WWWWWWWIN!”WWWWW”!: 134 047 THS IHIHHIIHHIIHIIllllllllllllllllllIlllllllllllllllllllllll 293 01812 7179 This is to certify that the dissertation entitled H. ‘_‘ ' r-ILIJ49 ~ / 1.511? [g/obd-pr a; All/£441.05»; 4L: A‘Qk" lyVonr— [/an1L)‘, é-“(ZVCQ’I'VAJV‘S presentedby £M7rm€ r4..30/(Léu' ( has been accepted towards fulfillment of the requirements for Pk D degree in M‘LdO/Wfité %@a%% ‘Major professor Date 4/5/{7 MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771 LIBRARY Michigan State University PLACE IN RETURN Box to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 1!” WWW“ FINITE-TIME BLOW-UP OF SOLUTIONS TO NONLINEAR WAVE EQUATIONS By Eugene A. Belchev A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1999 ABSTRACT FINITE—TIME BLOW-UP OF SOLUTIONS TO NONLINEAR WAVE EQUATIONS By Eugene A. Belchev This work studies the finite-time blow-up of solutions to the equation u“ — Au 2 F (u, Bu) in Minkowski space. Results available so far involve complicated analysis near the wave front. We develop a new technique which simplifies some of the ex- isting arguments. The approach we use is a modification of the so—called method of conformal compactification. In this we are motivated by the work of Christodoulou, and Baez, Segal, and Zhou on nonlinear wave equations, as well as the recent devel- opments in the rigorous theory of nonlinear quantum fields. In Chapter 3 we study the semilinear case u“ — Au = p’klull, where p is a conformal factor approaching 0 at infinity. We show that in this case the solutions blow up in finite time for small powers I, while having an arbitrarily long life-span for large Z. In Chapter 4 we prove finite-time blow-up for a class of quasilinear equations and develop a technique to generate more examples of finite-time blow-up. ACKNOWLEDGMENTS I would like to express my gratitude to my advisor, Professor Zhenfang Zhou, to whom I am deeply indebted for his constant help, guidance and excellent advice. I would also like to thank Professors Dennis Dunninger, Michael Frazier, Baisheng Yan, and William Sledd for their valuable suggestions. My warm thanks to Dr. Mariusz Kepka for being a great friend and colleague; I have benefited greatly from our many interesting discussions. I am grateful to my wife, Gera, for her love, care, and understanding. iii TABLE OF CONTENTS Introduction 1 Compactification of Minkowski Space 1.1 Conformal Transform ........................... 1.2 Transforming the d’Alembertian ..................... 2 Transforming the Equation into Einstein Universe 2.1 Modifying the Transform ......................... 2.2 Transformed Equation .......................... 3 Semilinear Case 3.1 Finite-Time Blow-Up ........................... 3.2 Comparing the equations [:lu = lul' and Clu = p"‘|u|l ......... 4 Quasilinear Case 4.1 Finite Time Blow-Up ........................... 4.2 Generating More Examples of Finite-Time Blow-Up .......... References iv 1 9 9 13 17 17 20 24 25 34 41 41 44 48 Introduction The Cauchy problem for a nonlinear evolution equation, when well-posed, determines a local-in-time solution. However, an important feature of nonlinear PDE, and nonlin- ear evolution equations in particular, is that their solutions may develop singularities as they move away from the initial state. First, a solution may contain singularities in its data, of which one would like to follow the propagation. Second, a solution may form a singularity in a given function space, but may still remain “regular” according to a weaker measure of regularity. Finally, beginning from smooth data, a solution may develop a singularity in finite time; this phenomenon is called blow-up, and we say that the solution blows up in finite time. The interpretation of blow-up in physical terms often poses dificulties; blow- up may indicate a real phenomenon, but it may also be a failure of the physical model. A physical example of a finite time blow-up is the solution of the semilinear Schrodinger equation in space dimension one, . _ (—1 mt — um —— |u| u, which blows up in finite time at, it is believed, only a single point. This corresponds to the focusing of a laser beam (see [16]). In this dissertation we concentrate on the finite-time blow-up of solutions to the nonlinear wave equation u“ — Au 2 F(u, Bu) (1) for (t,x) 6 IR X R", where A is the Laplacian in x and 0 is any derivative. This equation arises in different areas of applied mathematics, physics, and engineering, and describes such familiar and important processes as the movement of vibrating strings, drum heads, sound and electromagnetic waves, etc. In 19503 nonlinearities like F(u,8u) = mu + u3, m _>_ 0, were preposed as models in relativistic quantum mechanics with local self-interaction (see [26] and [27]). To model effects thought to arise in the case, for instance, of spinor fields u, equation (1) also has been considered in space dimensions n 2 3 (see [27]). The so-called a-model for the pure gauge solutions of the Yang-Mills equations involves nonlinearities of the type F(u,au) = u (lutlz — War”), where Vu is the spatial gradient of u (see [31]). Many blow-up results have been proved for nonlinear wave equations beginning with Keller [14] in 1957. Consider first the Cauchy problem for the semilinear wave equation u“ — Au = F(u), (2) ”(01): fr ut(0i ) = g' (3) Early results by Keller [14] in 1957, Jorgens [12] in 1961, Glassey [4] in 1973, and Levine [17] in 1974, showed that this problem does not admit a global solution when the initial data are large in some sense. On the other hand, John [9] proved in 1979 that in three space dimensions there are always global solutions of the problem u“ — A“ = lull, (4) “(0’ ) = f, 714(0)) : g) (5) forl > 1 + «2 and suitably small initial data, whereas forl < 1 + \/2 a global solution does not exist for any smooth non-trivial data with compact support. Thus he was the first to show that even small (in L°° norm) solutions could blow up. John’s result led Strauss [35] to conjecture in 1981 that in dimensions n 2 2 the critical power, dividing between global existence and finite-time blow-up, for the problem (4)-(5) should be the positive root lo(n) of the polynomial (n — 1)l2 — (n + 1)l — 2. Interestingly enough, the critical power lo(n) plays a prominent role in the scattering theory of nonlinear Schrtidinger equations; it appears for the first time explicitly in Strauss [34]. Subsequently, Glassey [5, 6] in 1981 verified the conjecture in two dimensions by showing that 10(2) = %(3 + x/fi); in addition, Schaeffer [24] proved in 1985 finite- time blow-up for the critical power. For higher dimensions 11. > 3, Sideris [33] proved finite-time blow-up for initial data satisfying a certain positivity condition while global existence was shown in 1996 by Lindblad and Sogge [18] but for spherically symmetric initial data only; for general initial data they were able to prove global existence only in dimensions n _<_ 8. There are also many blow-up results for the Cauchy problem for the quasilinear wave equation ”(01): f: at“)! ) = g' (7) For instance, the equation Utt — A“ = a“? + b|Au|2, a + b 7'é 0: for dimension three has arbitrarily small solutions which blow up in finite time (see [11] and [7]). If we restrict the nonlinearity in equation (6) to time and radial derivatives of u only, we obtain a class of quasilinear problems governed, as in the semilinear case discussed above, by a critical power. Thus it was conjectured for the equation u“ — Au 2 [ut|', |u,.|l that the critical power is l1(n) = E-i- for n 2 2. Toward verifying this conjecture, for 11(2) = 3 finite-time blow-up has been shown by John [10] for nonlinearity lutl3 and by Schaefl'er [25] for nonlinearity |u,.|3. The case ll (3) = 2 was verified by Sideris [32], who proved in 1983 finite-time blow-up for nonlinearity |u,.|2 and global existence of small radially symmetric solutions for nonlinearity lutI', l > 2. For 11(5) = %, Schaeffer [23] proved in 1983 finite-time blow-up for fairly large class of initial data. In high dimensions, Rammaha [22] proved in 1987 finite-time blow-up for spherically symmetric solutions; his result also included the critical power for n odd, whereas J iao [8] showed in 1996 blow-up for the critical power in even dimensions. In addition, global existence for small initial data was proved by Klainerman and Ponce [15] in 1983 but only(!) for powers I > 5342;1- > 11(n). Most proofs of blow-up reduce the PDE to an ordinary differential inequality for some functional H (u(t)) of a solution u. The inequality is then solved, subject to appropriate initial conditions, so as to obtain a lower bound for H (u(t)) that blows up at some finite time. If the definition of H assures that it is finite for globally existing u, then the blow-up of the functional establishes the nonexistence of u beyond a finite time. The typical local-in-time existence theorem (see Segal [28] and Kato [13]) asserts that either a solution u exists for all time or else some norm of u becomes unbounded as t approaches some finite time T‘. Thus we obtain that u blows up in time T’. The approach we are going to use is a modification of the so—called method of conformal compactification. In this we are motivated by the work of Christodoulou [3], and Baez, Segal, and Zhou [1] on nonlinear wave equations, as well as the recent developments in the rigorous theory of nonlinear quantum fields (see [19, 20, 29, 30]). The method of conformal compactification is based on an idea by Penrose [21] dating back to 1963. In order to study the nature of infinity in the various cosmological models, he suggested that a given physical space-time be compactified by conformally embedding it into a compact subset of the Einstein universe E = IR x S“; the “finite” boundary 0 of this subset would thus represent the“infinity” of the space-time. To be more specific, let us introduce coordinates on S” by regarding it as a unit sphere in 113"“: 1’12 + Y; + - ~ - + Y3“ = 1; thus a point in E is represented by (T, Y1, . . . ,Yn+1), T being the Einstein time. Define the map c : Mo —> E by c(t,x) = c(t,:cl,...,:r,,) = (T,Y1,...,Y,,+1), where t2—x2 sinT=pt, cosT=p(l— 4 ), TE(—7T,7T); t2—x2 Kiszji j:1:'°-in; Yn+1:p(1+ 4 )i with _l t2—x2 2 2 = t2 1— . We take the point of observation to be the north pole T 2 Y1 = Y2 = - - . = Y,, = 0, Yn+1 = 1 and denote by p E [0, 7r) the distance on S" from that point. It is easy to see that theimage of M0 under c is C(Mo)={p—7r too. The vertices I i of Ci represent past and future infinities, whereas the point Io : T = 0, p = 1r represents spatial infinity. Consider on M0 the Minkowski metric g = dt2 — dx2 = dt2 — idm? i=1 and on E the metric g = dT2 — d82, where dS2 is the canonical metric on S". The map c is a conformal map between the Lorentz manifolds (Mo, 9) and (E, g) with a conformal factor p, i.e., CE = p29. Let E] = a: -— A and If] = 0%. — Asa be the d’Alembertians relative to g and g respectively, A31: is the Laplace-Beltrami operator on S". We will see that the operators El and CL, = El + 32, s = "—g—l, are conformally covariant; for this reason [:16 is called the conformal d’Alembertian. In fact, the solutions of the wave equation on Mo and the conformal wave equation on E are in one-to-one correspondence via the relation u r—-> P’UIC(M0)- We will modify the conformal transform c by composing it with a one-parameter family of dilations thus obtaining a one-parameter family of conformal transforma- tions. We will then use these mappings to transform into the Einstein universe E the equation Du = p'k|Lu|', l> 1, (8) where k = 31 — £233, and Lu is defined by Lu :2 a(t, r)u¢ + b(t, r)u,. + C(t, r)u, r 2 |x|. In order to be able to solve this equation we have to prescribe initial data over some space-like hypersurface. We choose the hypersurface given by the equation t = O and prescribe “(0,30 = f(X), 111(0)") : 90‘): x E Rn' (9) We will prove that for a fairly large class of nonlinear interaction terms and com- pactly supported initial data every solution of the Cauchy problem (8)-(9) blows up in finite time. The factor p"c in equation (8) shows up there for a rather technical reason; it allows us to avoid having to deal with singularities along the boundary in E of the compactified Minkowski space C(Mo). However, as the parameter R increases with- out bound, this factor tends to 1 uniformly on any compact subset of Mo. In the semilinear case Lu = u, this renders equation (8) as a good approximation for the “classical” equation Bu 2 lull. In fact, we will see that in this case equation (8) is also governed by a critical power. The parameter R plays an important role in proving the blow-up results. We use the fact that as R increases, the support of the initial data for the transformed in E equation decreases; therefore, for any Einstein time T < g, we may choose R large enough so that the support of the solution at time T is contained in C(Mo). Thus a finite-time blow-up in E would imply a finite-time blow-up in Minkowski space. Our work is organized as follows. In Chapter 1 we discuss the compactification of Minkowski space with a conformal transformation c and prove some of the properties of this compactification. In particular, we show that the d’Alembertian El transforms “nicely” under c into the conformal d’Alembertian Dc. In Chapter 2 we modify the transform c and then use it to transform the Minkowski space Mo into the Einstein universe E. In Chapter 3 we study the semilinear case Lu = u and show that in this case the solutions of (8) blow up in finite time for small powers I, while having an arbitrarily long life-span for large I. Finally, in Chapter 4 we prove finite-time blow-up for a class of quasilinear equations and develop a technique to generate more examples of finite-time blow-up. Our main results are the two finite-time blow-up theorems stated as Theorem 3.1 and Theorem 4.1. Chapter 1 Compactification of Minkowski Space Penrose [21] suggested that to study infinity in Minkowski space Mo = IR x IR“, it would be beneficial to conformally embed Mo into a compact subset of the Einstein universe E = R x S“. The conformal map c : Mo —> E be proposed is the Lorentzian metric analog of the usual stereographic transformation from R" to S". In this chapter we define the map c and prove some of its basic properties. 1 .1 Conformal Transform We consider the (n + 1)—dimensiona1 Minkowski space Mo, which is IR x IR" endowed with the metric g = ah:2 — dx2 = dt2 — def. i=1 We also define the Einstein universe E to be R x S ” endowed with the metric g = de — d52, where dS2 is the canonical metric on S". We introduce coordinates on S” by regarding it as a unit sphere in RV”: Y,2 + Y} + - - - + Y3“ = 1. Thus a point in E is represented by (T,Y1, . . . ,Yn+1); we will call T the Einstein time. Definition 1.1. The map c : M0 —+ E is defined by c(t,x) = c(t,:1:1,...,a:,,) = (T,Y1,...,Y,,+1), where t2_x2 sinszt, cosT=p(1— 4 ), T€(—7r,7r); t2—x2 l/jszji j=1i°°°ini Yn+1=p(1+ 4 ); with NIH p=lt2+(1-”2"”>1 One checks that t2— 2 2 p2t2+p2(1— 4x) :1, which implies that T is well-defined. Also, C(t, x) E E since 2:: Y]? = 1. It is often more convenient to write the map c in spherical coordinates. To do that we represent a point (t, x) 6 Mo as (t, r,w), where r = [x] and w = 1% E Sm‘l. For a point (T, Y1, . . . , n+1) E E we write (T, p,w), where p E [0,1r) is the distance on S" from the north pole; thus p and w are defined by the embedding [0,1r) x S”-1 ——> 5” given by 10 (w,p) H (Y11Y2, - - - ,Yn+1), where (Y1, . . . ,Yn) = sinp - w, and Yn+1= cosp. (1.1) Definition 1.2. In spherical coordinates the map c : Mo —+ E is defined by c(t,r,w) = (T, p,w), where t2—‘7'2 sinT=pt, cosT=p(1— 4 ), Te(—7r,7r); 2 2 sinpzpr, cosp=p<1+ 4r), pE[0,1r); with the angular variables w E 5'”—1 unchanged and p=lt2+<1-”2’"”>1 In the following proposition we prove the conformality of c. ”Iv-l Proposition 1.1. The map c is conformal. More precisely, 6'57 2 1329. Proof. Denote by at, 8,, BT, and 8,, the respective partial derivative operators. Considered as sections of the tangent bundle TMo of Mo, 6; and 3,. induce vector fields on the compactified Minkowski space C(Mo) defined by 5, = c.6t and 5,. = c.8p. The metric g in M0 is given in spherical coordinates by g = dt2 — dr2 - rzdw2, (1.3) 11 where dam2 is the canonical metric on 3””. On the other hand, from formulae (1.1) we have for the metric g on E, [y = de — dp2 — sin2 pdw2 = dT2 — dp2 — p2r2dw2. (1.4) We compare (1.3) and (1.4) and take into consideration that TMO decomposes orthogonally into TS "'1 and the span of {6“ 0,}, to conclude that it suffices to show, a(5.,ét)=p2, anew—p", and 9(5t,5r)=0. (1.5) To this end, we first write ~_0T 0p ~_€9T 3P a- atafiatap and a._ mafigap, and then differentiate equations (1.2) to calculate 5%, 535;, g, and 3?. Thus we arrive at the following expressions for 5, and 5,: .. 1 1 t:—(1+cosTcosp)BT—55inTsinp8p, (15) 1 1 —-sinTsinp0T + -2-(1 + cosTcosp) 6p. r Q9: Now, using the fact that BT and 6,, are orthonormal in the metric g, the verification of (1.5) is straightforward. D It follows from (1.2) that in spherical coordinates 1 p = §(cosT + cos p), which shows that p can be extended on the whole Einstein universe E. As we men- tioned in the Introduction, the compactified Minkowski space C(Mo) is bounded by 12 the lightcones Ci = {:tT + p = 1r}. Therefore p vanishes on the boundary of C(Mo), in other words, this boundary is given by the equation p = 0. 1.2 Transforming the d’Alembertian We consider in M0 the d’Alembertian [:1 relative to the metric g and in E the operator [:1c 2 [:1 + 32, s = "—3, where [:1 is the d’Alembertian relative to the metric g. We will prove that Cl and Dc are conformally covariant. Let us first agree on the following notation: throughout the remainder of this chapter we will use u or to to denote a function in M0 and d; to denote a function in E. We will always assume that these functions have enough smoothness so that the derivatives invoked exist. Proposition 1.2. Let u and (13 be related by u = p‘(q3 o c). Then (Beth) o c = p'n—i‘s'lju. (1.7) To prove this proposition we will need the following two lemmas. Lemma 1.1. If 45 2 do c then the following formula is true: ([343) 0 c = 194% + (n - 1)p'3g(dp, d¢)- Proof. Here we adopt the summation convention whereby a repeated index implies summation over all values of that index. The indices used have their values in the set {1, 2, . . . ,n + 1}. Let {01, . . . ,6,,+1} be a local basis for TMo. Denote by 5,- : c..(6,~) the vector field induced on c(Mo) by 6,. Then {51, . . . ,5n+1} forms a local basis for the restriction of TE to c(M0). As customary, we denote g,,- = g(6,-,0j) and g.,- = g(5,,5,); then (9") denotes the inverse matrix of (95,-) and similarly, (g‘i) denotes the inverse matrix of (§,,). 13 Analogously to the Riemannian case, we have in terms of local coordinates fig. (safest), (1.8) [343: where C = det 57,- . Since c‘g = p29, we have gi-oc = ngg- and hence, g‘joc = p"2g‘j J .1 J and C o c = p”+lG, G being |det(g,-,-)|. Therefore using (1.8) we obtain .. - 1 1 _ i. n (W) ° C = p... W“ (p 292,, “net-ct). After differentiating the expression in the parentheses above, we get ([343) ° C = 515- - 5—5 a. (fit/5 as) + (n — 1)p-3g‘j(a.~p)(a.¢) = 1941125 + (n - 1)p‘3g(dp, d¢)- D For the second lemma we use the fact that in terms of the coordinates (t,x) in M0 the operator [3 is simply E] = 6,2 — A, A being the Laplacian in x. Lemma 1.2. The following formula is true: Dp‘ = szpy‘il. Proof. Observe first that Clp=sP3— ”—3'p3(t2-X2) Hence, 14 which combined with 9(dp,dp) = -— implies the result. [3 Proof of Proposition 1.2. Assume that u = p‘(d§ o c). From Lemma 1.1 we have ni3 n 3 p 2 (on) o c = 10‘ch + (n — 1)p"—2‘-’g(dp, dc») +st2¢. (1.9) On the other hand, applying the identity [3(45 ' i/J) = ¢U¢ + ¢D¢ + 290145, dw), we obtain EKIM) = 1”ch + (Dp‘)¢ + 29(dp’, do), which, using Lemma 1.2, implies BUM) = p‘D¢ + 3213;3345 + (n - lipnT49(dp, d¢)- (1.10) Combining (1.9) and (1.10), we obtain pl?” (£1.43) 0 c = D(p‘¢)- [:1 Remark. A well-known result in Riemannian geometry, generalized for the case of Lorentzian metrics, states that for a conformal transform c : (M, g) —> (N, 9) between two Lorentz manifolds of dimension n + 1 we have the following formula for the 15 conformal d’Alembertians on M and N: [(lj —Rn4;1)¢7>] oc=p“n_2fl (Cl —Rn4;l) (pnT—lu). Here R, R and C], [I] are respectively the scalar curvatures and the d’Alembertians for M and N. Taking into account that (M0, 9) is flat, i.e., its curvature is R = 0, and that the curvature of (E, g) is R = —n(n — 1), the above formula simplifies to formula (1.7). 16 Chapter 2 Transforming the Equation into Einstein Universe For a function u in M0 we denote Lu :2 a(t, r)ut + b(t, r)u, + c(t, r)u and consider the equation Clu =p—leull, l> 1, (2.1) n+3 where k = sl — 2 In this chapter we modify the conformal transform c and then use it to transform equation (2.1) into the Einstein universe E. 2.1 Modifying the Transform In this section we will compose the conformal transform c with an one-parameter family of dilations thus obtaining an one-parameter family of conformal transforms CR. 17 We will then reformulate Propositions 1.1 and 1.2 in terms of cR. We consider the one-parameter family of dilations on M0 t dR ; (t,x) H (E’ ), R>0. :olx We first note that dB is a conformal map, i.e., dfig = R“2g. Indeed, for a vector field X E TMo and a function 43 in M0 we have, using the Chain Rule, (dam = X(¢ 0 d3) = gym, which implies for X, Y E TMO, 1 3290(3) = gldaa-Xflmyl = 7239 (X, Y). With this in mind, we are ready now to restate Proposition 1.1. Proposition 2.1. The map 63, R > 0, is conformal. More precisely, egg 2 R"2ng. Proof. It follows immediately from the definition of the pull-back and the Chain Rule that a composition of two conformal maps with conformal factors A and p is also a conformal map with a conformal factor Ap. [:1 Remark. Note that in the above proposition p should be understood as t2 t2—X2 2 __ 1..— R2+( 4R2) From now on we will consistently suppress R in pg to avoid unnecessary pile—up of -1 2 PR=P°dR= notation. We now restate Proposition 1.2. 18 Proposition 2.2. Let u and gb be related by u = Rzp‘((f) 0 63). Then Proof. Assume that u = R2p’()- (2-4) 19 Combining (2.2) and (2.4), we obtain pair—3 (E1413) 0 on = E1(R2iv“r25)- which completes the proof of the proposition. Ci 2.2 Transformed Equation For a function u in E, let us denote Av :2 A(T, p)vT + B(T, p)v,, + C(T, p)v. (2.5) Identifying Mo and cR(Mo), we have the following transformation relation between Lu and Au. Proposition 2.3. Let u and v be related by u = R‘ ng‘v. Then, Av = RfipflLu, (2.6) where the relation between the coefficients a, b, and c of L and A, B, and C of A is given by t2+r2 tr a—A(R+ 4R )+Bfi’ tr t2+r2 b—Afi+B(R+ 4R ), st sr C—A'é-é‘l'BER‘l'C. Proof. Since, as we already noted in Section 2.1, all; maps a vector field X E TMO 20 to fiX, we have from formulae (1.6) on page 12, l 1 8t (1+cosTcosp)8T— fisinTsinp8p, zfi 1 , , l 8, — -—2—I—zsstmp8T + $0 + cosTcosp)8p, where we identify on cR(Mo) the vector fields 8, = c.8t E 8, and 8, = c.8, E 8,. We solve for 81 and 8p to get 6T = Ep-2 [(1 + cosTcos p)0¢ + sinTsinpar], 2 (2-7) 8,, = gp‘z [sinTsinp8t + (1 + cosTcos p)8,]. We next employ the fact that C}; is given by the formulae t t2— 2 sinT=%, COST:p(1——TR2L)’ T€(—7r,7r), t2— 2 sinpzzi-g, cosp=p(1+fi7r—), p€[0,7r), to obtain t2_ 2 t2_ 2 1+cosTcosp=1+p2(1— 4R:)(1+ 412:) t2 tz—r2 2 t2—r2 tfl—r2 E+(l— 4R2)+(1_ 4R2 )(1+ 412?) :p2 and 2 t sinTsinp = 13R; 21 which, substituted in (2.7), gives us t2+r2 tr 87—(R+:—— 4R ———)8t+—R8,, t__2+r2 Therefore, R; I)” +t2-21‘2 tr R1.” tr t2--l-'I'2 1" [ATRI‘LBE‘L 4R )l“ _2_1 _ +t2+r2 tr R‘ P 8".{319 AR+[( 4R )atP‘l‘fi'arP] t2 —sp’lB [(R+4R +1. ) +2R Lap] +C}u. Finally, we use that and to complete the proof. [3 Proposition 2.4. The equation Elu = p‘leu]l 22 transforms under CR into the equation Dev 2 |Av|', where u and v are related by u = R-T-ll’p‘v, and k = 31 — 13—3. Proof. Recall that from Proposition 2.2 we have for ¢ = q; 0 CR, [165 = p‘fingzp‘cbl Consequently, n+3 n+3 p—z—Elcv = pTElc (R&p_‘u) = Rl-z-Lllju = 112%p4‘lLull = piglAvII where in the last equality we used equation (2.6). E] 23 Chapter 3 Semilinear Case In this chapter we will prove finite-time blow-up for the solutions of the following Cauchy problem in M0: Du = 19"‘IUI‘, (3-1) “(or ) = f: 214(0, ) = g, (32) where l > 1 and k = 31 — 9—33. We assume that f, g E X where the space X is defined by X:={f|f608°(R"), f20, }- Let us first fix some notation and recommit ourselves to some already introduced one. In what follows we will identify, for the sake of brevity, Mo and c(Mo); thus, for example, the expression lilo; will be understood where appropriate as (flag) 0 c. We will continue to suppress R in pH = p 0 dB; in addition, we will suppress R in 24 CR = c 0 d3. Accordingly, the map c : Mo —§ E is defined in spherical coordinates by t 2 _ 2 sinT=p—, cosT=p 1—t r , TE (—1r,7r); R 4R2 2 2 (3.3) sinp : p—R" ,. COSp=p(1+ 4R2 ) p€[0,7r); with the angular variables unchanged and t2 tz—r2 2 _ 1-.— R2+( 4R2) 3.1 Finite-Time Blow-Up NI»: 1):: To prove finite-time blow-up we will transform the problem (3.1)-(3.2) to the Einstein universe E. Let us define f = f o C“, 9 = g 0 0-1. We will show that under the transform 0 problem (3.1)-(3.2) transforms into the following Cauchy problem in E: 133196 (“mi (35) where u and v are related by u = R‘fip‘v and p0 = cos2 g. Indeed, equation (3.1) transforms into equation (3.4) by virtue of Proposition 1.2 applied for L and A being the identity maps. To verify the first initial condition it is enough to note only that 1 p p0 := p(0, ) = -2-(cosT + cos p)lr=o = cos2 5. 25 For the second initial condition we use that - 1 Bilbo 2 E“ + cosp) arlT:o = %3TIT=0, and, consequently, dtl i=0 p = 0. Therefore, We now state the first main blow-up result. Theorem 3.1. Let 1 < l < %+ 1, and u be a solution of (3.1)-(3.2) with initial data f, g E X. Then u blows up in finite time. To prove this theorem we will need the following proposition which, although stronger than what we need for the proof of Theorem 3.1, we believe is interesting by itself to warrant stating in its full power. Lemma 3.1. Consider the initial-value problem y: = \/yl+1 _ Ml+1 + M2, (3.6) W) = M, where the constant M Z 2. Let Ly be the life-span of the solution y of (3.6). Then L, ~ M-“T‘. Proof. We set z(t) = M‘1y(M‘l_Tlt). Hence, z' : M—‘ig‘ly' : M—52L1\/yl+1 _ Mz+1 + M2 : \/(M‘1y)l+1 _ 1 + Ml—l. 26 Therefore 2: is a solution of z’ = \/2’+1 — 1 + M1", (3.7) 2(0) 2 1. Denote by L, the life-span of 2:. It suffices to show that L2 ~ 1, i.e., c1 3 L, 3 c2. Step 1. L; S C2. By the Mean Value Theorem, we have z'“ — 1+ M1“ 2 (l+1)(z —1)6‘+ M1" 2 (z — 1) + M“. Setting w = z — 1 we obtain w' — z' 2 \/w+M1—‘, or equivalently, x/w+M1"l _>— 1, which, after integration, gives 2\/w+M1"—2MIT_l 2 t. Hence 2 w+MHZ%+MH, 27 from where it follows that Therefore for t 2 2, we have Zz+1 _1+ Ml-l Z zl+1 _ 1 2 A21“, where we set A = 1 — 2‘"1 > 0. Hence we have 2’ 2 t/Azl“, which we integrate between 2 and t to obtain 131(Z¥_2¥)2fi(t—2)' This is equivalent to [— 1 z 2 IV which implies L, g 2 + cst. = c2. Step 2. L, 2 c1 > 0. Assume z 5 2 on [0, 6]. Hence, z’“ — 1 + M1"1 3 (l + 1)(z — 1)0’ + M1"1 3 cw + 1, since 6’ 3 2l and therefore w’ E \/cw +1. We integrate the above inequality 28 between 0 and t and arrive at c Thus for any t g E, we have Vow+132 or, consequently, 2 6’ Hence for t g 3, we have 2 S 2, which implies that 2 can not blow up before c1 = i.e., L; 2 C1. B Let us now prove Theorem 3.1. Throughout the proof we will use for convenience C as a generic name for a (strictly) positive constant—its values may be different in the various places it appears; what matters is that it is always independent of the Einstein coordinates, as well as the parameter R. Proof. Let u be a solution of the Cauchy problem (3.4)-(3.5). We define the function H (T) by H(T) := f" v(T, -) dS. Observe that the definition of the functional H assures that it is finite for globally existing 1); therefore, the blow-up of H would establish the nonexistence of 1) beyond a finite time. Observe also that as the parameter R increases, the support of the initial data (3.5) decreases at a rate of %. Therefore, for any Einstein time T < 12‘, we may choose R large enough so that the support of v(T, ) is contained in c(Mo). 29 Thus we have that a blow-up for H (T) at a finite time T < % implies a finite-time blow-up for the solution u of (3.1)—(3.2) in Minkowski space. Remark. Note that had we not included the factor 19* in the interaction term of equation (3.1), we would have had a factor involving some power of p in the trans- formed equation in E. This would have made the functional H singular along the boundary in E of the compactified Minkowski space c(Mo). We integrate equation (3.4) on S" and, noticing that by the Divergence Theorem ASH?) d8 = 0, Sn arrive at H”(T) + 32H(T) = |v(T, -)|'dS. (3.8) Sn We use Holder inequality to estimate 1 |v(T, -)|‘d.S' 2 O / v(T, ) dS , S" S" which, combined with equation (3.8), gives us H” + 32H 2 ClHl’. Writing the latter inequality as H” _>_ ClHl' + (CIHl’ — 32H), 30 we obtain H” _>_ CIHI' + (CIHIH — .92) |H|. (3.9) As we showed, the transform c is a conformal transform with a conformal factor 1%, i.e., we have in local coordinates p2 §0B=Egafiv aifi:oila'°'an' If we restrict ourselves only to Minkowski time t = O (or equivalently, to Einstein time T = 0), we have the following relation between the determinants of the metrics g and g on {O} x S" and {O} x R" respectively, 2n det gag = % det gag. From this we can deduce that the corresponding volume forms are related by d5 = %dx. (3.10) Consequently, for R > 1 we have from the assumption for the initial values H0 := H(O) and H6 := H’(0) of H(T), 2 ~ 2 r2 2 H =Rm “f dS=RI‘-T'"/ (1+—) fda: ° 5.090 ) . 4R2 2-11:}. 2 RI—fT’"/ (1 + T?) fdz = CHI-37‘", 31 and 11—] H’ = Rig] (RP—(”I’m d5 = Rid-"f 1 + 3—2— _T gdx ° .. ° . 4122 La. 7‘2 A? 1:1- 2 RI—I "/ (1 + I) gda: = CRz—x ". Since g; > n, we have, for R >> 1, in the right-hand side of equation (3.9) C|H(T)|"1 — s2 2 o, for all T 2 0. Thus inequality (3.9) leads to H” 2 CH‘. We integrate this inequality to get H'2 2 C (H'H _ Hb“) + H62, which, setting y(T) = HJ‘H (C’iHJTT). implies that y is a solution of the initial-value problem 31,2 \/yl+1 _ 1 + CMl—l, where M = Rig—1’". Choosing again R large enough so that M 2 2, we employ Lemma 3.1 to obtain 32 Ly S C for the life-span Ly of y. Consequently, we have for the life-span T0 of H (T) To 3 CM-“T‘ .—_ 0124—1“? -1. It is important to notice that To decreases at a slower rate compared with the support of the initial data. This allows us to choose R >> 1 so that v blows up at a point inside of c(Mo). Finally, it follows from the definition of the map c that for the corresponding Minkowski time we have t0 ~ RTO 3 0124—1"?! 3 CR. :1 Remark. As we know from the Introduction, I < % + 1 is not a sharp critical power dividing between global existence and finite-time blow-up. Having said that, we must note that Theorem 3.1 has the following advantage over the existing blow-up results: the nonlinearity in equation (3.1) is tempered at infinity by the factor p‘“, which for l < %+ 1 approaches 0, as |x| ——> 00; yet, this nonlinearity still produces a finite-time blow-up. As a generalization of Theorem 3.1, we state the following theorem. Theorem 3.2. Let 1 < l < % + 1, and u be a solution of the Cauchy problem Bu 2 F(t,x, u), (3 11) H(O, ') = f, 111(01): 9) where F(t,x, u) 2 c|u|’ and the initial data f, g E X. Then u blows up in finite time. 33 Proof. We write the right-hand side of the above equation as F(t,x,u) = p‘kka(t, x, u) 2 cp—k|u|', then observe that the proof of Theorem 3.1 can be ad lib. modified to accomodate this case. 3.2 Comparing the equations Bu 2 |u|l and Bu 2 19”’°IUIl Consider the equations Bu 2 p—klull, R > 1, (3.12) and Clu = lull, (3-13) and the initial conditions u(0,x) = f(x), u,(0,x) = g(x), x 6 IR", (3.14) where the Cauchy data f, g E X. Let uR be the solution of the Cauchy problem (3.12)-(3.14) and u be the solution of the Cauchy problem (3.13)-(3.14). We will obtain an estimate for uR and u for which we will need the following lemma. Throughout this section, we will denote by W“? = Wm’q(lR") and H m = Wm'2(lR“) the usual Sobolev spaces. 34 Lemma 3.2. Let u 6 Wm'q(R") have a compact support, q 2 1, and m > 2. Then for every function h E Cm(IR), h(0) = 0, there exists a function C such that ||h(U)||wm~ S C(llUwal When q = 2, this result is well-known; for other values of q, see [2]. The following estimate for u R and u is true. Proposition 3.1. Letl > m > 121. Then “19411143l — IUIIIIHWI S C(HUHHm. IIURHH'") ll“ — uRlle + 5(3), where 5(R) —> 0, as R —> 00. Proof. We first observe that llp‘kluRl' - lul’lle—l S l|P"° (luall - lul') HHm—x + l||u|’(P“" - 1)lle-x- Denoting : — —— —_._ —— — 1 ’U — 0 (811.3 +( — s)u) d8 = [01 a'(suR + (1 —- s)u) ds, where a(x) = lmll, we have for the first term in (3.16), ”19‘“ (IURI' - lul’) IIHm—x = HP"? (an — tally--. = Z “V"“p-lc Vm'fi Vm3(uR — u)||L,. 05mgSm-1 i=l,2,3 35 (3.15) (3.16) Using Hiilder’s inequality, we estimate every term in the above sum by ||vm1p"° V'"“b vm3(uR — u)HL, s llvm‘p‘kllL... Ile’fiVm’wR — u)llL2 so nvmzavmsmfl — mum. We choose numbers r,s 6 (2,00) so that § > max{0,-;- — 9—331}, 1 > max{0,% — £121}, and i— + % = %. This is possible since by the assumption we have 1 m—m2 1 m-m3 2m+m2+m3 m n /\ [\DII-ul and also Using again Holder’s inequality we obtain IIVm’fiVm3(uR — mum s ”Wan... vawua — um... Observe that since 1 > 1 — 3511, we have by Lemma 3.2 and the Sobolev embedding ? 2 theorem 1 % ”Wan... _<. [f |l0’(suR + (1 — 3)u)lllym..r as] 0 1 s ( 01||0’(sun +(1— s)u)”;p. ds) S C(lluRIIHm. IIUIIHm)- 36 On the other hand, since -} > %— #1, we obtain by the Sobolev embedding theorem nvmscun — on... < Cum. — uHHm. We now estimate the second term in (3.16). |l|u|'(P"‘ - 1)llgm-1 = Z IIVWIl Vm’ (P‘k - 1) Hu- 03m;Sm—l i=1,2 For every term in the above sum we choose numbers r, s 6 (2,00) as before and again employ the Sobolev embedding theorem and Lemma 3.2 to obtain ”‘7'“ lul’ Vm’ (P"° - 1) Ilia Vm’ (P"‘ - 1) IILs SC lllul’llwm... ||p“" — IIIHm s Ilvmllul' LT SC(|IUlle) HIT" - 1|le = EU?)- This completes the proof of Proposition 3.1. CI Assume now that, for l > m > 3%, the solution u of problem (3.13)-(3.14) in H m is defined globally in time, whereas the solution uR of problem (3.12)-(3.14) in the same space blows up in finite time To for arbitrarily large R. The latter means that ”UR(t)||Hm increases without bound as t —> To. Therefore we may choose 6 > 0 so that “UR(T0 — 6) — H(To — 6“le > 1. (3.17) 37 Denoting U = and UR = u" , u it}; we have by the Duhamel’s principle U(t) = S(t)U(0) + ft S(t — s) a(U(s)) ds, (3.18) UR(t) = S(t)UR(O) +/0 S(t — s) H(UR(s)) ds. (3.19) Here S (t) are the linear bounded operators generated by the linear wave equation, and O 0 a(U) : : [B(UR) : ’ lull P"|u1rz|l respectively. We let 01 = sup ||S( )ll, OStSTo and 02: SUP 010(lluR(t)||Hm.IIU(t)||Hm)a ogthO—a and use 0 < T1 < T; < < T1,..1 < To — 5 to subdivide the interval [0,T0 - 6] into R: subintervals, each of length at most 5%.? We next choose R >> 1 so that k —1 e < (Z 2‘01) . i=1 38 Subtracting equations (3.18) and (3.19) we obtain t . EiUR — U H"- S «U — L'meeHm-l S C1] .EP—k “R I - “UH”: d5~ 0 which, by virtue of Proposition 3.1. leads us to SUP IVCUR — 113an S C2T1 Sllp :‘UR — U ‘Hm 't' C15(R). OS’STI 03551} Since C21] < %. we obtain from the last inequality sup tug — UZIHM g 2C15(R). OSSST1 and hence, IIUR(T1) - U(TlllleeHmrl S C2T1 SUP FUR — U Hm + C15(R)S 205(3)» (330) OSISTI Using T1 as a starting point, we rewrite equations (3.18) and (3.19) as S(t)L-"(T1) + tS(t — s) a(L'(s)) ds, T1 U3“) 2 S(t)UR(T1) + 5(t — s) 3(Ug(s)) ds. T1 U(t) Thus, for T1 3 t 3 T2, we have Hun ’ "1le S HUB — UlleeHm-l S ClllUR(T1) — Lr(Tl)llH"‘eH"‘-1 + C2(t — T1) sup 3:11}; — ungm + C1€(R). T1 SET: 39 Consequently, using inequality (3.20), we obtain sup [lug — UIIHm 3 201250?) + 02(T2 — T1) sup Hun — UHHm + C15(R), TlgtSTz TIStSTz which implies sup Hug — UHHm S (2C1 + 22C?) €(R), T1 StSTz and hence, IIUR(T2) — U(T2lllumeHm-1 S (201 + 22012) €(R)- We continue in this fashion and after I: — 2 steps arrive at i=1 k —1 IIUR(To - 5) - U(To - 6)lle@Hm-1 S (2 2‘0?) EU?) S 1- Finally, we obtain HUR(T0 — 5) — “(To — 5)“ S IIUR(T0 — 5) — U(To — 5)||HmaHm-1 S 1, which contradicts (3.17). Thus we proved the following result for the Cauchy problem (3.12)-(3.14). Theorem 3.3. If the Cauchy problem (3.13)-(3.14) admits a global solution in some Sobolev space H’", l > m > g, then the solution uR of the Cauchy problem {3.12)- (3.14) exists at least on the interval [0,TR], with TR ——> 00, as R ———> oo. 40 Chapter 4 Quasilinear Case In this chapter we will prove finite-time blow-up for the solutions of a certain type of quasilinear equations. We use the same notation and conventions as in the previous Chapter 3. 4.1 Finite Time Blow-Up We consider the following Cauchy problem in Minkowski space M0: Du = p'leul', “(0) ) = f: at“): ) = 9’ where t2 + 1'2 4R Lu=A(R+ 2R Here in Einstein coordinates, A = sin 3T, and C = —s cos 3T. 41 tr st )u, + A-Z—R-u, + (A— + C) u. (4.1) (4.2) We assume that f E Y, where the space Y is defined by Y=={f|f€CS°(IR"). me}. We are ready now to state our main quasilinear finite-time blow-up result. Theorem 4.1. Let 1 < l < % + 1, and u be a solution of (4.1)-(4.2) with Lu given by {4.3), and the initial function f E Y. Then u blows up in finite time. Remark. In the spirit of the remark we made on page 33, we observe that for l < LL—fi < §+ l, the coefficients of the nonlinearity of equation (4.1) approach 0, as |x| —> 00. Thus for such powers this nonlinearity, although tempered at infinity, still produces finite-time blow-up. In contrast, Rammaha in [22] proves a sharp result for l, but uses nonlinearities that are not tempered at infinity. Proof. Let u = Rfip'm. Since u satisfies (4.1), by Proposition 2.4 it follows that Elev = |Avl', (4.4) where, using Proposition 2.3, Av = (sin sT)vT — (3 cos sT)u. Define a function H (T) by H(T) = [a Av d3 = [(sin sT)vT — (3 cos sT)v] dS Sn From the Divergence Theorem we have, A5911) d5 = 0, Sn 42 hence, H'(T)=/ sinsT.El,udS. Therefore, multiplying both sides of equation (4.4) by sin sT and integrating on S", we obtain for T E [0, 11—1), H'(T) 2 sin sT|H(T)|l. Another integration yields 1 1 1 1.23 a(H—fl-a—filzgsm 5"”, Where H°=H(°” or equivalently, 1 1 Hl—l — Hé—l S —C ' 2—T, sm 2 where C is a constant depending only upon n and l. The latter inequality implies that v blows up at most in time To, where To is the smallest solution of H04“) = 03111237". (4.5) Using relation (3.10) for the volume forms, we observe that _n_+_1 2 2 2 H. = —s /. um. )45 = ~sRl’-‘1’" f . (1 + 4%,) fdx = C'RIZ’T‘". where, according to the assumption, 0' is a positive constant. This and equation (4.5) imply that T0(R) -_- om-Hfl‘afl). As in the proof of Theorem 3.1, we observe that the support of the initial data (4.2) 43 when transformed by c, decreases at a rate of i, as the parameter R increases. This rate is slower compared with the rate of To; therefore we may choose R >> 1 so that v blows up inside the compactified Minkowski space c(Mo). Finally, it follows that the solution u of the Cauchy problem (4.1)-(4.2) blows up n l—l in finite time bounded by to ~ RTO = 0(R 2 ). E] 4.2 Generating More Examples of Finite-Time Blow-Up Let us consider the equation in E [1,22 = |Avll, (4.6) where Av = AvT + va + Co is defined as in Section 2.2. We multiply both sides by [B(T, p) and integrate on S" to obtain 1 fflpzmcvds 2 [nuiAvdS . Therefore a choice of A, B, C, and #2 so that 3 3 2 a([flfl'AvdS) Z/nchvdS (4.7) would result in H(T) Z |H(T)ll, 44 where we set H(T)==/ ulAvdS. Thus a choice of A, B, C, and #2 satisfying condition (4.7) would imply a finite time blow-up for a solution 12 of equation (4.6). This, employing the transformation formulae from Proposition 2.3 for the nonlinear part Au, will translate into a finite time blow-up for a solution u of the corresponding equation in Minkowski space. After an integration by parts, we see that inequality (4.7) on A, B, C, and ,u2 is equivalent to f" { (Ali/1):)” + (u%B)Tv, + 0&0)“, + [uiAT +ulC — (n — 1)cotpu%B — (uiB) JUT} d3 p 2 / [uzvrr+2(li2)pvp+(Asn#2+82M2)v] ds. (4.8) Restricting (4.8) to an equality we obtain #2 : Ill-A1 (4 9) 2 _ 2 2m ). — #73)T, (4.10) Asa/1,2 + 32;} — (pic) , (4.11) ,uiAT + ulC = (n — 1)(cot p)u%B + (uiB) . (4.12) p After a differentiation with respect to T, equation (4.9) yields 2 l — I WAT = T‘(I-‘2)T- (4-13) Differentiating equation (4.12) with respect to T, and using (4.13) to substitute for 45 (ulAT)T, equation (4.11) to substitute for (ulC)T, equation (4.10) to substitute for (ulB)T, and again equation (4.10), differentiated this time with respect to p, to substitute for the mixed derivative (uiB)T , we arrive at p l‘ 1 2 2 2 2 2 2 -l—(u )TT + As" (it )+ s u = 2(n - 1)(cotp)(u ), + 204 M- This, considering the fact that A511 restricted to a function depending only upon p is given by (n — 1) cot p6,, + 63p, leads us to the conformal wave equation l — 1 T012)” — A9103) + Szflz = 0 46 REFERENCES 47 References [1] J .C. Baez, I. Segal, and Z. Zhou, The global Goursat problem and scattering for nonlinear wave equations, J. of Functional Analysis 93 (2) (1990), 239-269. [2] E. Belchev, M. Kepka, Z. Zhou, Global Existence of Solutions to Nonlinear Wave Equations, Comm. 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