» 0 The constant C in the condition (h2) is independent of 51:. When q = 2, this result is well-known (see, for example, [18]). PROOF. First, let us make the following observation. For any 0 < j S m, we can 2 Cl(k1, . . . , k,)Vk‘u. . . Vk‘u. (4.2) write j . Emu) i=1 k1+...+k,=j 1:121 (4.3) ‘WUWD= We claim that whenever i > 1 or j < m, we have m, — k‘() 1 < —a q :G- ., for any 1 S s g i. In fact, i 1 m — k1 _ 1 m q n — q n , , 1 m 1 . +(i—s) (———) ——(A:0+k1+...+k,,_1), \ q n n 1 4—+ q n where he is understood as 0. Note that the last two terms are nonpositive. Hence, the claim is trivial for j < m. For j = m, one of the last two terms must be negative fii>L To complete the proof of the proposition, we need to estimate the L" norm of the right hand side of equality (4.2). We will divide this task into several lemmas. 35 Lemma 4.1 1 , 1 k- 1) nvkunu g 02(k,7~,m,q)nuum, if — — 3 < — — —, r e [1, oo], 0 g k g m, q n T n 2) ||f(i)(1t)lltoo S M(iamaq, IIUIImq), where M(i, m, q,:z:) :2 sup Ifm(y)|, i = 0,1,...,m. lyl SC? (0,00,1n,q)a: PROOF. The first part follows from the Sobolev Embedding Theorem. The second part holds since ||u||Loo 3 02(0, 00, m, q)||u||m,q, from m > g and the first estimatefl Lemma 4.2 The norm N f (u)||m,q is bounded by m j ||f(’U)||La+ZZ|lf(')(U)l|Loo Z: 01(k1,-~,killlvklullm'---'||Vk‘U||Lm j=1 i=1 k1+...+k,-=j 1:121 where, for any choice of (k1,...,k,-) as specified above, we can choose numbers r1, . . . , r,- such that Tl=qa (4'4) ifj = m and i = 1; otherwise, q_ 0 }. Choose 7‘1 2 = T's_1 2 00. Because of the inequality (4.3), r,, s S l S i, can be chosen to satisfy conditions (4.5). D Combining Lemmas 4.1 and 4.2, we conclude the next lemma. Lemma 4.3 “f(ulllmq S h'm.q(|lullm.q)a where m h,,,,(3: ) =02“) q,0 oo)M(0 m, q,3: 3:)+ ZM(i,m,q,:L‘) (4.6) i=1 m x Z Z 0103,...,men/33,33,m,q)...C3(k..r.~,m,q) 22". j=i k1+...+k,‘=j MEI PROOF. By Lemma 4.1 we have ||f(u)l|m S C2(0, (1:0: 00)l|f(U)|lL°o S 02(036130: 00) - M(Oa‘maq: IIUIlm.3)- By Lemma 4.2, the Sobolev Embedding Theorem gives us llvk'ullm S C2061:7‘13m3Cllll’ullrn.q- Hence =1 K). m J ”not... 3 C3M X Z Cl(k1,...,k¢)C2(k71,7'1,7 m, (I k1+...+k,=j 012(k)“ Ti, 77? ,Q)Hu“i Tn ,‘q 37 Lemma 4.4 hm, satisfies conditions (Hf-(hf!) of (4.1). PROOF. To prove (hl), observe that if :1: g y, then M(i,m,q,:r) S M(i,m,q,y), from the formula for 1W (i, m, q, 2;) in Lemma 4.1. Therefore, using equality (4.6), we conclude that the condition (111) is satisfied. The proof of (h2) is simillar. Let C be a constant, such that C2(kl,Tl,Th,q) _<_ C - C2(k,,r1,m,q), l = 1,. . . ,i; C2(O,oo,fit,q) S C - C2(0, oo,m, q); where kl’s, 73’s, and rl’s are as in Lemma 4.2, and C2(lcz,rl,m,q), C2(0,oo,m,q) are the Sobolev constants from Lemma 4.1. Then, using as before the formula for [M(i, m, q, 1'), we get the estimate Mamie: sup |f“’(z)|< sup If“’(z)l=M(z‘.m.q,Cz-)- IZISC'2(0~OO~77I:(?)$ — IZISCz(0,00.m-:Q)(C~T) This allows us to estimate the first part of the right hand side of the equality (4.6) by CC3(0, q, 0, oo)]W(0, Tn, q, Cr), while the remaining part may be estimated by i:M(i,m,q,C$) j=i i=1 X Z C1(k1,...,k,)C2(k1,T1,m,q)...C2(k,-,r,-,m,q)(Ca:)'. k1+...+ki=j k121 The conclusion follows. The proof of (h3) is obvious, since M (i, m, q, 1‘) is bounded in a neighborhood of :1: = 0, i = 1, ...,m; while f(0) = 0 implies M(O,m,q,a:) —> 0 as :2: —-+ 0. This ends the proof of Proposition 4.1. I] Now we solve the abstract ODE associated with our equation. 38 Theorem 4.1 Let X = H’"(S“) {B Hm_1(S"). Suppose the following conditions are satisfied. (A) l > max n + m,m , m > E if n 2 2; n — 1 2 (B) l: 5, m = 1, 2,3,4 if n = 2; (4.7) (C) l=3, m=2 if n=3. Then the equation 111’: At + 1201:), ‘1’(0)= (F30), has a weak solution \II E C([0, 7r], X) provided ||(F, C)||Xis small enough. Here, 0 1 v 0 _ _ ‘ A: , R = , k:l(n 1)2(n+3). 3 — 72 0 w pklvll PROOF. In order to solve equation (4.8), we want to use Theorem 3.1. We already know that the operator A generates a Co-semigroup of continuous operators (Propo- sition 3.3 and Proposition 3.4). Thus, it remains only to prove that R is a locally Lipschitz transformation. We do this in Lemma 4.10, where we estimate the norm ”Btu—mm: Z ||vmlpk-Vm'23~:-vm3(3,—3:2)||L3. (4.9) 1111,1n2,m3Sm-1 771120 Here, we use the following notation. ’ 1— MI 1 1 d i7 : = M = —— / —o(sv1 + (1 — s)o2) (13 (4.10) 0 '01 — '02 '01 — '02 . ds 1 = / o'(svl + (1 — s)v2) ds, 0 39 where o(:r) = |:1:|’. To prepare ourselves for this task, we obtain a series of estimates for the different. parts of the right hand side of the equality (4.9). First, we need some estimates of the L2 and L00 norms of V’mpk. Lemma 4.5 Let p be the conformal factor of the Penrose transform, introduced in Definition 1.3. Then , ,t ‘ va‘lvkllz.2 < 00 if m1: m — 1, llvmlpklle < 00 if m1 < m — 1. PROOF. It is enough to observe, that. by Claim 1.3, p 2 (cost + cos p) /2. Thus pk‘m‘ E L2({t} x S"), since from the assumption (4.7) it follows that k — m1 > —1/2 if m1 3 m — 1. This impies the first inequality. The second one is obvious, since for ml < m — 1, we have k: — m1 > 0 and therefore p"“""1 is a smooth function. [:1 Throughout the proofs of the next four lemmas we will use Lebesgue norms of degree T or s where T and s will be required to satisfy the conditions stated in the following. Lemma 4.6 I 0 S mg, "in; < m. 0 3 mg + m3 3 m. 7712 Z 1 and m > 3, then one x I z 2 can find real numbers T and s, with r, s 6 (2,00) such that max{(), t -- #2} < ,1: < min{::-,-"1—f}; max{0,%— #1} < i < %; + ‘il—i {alt—- NIH 40 PROOF. The possibility of such a choice of T and 8 follows from the following facts. i) Since m — m,- 2 0, we have , i:2, 3. [\DIi—t 2 (Oh—t ii) By the assumption, we have the inequalities mg + m3 3 m and T7 > %, which allow us to conclude that 1_m—m2+1_m—m3=1__2_m+m2+m3Sl-m 2 n 2 —S n n n n [\DIr—a In the next four lemmas we will obtain estimates for the parts of the right hand side of equality (4.12). The most important will be the estimates of ’0 in terms of v1 and 112 as defined in (4.10). We begin with the next fact. Lemma 4.7 73) II’DIILwS C(llvlllum + l|v2l ) an)l—l Zf m > 33> “3“,, s C(H'Ullle + 11331133114 if m _>_ NI: NI: ,lgr$,l>j+1,r21, llei'l Lr S his (llvllljm + lll’zlljm): where h], is the function, described in Proposition 4.1. 42 PROOF. From the expression for 3*: and Proposition 4.1, we have -1 IlevHLr : I Vjo'(sv1+(1— s)v-2) ds . 0 Lr - 1 3 g = / / Vjo’(svl + (1 — s)v2) ds d57'] _ s" 0 . 1 _ l S / “Wet + <1 — smut 3. - 0 .. r l t S / (hp-(H801 + (1 — Sl'li’zlljmllr dsl _ 0 S ’13.:(II’UIII33 + ll'v2ll3'3): which finishes the proof of the lemma. [:1 The next lemma states estimates for the gradients of 37 and vl — 112. Lemma 4.9 If0 S m2,m3 < m, m; + m3 3 m, m > 73", and l > m, then (')||V’"2~V"‘3( )H < C(HmHHm + ||vg||Hn,)l—1HU1 _ ”2|le if m2 = 0; Chm.2(C(lll"1lle + II’U‘ZHHmll HUI 7" ”2| (33)||37(3:, — U2)HLoo g C(Hvlllnm + HU‘ZHHm)l_1||U1 — v2||Hm. Moreover, if m 2 %, then. (iii) ||D(vl — v2)HL2 S C(||v1||Hm + ||L22||Hm)”1||v1 — i22| If": . Remark: Lemma 4.9 is stronger than we need for the proof of Lemma 4.10. For us, mg + m3 3 m — 1 is enough but we will use the stronger version in Chapter 5. PROOF. Let us start the proof of the first inequality. As in the previous proofs, we need to consider several possibilities for the values of 7712 and m3. 43 Case 1: m2 = 0. Then, Ilez’FJVval — 122) “L2 Sll’i’llmo ' IIV'"3(v1 - ”2“le SC(H‘“1| H... + ||222||,.,m)"1 ' H111 - v2”va by the Holder’s inequality and Lemma 4.7. Case 2: m2 > 0. Here, we can choose numbers r and s as in Lemma 4.6 so that by the Helder’s inequality we have ||V7"'217V"'3(v1 — v2)||L2 S ||V""217||Lr - ||V’"3(v1 —- 122)| Ls. Observe, that since m2 > 'f, l > m 2 772.2 + 1, T Z 2, we have, by Lemma 4.8 and property (112) of Proposition 4.1, llezfillu S hm3.r(l|‘vl||m3.r + III/'2llm33) S Chm.2(C(l|v1||Hm + llvzllam))- On the other hand, by the Sobolev Embedding Theorem and i > % — ”‘7’”, we have the inequality “V"“(vl — v2)||L3 S Cllvl — vQHHm. This concludes the proof of Case 2. The second inequality of our Lemma follows from Lemma 4.7 and the Sobolev Embedding Theorem for ”v, — v.2“ L33. To prove the third inequality of the Lemma, let us observe first that by the Holder inequality and Lemma 4.7 1.2 S C(H’lhllnm + ll'll’2llllmll‘1lll’l ‘ ""2“” ||v(v1 — v.2)| 44 Therefore, the conclusion follows since m 2 g and the Sobolev Embedding Theorem gives us the estimate ||vl — 212||L4 S Cllvl — 222||H3n. C] Now we are ready to obtain the crucial lemma for Theorem 4.1. Lemma 4.10 R is locally Lipschitz with Lipschitz constant L(||\I’1||X, ||\I'2||x), which is small when ||\I11||X + ||1142||x33 small. PROOF. Let us start with the proof if l and m satisfy condition (A) in Theorem 4.1. This is the case where we assume that l > max{"—'::2-(fl,m} , m > g, n 2 2. U. Let \II, = 1 , i = 1, 2. To check the Lipschitz condition we need to estimate the wt norm of the difference ||R\I!1—R\112||x= Z valpk.vmzt-va,—3:2)||L3. (4.12) ml .1112.1n3S1n —1 771120 As before, we need to consider several cases depending on m1, m2, and Th3. Namely, we have Case 1: m1 = , — 1 (m2 = m3 = 0). Then IIV’mp“ - vma - We: — 33mm |/\ va—lpk [’2 ' Hf} ' (1)1 — U2)HLoc S C(ll’U1lle +l|v2l|Hm)l-1H7’1 - WHH'": where we used the Holder’s inequality, Lemma 4.5 and Lemma 4.9. 45 Case 2: m1 < m — 1, m2 = 0. Here we also use the Holder’s inequality, Lemma 4.5 and Lemma 4.9, to obtain estimates valpk ' szv ° V7713 (’Ul — ”‘2”le |/\ HlelUkllLoo ' ”sz‘f) ' vmsm — ’02)HL2 |/\ C(“vllle + ”IU'ZI H’")l—1|lvl _ IUQHHM. Case 3: m1 < m — 1, m2 > 0. As in the previous case ||Vm1pk-Vm237-vm3(v, — v2)l|L2 S lle‘Pklle ' “V7712?” Vmstvl — ’U2)HL2 S Chm,2(C(||v1| Hm + H’U2HHml) ' II’U1 — “2|le- This completes the proof of part (A). The proofs of the other two parts are simpler because we do not have factors of the form lepk on the right hand side of (4.12) since there I: = W = 0. But we have to be careful here, since m is not strictly bigger than %, which is crucial for the embedding of the form ||u||00 S C llulIHm. Let us begin the proof of part (B). In fact, first we consider only (Bl)l=5,m=1,n=2. In this situation we cannot use the previous approach because, as we pointed out, m is not big enough. But we can still proceed since HV’nlpk ‘ sz'D ‘ Vm3 (U1 “"" 'l)2)”L2 = “17(111 - villlm S C(IIU1IIH1+||v2||111)l'1||v1— ’U‘zllHI: by the Holder’s inequality and Lemma 4.9. To complete the proof of part (B), we consider the case (B2) l= 5, m = 2,3,4, n = 2. 46 In this case, l > m and m > 325. Consequently, we can proceed as in part (A). The proof of part (C) is completely analogous to the proof of part (B) and therefore will be omitted here. We have completed the proof of Lemma 4.10 and at the same time, the proof of Theorem 4.1. E] v Having a solution \11 = of equation (4.8) we will show that actually v is a w classical solution of the equation (2.2) we are seeking. More precisely, we have the following assertion. Theorem 4.2 If (F, C) E Hm(S") EB Hm‘1(S”) with l satisfying conditions (4.7), m > I; + 2 and ||(F, G)||Xsmall, then the Cauchy problem ('3 + 72)?) = Pklvll, v(0, ) = F, (4.13) [01(0) ) = G: has a solution v E C2((0, 7r) x S"). PROOF. By Theorem 4.1 we have (v, w) E C([0,7r), X) such that r v’ = w, w’ = (A — 72)?) + pk|v|’, (4.14) Since v E C([0,7r),H’"), v E C([0,7r),C2(S")) and v E C2([0,7r),C(S")). Also to E C([0,7r),H’""1). Therefore v’ E C([0,7r), C1(S")) and v E C1([0,7r), C1(S")) and thus v E C2([0,7r) x S"). D 47 4.2 Solutions on Minkowski space We have found solutions to the transformed equations on the Einstein universe. Now we would like to find solutions of the original equations on the Minkowski space. To do this, we first specify a space of functions to which the initial data (f, g) of our Cauchy problem belong. In short, it is a“pull back” by the Penrose transform of space X defined on the Einstein universe. More precisely, we have this definition. Definition 4.1 In the following, we will use notation f,g for functions defined on IR". We will write F == (p‘lf) 06”. G == (p‘l‘lg) 0 C", where c“1 should be understood here as (c,,0,x,,,)-1 : {0} x S" ——+ {0} x IR". It should be emphasized that for the Minkowski time 5170 = 0 (or equivalently, the Einstein time t = 0) the Penrose transform becomes the classical conformal projection of IR" into S" and its formula becomes much simpler. We are going to use this fact in our calculations. Definition 4.2 We define X as a set {(f:.(1) | (KG) 6 X}: X = H’"(S") esH""1(S"), m = 1,2,.... We also define space X0 as the subspace of X with compact support. i.e., X0 :2 {(f,g) E X | f,g have compact support}. The norm on X is naturally defined as a“pull back” of the norm on X. 48 Definition 4.3 For every (f,g) E X we define l|(f,g)llx := ||(F3G)llx. If m = 1, the norm H ' ”x is equivalent to the norm given by the so called Einstein energy E(f,.Q). i-e-, Definition 4.4 For (f,g) E X with m = 1, E(fla) == [IGI2 + IVFI2 + 72172] d5"- Sn Actually, we can express E ( f, g) as an integral on IR X 1R". Proposition 4.2 When f and 9 have compact support and m = 1, I E(fig) = [Rn 17-1012 + IVfl2) — gfzdzr. PROOF. In view of Definition 4.4, it is enough to show that a) G2dS" :/ p"1g2d:1:, JR" Sn b)/ F2313" =/ prdx, c) IVFIdS"= / [—(%+72p)f2+p“lvfl2] 333. Sn Rn We have proved in Chapter 1 that the Penrose transform 0 is a conformal trans- formation with a conformal factor p2; therefore, we can write gag 2 p290); (1,3 = 0,1,...,n. 49 If we restrict ourselves only to the Minkowski time 2:0 = 0 (or equivalently, Einstein time t = 0), we have the following relation between the determinants of the metrics g and g on {0} x S" and {0} x R" respectively. 271 detg,j = p deth. From this we may deduce the volume forms are related by d5" = p"d:1: = p27+1d.r. Consequently, 02 (1811 = / p—2'y—292p2'y+1 (11. = / p-lg‘l d3), n 1R7: Sn which proves (a). Similarly, we obtain / F2 (1511 = / p—‘Z'yf‘21)T2')'+ldI : / pf'z (1113, S" n . n which establishes (b). To get (c), we first calculate the length of VF in terms of the Minkowski coordinates. By Definition 4.1 and conformality of the Penrose transform we have 1 .3 _. 1’1‘~,_ _ .3".-. IVFI2 =§(dF3dF) =. @9041) “’f)~.d(19 "f)) =1) 22pr ”f)|2. 3:1 Using (ix) of Proposition 1.1 we can conclude that 8,(p"7f) = —yp'7—1(8,-p)f +p‘78,f = gp'lflrj +p°(9,-f. 50 Therefore, / lVFlgdSn : / pflZP—inlnif+3z'fl21)27+1d33 Sn . IR 2 2 = /112 p” [%p2$?f2+wx.f03f+(3.02] d1: ' i=1 (- . $2 ‘ n . n . = ./IR" izpl-4l—f2+Z-gr363(f2) +P‘1 (d-fV] d9: i=1 i=1 b . . _ . 7 . _ . 323(3) 1- 1)f2 — 5-an +19 1IVflzl dr r 7 . . _ . ~53” — 3232f? +p ‘IVflzl 33:, which is exactly (c). D It should be pointed out that it is not easy to see directly that E (f, g) in Propo— sition 4.2 is a norm because of the negative term involving f. However E( f, g) is equivalent to the first two positive terms in its expression, which looks like a norm for (f, g) E X0 with m = 1. This is exactly what Klainerman [18] used to study the global existence. He started from the consideration of covariance of the linear wave operator under the translation and Lorentz transformations. The problem is that the norm with two positive terms in E (f, g) is not quite conservative. In any case, let us state what. we discussed as a proposition. Proposition 4.3 The Einstein energy E(f, g) is equivalent to / <1 + 141%? + IVfl‘Z) drr . R" PROOF. Definition 1.3 for the Minkowski time 2:0 = 0 implies p"l :- 1+ |:1:|2/4, and the previous proposition gives us E(f,g) 3] R1; p-1(.32+ IVfl2)d:v s / <1 + 1312x312 + 1W) 44:. (Rn. 1' ,’ Therefore, the rest of the proof will be devoted to showing that E(f,.q) 2 0 f1, 1271(5)? i’lVfl2) .33. (4.15) We need two technical lemmas. Lemma 4.11 Let n > 2. There exists a 6 > 0 such that (n—1+6)/ f2dIS/ |$|2|Vf|2d$, IR" IR" for every f E CHER"). PROOF. By the homogeneity of the estimate, it suffices to prove this lemma for f E C(}(B(0,1)), where B(0, 1) is the unit ball in IR" centered at the origin 0. In fact, for any f E C(KIR"), there is a R > 0 such that the support of f is in B (0, R). Therefore f E C3(B(0,R)). Let f(zr) := f(Rm). Then f E C6(B(0, 1)). If the lemma holds for this f, we have (71.- 1+ 6) f2(a:) (1:17 S / I11:|2|Vf|2da:. Rn Rn Substituting y = Br we get (n — 1 + (5)/1R" f2(y)dy = (n — 1+ (5)1?" IR“ f2(:l:)da: (4.16) 5 en] lxl2|Vf|2(m)d-r= / 13121vf12<3>d3 (4.17) Thus we only need prove the lemma for f E C6(B (0, 1)). Let us start by proving this fact for f spherically symmetric. That is, if f E C(l(B(0,1)) and f = f(r), n > 2, then 00 ‘ CD ‘ (n — 1+ 6)/ f2(7“)7'"—1(l7" S / ffrn+1(lr. 0 . 0 By integration by parts, we have 1 1 1 / 2r"ffr dr =/ 7""(f2)r dr 2 Tnf'zl; —/ nT""1f2 dr. 0 0 0 Hence, 1 f 23" f f, 33 0 1 1 . 1 . S / er"_1f2dr +/ —r"+1f,.2 dr, 0 0 5 1 / nrn‘lf?‘ dr = 0 I 1 f0 2 0. We rewrite the above inequality in the following form. 1 1 e(n — e) / flzr’“l dr S / f,gr"+1 dr. (4.18) . 0 0 We are interested in 1 < 3)} "'< 1") max 5 n — = — n — — = —. e>0 2 2 4 In this case, Inequality (4.18) becomes 7,2 00 1 —4— / fZT"_1dr S / ff'r"'1(lr. . 0 0 This completes the proof for symmetric f since a: = n — 1 + (":2)2 = n — 1 + 6 with 6=(1'—.;'—2)2>Oifn>2. 53 We can now finish the proof of the lemma by establishing the estimate for general f E C6(B(0, 1)), which may not be necessarily radially symmetric. Let us write f in terms of spherical harmonics qfij [24]. f(Jr) = f(w) = Z f3(r) - (to). 3:1 Then, by using the estimate for symmetric f ,(T) derived earlier and the orthogonality of qu 011 5"“, we get 00 1 f2 (1117 =/ f2(7":w)r"_ldwdr =/ f2(r,w)r"_l dwdr IR" 0 S"-1 o 571-1 1 00 0° 1 _ ‘2 . .n- , _ , n— _/0 ijh)? 1(1r_Z/0 fJ2(1)r ldT j=1 51:1 1 1 . 1 1 . < - 81' ‘2 n+ll = - n+1 2d, 1 _Ej ——-n_1+0/0( f3)?“ (T _n—l+c)/OT .Sflf, ..wr 1 22 1 f 2 =—-—. d <———-——- 1: V d'. n—l+b/nrfr x_n—1+6 Rnllll fl I This ends the proof of lemma 4.11. [3 Using this lemma, we prove the following. Lemma 4.12 There is a positive constant C such that for any f E CMR"), fgdzz: S C/ p"1|Vf|2 — gfzdr. .11" R" I 2 u D PROOF. Note that, s1nce p‘1 = 1 + L1}, 1t suffices to prove that /(I+%)f2d.rS/ g|Jrl2|Vf|2dL 54 which is equivalent to 4 , . . £1301] fldxg/ Ia'IZIVflzda‘. C Rn Rn From Lemma 4.11 we know that the inequality above is possible if 4 QC - 4 —%——7=n—1+o.1..e, C=g>0. This finishes the proof of the lemma. C] Now we can finish the proof of Proposition 4.3 using Lemma 4.12 and Proposition 4.2. In fact, it.“ + Irena + IVfI'Z) dz: _ . _ 1 z s 4/Rflp1(9+|Vf|2)d~E—4[E(f,g)+ 2[WW] = 4 [E(M) + game] = 4(1+§C) E(f,9)- a We arrive at the final phase of Chapter 4, where we prove the main result for the existence of global solutions in Minkowski space. Theorem 4.3 Ifl satisfies conditions (4.7), m > §+ 2, and ||(f,g)||;( is small enough, then the Cauchy problem Du = lul‘, ”(03 ) _ fa 117.0(0, ) 2 g: has a global solution u E C2([O, 00) x IR”). PROOF. Let F and G be as in Definition 4.1. Then by Definition 4.3 and the assumption about the size of ( f , 9), we know that we can control how small the norm ||(F,G)||X is. Therefore, using Theorem 4.2, we can find i) E 02([0,7r) x S") such that. Then, if we let u = 1371), we see that Du = |u|l, as discussed in Chapter 2. Also, ”(07 ) = 1,740,) : p71? : fa ux0(0, ) = p - 8t(p7u) = p7+1ut(0) = 9. Finally, u E C 2([0, 00) x 1R"), because the Penrose transform is a smooth transforma- tion. This completes the proof of the theorem. [:1 CHAPTER 5 Global Existence for Quasilinear Equations In this chapter, we analyze quasilinear equations of the form Du = luxolla luTlli ”(09 ) = f7 are“): ) = 9 Although, they are different than the semilinear equations we have encountered in Chapter 4, our method of finding global solutions still works here. The reason for this is the possibility of transferring the equations to the Einstein universe and obtaining the following non—linear equations (51 + flv = pk|T(v)l’. 21(0, ') = F, (5.1) 1),,(0, ') = G, 57 where T(u) was defined in Theorem 2.4. As before, one changes equation (5.1) to the abstract ordinary differential equation VzAm+Rm) on the Hilbert space X = H '"(S")EBH "“1 (S n). The only difference between this ODE and the one we had in Chapter 4 is the form of the operation R. Nevertheless, we are again able to show that R is Lipschitz continuous with small Lipschitz constant and we can use Theorems 3.1 and 3.2 to claim the existence of a solution 1) E C ([0, 7r), X). Now, we would like to state the main theorem of this chapter. Theorem 5.1 Assume l is a real number greater that max{2%":+1—",m}, m > g + 1, and n 2 2. IfX is the space defined in Definition 4.2 and ||(f,g)||,1» is small enough, then the Cauchy problem Du = lumll, “'(Oa ) _ fa ”30(03 ) = g has a global solution u E C([O, 00) X IR"). Let us remark that we state our result only for the right hand side of the equation of the form Iumoll. The case when the right hand side of the equation is equal to Iu,.|l will also be covered by our proof. PROOF. We write our equation as W=AW+R 58 where A is defined by (3.1) and, by Proposition 3.4, is a, generator of a Co-semigroup {St}, and 0 R(\Il) 2' 1 ple(’U)I’ T(u) = %[—7 sin t cos pi) — sin t sin pup + (1 + COS t COS film]- In the rest of the proof we will use the following notation to simplify our arguments. 1 T; = —2-(h.1’Uz' ‘l‘ hQ'ULp + hgw,), h1,h2,h3 E Coo((—7T,7T) X S”), (5.2) where (uhwi) E X,i = 1,2. Analogous to introducing i) in the proof of Theorem 4.1, we define here ~ 1 1d T: — .T 1—.T« d .3 11—21,], d30(91+( .) 2) s (5) 1 =/ o’(sT1 + (1 — s)T2)ds, 0(1‘) = |$|l. 0 As we have already mentioned at the beginning of this chapter, the only difference between the results in Chapter 4 and here is the new form of the transformation R. Therefore, we will focus mainly on checking the Lipschitz condition for the transfor- mation R. First let us prove the following estimates. Lemma 5.1 For the operators T,, i : 1, 2, defined in (5.2) we have i) ”Til Hm~l S Cll‘I’lee ll) ”Tl _ Tzl I’m—I S CH‘I’I — \II‘ZIIX. 1““ ~"2 " ll ~ ll 4. " uJ‘uJJ, - J "Jq ( UJ" U“).- l 59 PROOF. To prove the first estimate, we use the fact that hl, h) and h3 are smooth functions, and we can easily see that IIT.| Hm-l = 23:? lleTz‘llL2 S C [Ex-"=0 llvjvilll.2 + 23:01 llvjwillL'z] = C lllvz‘llH'" +|lwi|le-1] = CH‘I’illx- The second estimate is obtained in a similar fashion from the linearity of T,- in terms Of \Illj E X. C] Now we arrive at the main result of our proof. Lemma 5.2 R is locally Lipschitz with Lipschitz constant L(||\111||X, ”\I’QHX) which is small if ||\111||x + ||\112||X is small, i.e., a: —+ 0+,y —+ 0+ implies L($, y) = O. PROOF. The proof is similar to the proof of Lemma 4.10. Note that IIR\P1— Manx = Z ||V’"‘(p") - vmzr - Wm — T2)||L2- m1+m2+m3£m—1 03171, We need to consider several possibilities for the values of m1, m2, and m3. Case 1: m1 = m — 1 ( m2 2 m3 = 0). Using Holder inequality, Lemmas 4.5, 4.9 and 5.1 give us ||vm1pk - vmzr . vmur1 — mum l/\ llV7"'—110kl|1.2 ' HT(T1 — T'zllloo C(ll‘I’IIlX +ll‘1’2llxll_1'l|‘1’1— ‘I’zllx- |/\ 60 Case 2: 7712 = m — 1 (ml 2 mg 2 0). In this case, we use the expression for T and Lemma 4.8 to get llvm119k'vm2T'V'n3lT1— Tzlllm llpk Chvn-l.2(llTllll-Im—1 + llT‘zlle-I) ' “Tl — Tzlle-l S Chm~1.2lll‘1’1llx +ll‘1’2llxlll‘1’1 — ‘1’2llx- |/\ 14°C ‘ IIV’"‘1TIIL2- ”Tl" 772'le |/\ Case 3: m3 = m — 1. Here we use Lemmas 4.5 and 4.7 to obtain ||vm1pk . Vm2T-Vm3(T1 — T2)||L2 S Hka Loo-1|Tllmo'llvm’lC’i-7172)||L2 < C'(ll‘1’1llx +ll‘1’2llx)1_lll‘1’1— ‘I’2llx. Case 4: m, < m — 1, for all i = 1, 2, 3,. By using Lemmas 4.5 and 4.9 (for m — 1 instead of m), we obtain the following estimates. ”vmlpk . vmgj'i . vmg (Tl _ T2)||L2 S ”Pk—ml H120 ' ||V""2TV'"3(T1 — T2)HL2 S L(llTll firm—l, HTzllllm—I)HT1 — TZHHm—l S Llll‘I’il Hm. II‘I’zHum) l‘I’l - \IIQHHm. The smallness of the Lipschitz constant when H‘I’iHHm and ||\112||Hm are small is ob- vious from the argument. This ends the proof of the theorem. D BIBLIOGRAPHY BIBLIOGRAPHY [1] Aubin, T., Nonlinear analysis on manifolds, Monge-Ampere equations, Springer- Verlag, Berlin, New York, 1992. [2] Baez, J. C., Segal, I. and Zhou, Z., The global Goursat problem and scattering for nonlinear wave equations, J. of Functional Analysis 93 (2) (1990), 239-269. [3] Brenner, P. and von Wahl, W., Global classical solutions of non-linear wave equations, Math. Z. 176 (1981), 87-121. [4] Choquet—Bruhat, Y. and Christodoulou, D., Existence of global solutions of Yang-Mills, Higgs and spinor field equations in 3+1 dimensions, Ann. Sci. Ecole Norm. Sup. 14 (1981), 481-506. [5] Christodoulou, D., Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), 267—282. [6] Eisenhart, L. P., Riemannian Geomtry, Princeton University Press, Princeton, " 1966. [7] Georgiev, V. and Schirmer, P. P., Global existence of low regularity solutions of non-linear wave equations, Math. Z. 219 (1995), 1-19. [8] Ginibre, J. and Soffer, A. and Velo, G., The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal. 110 (1992), 96-130. [9] Glassey, R. T., Finite-time blow-up for solutions of nonlinear wave equations, Math. Z. 177 (1981), 323-340. [10] Glassey, R. T., Existence in the large for Du = F (u) in two space dimensions, Math. Z. 178 (1981), 233-261. [11] Grillakis, M. G., Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Ann. of Math. 132 (1990), 485-509. 61 [12] [13] [14] [16] [17] [18] [19] [20] ’[21] [22] [23] i241 62 John, F., Blow-up of nonlinear wave equations in three space dimensions, Man. Math. 28 (1979), 235-268. John, F., Blow-up of quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math. 54 (1981), 29-51. John, F., Nonexistence of global solutions of [la 2 £17 (ut) in two and three space dimensions, Mathematical Research Center, Univ. of Wisconsin, Technical Summary Report No. 2715. John, F., Blow-up for quasi-linear wave equations in three space dimensions. Comm. Pure Appl. Math. 34 (1981), 29-51. John, F. and Klainerman, 8., Almost global existence to general nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math. 37 (1984), 443- 456. Jorgens, K., Das Angfangswertproblem im Grossen fiir eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77 (1961), 295-308. Klainerman, 8., Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), 43-101. Klainerman, 8., Global existence of small amplitude solutions to nonlinear Klein- Gordon equations in four space-time dimensions, Comm. Pure Appl. Math. 38 (1985), 631-641. Klainerman, S., The null condition and global existence to nonlinear wave equa- tions, Lectures in Appl. Math. vol. 23, 1986, 293-325. Klainerman, S. and Ponce, G., Small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36 (1983), 133-141. Lindblad, H. and Sogge, C. D., On existence and scattering with minimal regu- larity for semilinear wave equations, J. Funct. Annal. 130 (1995), 357-426. Lindblad, H. and Sogge, C. D., Long-time existence for small amplitude semilin- ear wave equations, American Journal of Mathematics 118 (1996), 1047-1135. Miiller, C., Spherical Harmonics, Lecture Notes in h/Iathematics No. 17, Springer- Verlag, Berlin, New York, 1966. [25] Pedersen, J. and Sega], I. E. and Zhou, Z., Massless a); quantum field theories and the nontriviality of 953, Nuclear Phys. B, 376 (1992), 129-142. 63 [26] Pedersen, J., Segal, I. E. and Zhou, Z., Nonlinear quantum Fields in 2 4 dimen- sions and cohomology of infinite-dimensional Heisenberg group, Trans. Amer. Math. Soc. 345 (1994), No. 1, 73-95. [27] Penrose, R., Conformal Treatment of Infinity, in Relativity, Groups and Topol- ogy, (B. DeWitt and C. DeWitt eds), Gordon and Breach, 1963. [28] Rauch, J ., The u5-Klein-Gordon equation, Nonlinear partial differential equa- tions and their applications (H. Brezis and J. L. Lions, eds), Pitman, Boston, _ 1982, 335-364. ‘ [29] )Schaeffer, J ., Wave equations with positive nonlinearities, Ph.D. thesis, Indiana 1 University, (1983). [30] Schaeffer, J ., The equation Bu 2 [ulp for the critical value of p, Proc. Roy. Soc. Edinburgh 101A (1985), 31-44. [31] J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Annals of Mathematics 138 (1993), 503-518. [32] Segal, I. E., and Zhou, Z., Convergence of massive nonlinear scalar quantum field theory in the Einstein universe, Annals of Physics 218 (1992), 279-292. [33] Segal, I. E., and Zhou, Z., Convergence of quantum electrodynamics in a curved deformation of Minkowski space, Annals of Physics 232 (1994), No. 1, 61-87. [34] Segal, I. E., Reduction of Scattering to an invariant finite displacement in an Ambient space-time, Proc. Nat. Acad. Sci. USA 81 (1984), 7266-7268. / [35] Sideris, T., Global behavior of solutions to nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations 8 (12) (1983), 1291- 1323. [36] Sideris, T., N onexistence of global solutions to semilinear wave equations in high dimensions, J. Difierential Equations 52 (1984), 378-406. [37] Strauss, W., Nonlinear scattering theory, Scattering theory in Mathematical Physics, NATO Adv. Study Inst. Sec. C: Math. Phys. Sci., vol. 9, Reidel, Boston, 1979, pp. 53-78. [38] Strauss, W., Nonlinear scattering at low energy, J. Funct. Annal. 41 (1981), 110-133. "‘1111111]lilillli