The Concentration-compactness/ Rigidity Method for Critical Dispersive and Wave Equations

The Concentration-compactness/ Rigidity Method for Critical Dispersive and Wave Equations Carlos E. Kenig Supported in part by NSF

3 In these lectures I will describe a program (which I will call the concentrationcompactness/rigidity method) that Frank Merle and I have been developing to study critical evolution problems. The issues studied center around global wellposedness and scattering. The method applies to non-linear dispersive and wave equations in both defocusing and focusing cases. The method can be divided into two parts. The first part ( the concentration-compactness part) is in some sense universal and works in similar ways for all critical problems. The second part ( the rigidity part) has a universal formulation, but needs to be established individually for each problem. The method is inspired by the elliptic work on the Yamabe problem and by works of Merle, Martel Merle and Merle Raphäel in the non-linear Schrödinger equation and generalized KdV equations. To focus on the issues, let us first concentrate on the energy critical nonlinear Schrödinger equation (NLS) and the energy critical non-linear wave equation (NLW). We thus have: i t u + u ± u 4/N u = 0, (x, t) R N R, u t=0 = u 0 H (R n ), N 3, () and t u u = ± u 4/N u, (x, t) R N R, u t=0 = u 0 H (R n ), t u t=0 = u L (R n ), N 3. In both cases, the sign corresponds to the defocusing case, while the + sign corresponds to the focusing case. For (), if u is a solution, so is u ( x λ N / λ, ) t λ. For (), if u is a solution, so is u ( x λ N / λ, λ) t. Both scalings leave invariant the energy spaces H, H L respectively, and that is why they are called energy critical. The energy which is conserved in this problem is E ± (u 0 ) = u 0 ± u 0, (NLS) () E ± ((u 0, u )) = u 0 + u ± u 0, (NLW) where = N = N N. The + corresponds to the defocusing case while the corresponds to the focusing case. In both problems, the theory of the local Cauchy problem has been understood for a while (in the case of (), through the work of Cazenave Weissler [7], while in the case of () through the works of Pecher [37], Ginibre Velo [4], Ginibre Velo Soffer [3], and many others, for instance [3], [0], [34], [4], etc.). These works show that, say for (), for any u 0 with u 0 H δ, there exists a

4 unique solution of () defined for all time and the solution scatters, i.e., there exist u + 0, u 0 in H such that lim u(t) e it u ± 0 H = 0. t ± A corresponding result holds for (). Moreover, given any initial data u 0 ((u 0, u )) in the energy space, there exist T + (u 0 ), T (u 0 ) such that there exists a unique solution in ( T (u 0 ), T + (u 0 )) and the interval is maximal (for (), ( T (u 0, u ), T + (u 0, u ))). In both problems, there exists a crucial space-time norm (or Strichartz norm ). For (), on a time interval I, we define while for () we have u S(I) = u L (N+)/N I u S(I) = u L (N+)/N I L (N+)/N x L (N+)/N x This norm is crucial, say for (), because, if T + (u 0 ) < +, we must have u S((0,T+(u 0))) = + ; moreover, if T + (u 0 ) = +, u scatters at + if and only if u S(0,+ ) < +. Similar results hold for (). The question that attracted people s attention here is: What happens for large data? The question was first studied for () in the defocusing case, through works of Struwe [44] in the radial case, Grillakis [6], [7] in the general case, for the preservation of smoothness, and in the terms described here in the works of Shatah Struwe [4], [4], Bahouri Shatah [3], Bahouri Gérard [], Kapitansky [0], etc. The summary of these works is that (this was achieved in the early 90 s), for any pair (u 0, u ) H L, in the defocusing case we have T ± (u 0, u ) = + and the solution scatters. The corresponding results for () in the defocusing case took much longer. The first result was established by Bourgain [4] in 998, who established the analogous result for u 0 radial, N = 3, 4, with Grillakis [8] showing preservation of smoothness for N = 3 and radial data. Tao extended these results to N 5, u 0 radial [48]. Finally, Colliander Kell Staffilani Takaoka Tao proved this for N = 3 and all data u 0 [8], with extensions to N = 4 by Ryckman Vişan [40] and to N 5 by Vişan [54] in 005. In the focusing case, these results do not hold. In fact, for () H. Levine [33] showed in 974 that in the focusing case, if (u 0, u ) Ḣ L, u 0 L and E((u 0, u )) < 0, there is always a break-down in finite time, i.e., T ± (u 0, u ) <. He showed this by an obstruction type of argument. Recently Krieger Schlag Tătaru [3] have constructed radial examples (N = 3), for which T ± (u 0, u ) <. For () a classical argument due to Zakharov and Glassey [5], based on the virial identity, shows the same result as H. Levine s if x u 0 <, E(u 0 ) < 0. Moreover, for both () and (), in the focusing case we have the following static solution: W (x) = ( + x ) (N )/ H N(N ) (R N ),,.

5 which solves the elliptic equation W + W 4/N W = 0. Thus, scattering need not occur for solutions that exist globally in time. The solution W plays an important role in the Yamabe problem (see [] for instance) and it does so once more here. The results in which I am going to concentrate here are: Theorem (Kenig Merle [5]). For the focusing energy critical (NLS), 3 N 6, consider u 0 H such that E(u 0 ) < E(W ), u 0 radial. Then: i) If u 0 H < W H, the solution exists for all time and scatters. ii) If u 0 L <, u 0 H > W H, then T + (u 0 ) < +, T (u 0 ) < +. Remark. Recently, Killip Vişan [9] have combined the ideas of the proof of Theorem, as applied to NLS in [0], with another important new idea, to extend Theorem to the non-radial case for N 5. The case where the radial assumption is not needed in dimensions 3 N 6 is the one of (). We have: Theorem (Kenig Merle [3]). For the focusing energy critical (NLW), where 3 N 6, consider (u 0, u ) Ḣ L such that E((u 0, u )) < E((W, 0)). Then: i) If u 0 H < W H, the solution exists for all time and scatters. ii) If u 0 H > W H, then T ± (u 0 ) < +. I will sketch the proofs of these two theorems and the outline of the general method in these lectures. The method has found other interesting applications: Mass Critical NLS: i t u + u ± u 4/N u = 0, (x, t) R N R, u t=0 = u 0, N 3. (3) Here, u 0 L is the critical norm. The analog of Theorem was obtained, for u 0 radial, by Tao Vişan Zhang [50], Killip Tao Vişan [8], Killip Vişan-Zhang [30], using our proof scheme for N. (In the focusing case one needs to assume u 0 L < Q L, where Q is the ground state, i.e., the non-negative solution of the elliptic equation Q + Q +4/N = Q.) The case N = is open.

6 Corotational wave maps into S, 4D Yang Mills in the radial case: wave map system u = A(u)(Du, Du) T u M Consider the where u = (u,..., u d ) : R R N M R d, where the target manifold M is isometrically embedded in R d, and A(u) is the second fundamental form for M at u. We consider the case M = S R 3. The critical space here is (u 0, u ) Ḣ N/ ḢN /, so that when N =, the critical space is H L. It is known that for small data in H L we have global existence and scattering (Tătaru [5], [53], Tao [47]). Moreover, Rodnianski Sterbenz [39] and Krieger Schlag Tătaru [3] showed that there can be finite time blow-up for large data. In earlier work, Struwe [45] had considered the case of co-rotational maps. These are maps which have a special form. Writing the metric on S in the form (ρ, θ), ρ > 0, θ S, with ds = dρ + g(ρ) dθ, where g(ρ) = sin ρ, we consider, using (r, φ) as polar coordinates in R, maps of the form ρ = v(r, t), θ = φ. These are the co-rotational maps and Krieger Schlag Tătaru [3] exhibited blow-up for corotational maps. There is a stationary solution Q, which is a non-constant harmonic map of least energy. Struwe proved that if E(v) E(Q), then v and the corresponding wave map u are global in time. Using our method, in joint work of Cote Kenig Merle [9] we show that, in addition, there is an alternative: v Q or the solution scatters. We also prove the corresponding results for radial solutions of the Yang Mills equations in the critical energy space in R 4 (see [9]). Cubic NLS in 3D: Consider the classic cubic NLS in 3D: i t u + u u u = 0, (x, t) R 3 R, u t=0 = u 0 Ḣ/ (R 3 ). Here Ḣ/ is the critical space, corresponds to defocusing and + to focusing. In the focusing case, Duyckaerts Holmer Roudenko [0] adapted our method to show that if u 0 H (R 3 ) and M(u 0 )E(u 0 ) < M(Q)E(Q), where M(u 0 ) = u 0, E(u 0 ) = u 0 u 0 4, 4 and Q is the ground state, i.e., the positive solution to the elliptic equation Q + Q + Q Q = 0, then if u 0 L u 0 L > Q L Q L, we have blow-up in finite time, while if u 0 L u 0 L < Q L Q L, then u exists for all time and scatters. In joint work with Merle [4] we have considered the defocusing case. We have shown, using this circle of ideas, that if sup 0<t<T+(u 0) u(t) Ḣ/ <, then T + (u 0 ) = + and u scatters. We would like to point out that the fact that T + (u 0 ) = + is analogous to the L 3, result of Escauriaza Seregin Sverak for Navier Stokes [].

7 We now turn to the proofs of Theorems and. We start with Theorem. We are thus considering i t u + u + u 4/N u = 0, (x, t) R N R, u t=0 = u 0 H. Let us start with a quick review of the local Cauchy problem theory. Besides the norm f S(I) = f (N+)/N L introduced earlier, we need the norm I Lx (N+)/N f W (I) = f (N+)/N L. I L (N+)/N +4 x Theorem 3 ([7], [5]). Assume that u 0 H (R N ), u 0 H A. Then, for 3 N 6, there exists δ = δ(a) > 0 such that if e it u S(I) 0 δ, 0 I, there exists a unique solution to (4) in R N I, with u C(I; H ) and u W (I) < +, u S(I) δ. Moreover, the mapping u 0 H (R N ) u C(I; H ) is Lipschitz. The proof is by fixed point. The key ingredients are the following Strichartz estimates [43], []: e it u W 0 (,+ ) C u 0 H t 0 ei(t t ) g(, t )dt W (,+ ) C g L t L N/N+ x sup t t 0 ei(t t ) g(, t )dt L C g L t L N/N+ x and the following Sobolev embedding (4) (5) v S(I) C v W (I), (6) and the observation that ( u 4/N u) C u u 4/N, so that ( u 4/N u) u 4/N L I L N/N+ S(I) u W (I). x Remark. Because of (5), (6), there exists δ such that if u 0 H δ, the hypothesis of the Theorem is verified for I = (, + ). Moreover, given u 0 H, we can find I such that e it u S(I) 0 < δ, so that the Theorem applies. It is then easy to see that given u 0 H, there exists a maximal interval I = ( T (u 0 ), T + (u 0 )) where u C(I ; H ) { u W (I )} of all I I is defined. We call I the maximal interval of existence. It is easy to see that for all t I, we have E(u(t)) = u(t) u = E(u 0 ). We also have the standard finite time blow-up criterion : if T + (u 0 ) <, then u S([0,T+(u 0)) = +.

8 We next turn to another fundamental result in the local Cauchy theory, the so called Perturbation Theorem. Perturbation Theorem 4 (see [49], [5], []). Let I = [0, L), L +, and ũ defined on R N I be such that sup ũ H A, ũ t I S(I) M, ũ W (I) < + and verify (in the sense of the integral equation) i t ũ + ũ + ũ 4/N ũ = e on R N I, and let u 0 H be such that u 0 ũ(0) H A. Then there exists ɛ 0 = ɛ 0 (M, A, A ) such that, if 0 ɛ ɛ 0 and e L ɛ, e it [u I L N/N+ 0 ũ(0)] x S(I) ɛ, then there exists a unique solution u to (4) on R N I, such that u S(I) C(A, A, M) and sup u(t) ũ(t) H C(A, A, M)(A + ɛ) β, t I where β > 0. For the details of the proof, see []. This result has several important consequences: Corollary. Let K H be such that K is compact. Then there exist T +,K, T,K such that for all u 0 K we have T + (u 0 ) T +,K, T (u 0 ) T,K. Corollary. Let ũ 0 H, ũ 0 H A, and let ũ be the solution of (4), with maximal interval ( T (ũ 0 ), T + (ũ 0 )). Assume that u 0,n ũ 0 in H, with corresponding solution u n. Then T + (ũ 0 ) lim T + (u 0,n ), T (ũ 0 ) lim T (u 0,n ) and for t ( T (ũ 0 ), T + (ũ 0 )), u n (t) ũ(t) in H. Before we start with our sketch of the proof of Theorem, we will review the classic argument of Glassey [5] for blow-up in finite time. Thus, assume u 0 H, x u 0 (x) dx < and E(u 0 ) < 0. Let I be the maximal interval of existence. One easily shows that, for t I, y(t) = x u(x, t) dx < +. In fact, [ ] y (t) = 4 Im u u x, and y (t) = 8 u(x, t) u(x, t). Hence, if E(u 0 ) < 0, E(u(t)) = E(u 0 ) < 0, so that ( u(t) u(t) = E(u 0 ) + ) u(t) E(u 0 ) < 0, and y (t) < 0. But then, if I is infinite, since y(t) > 0 we obtain a contradiction. We now start with our sketch of the proof of Theorem.

Step : Variational estimates. (These are not needed in defocusing problems.) Recall that W (x) = ( + x /N(N )) (N )/ is a stationary solution of (4). It solves the elliptic equation W + W 4/N W = 0, W 0, W is radially decreasing, W H. By the invariances of the equation, W θ0,x 0,λ 0 (x) = e iθ0 λ N / 0 W (λ 0 (x x 0 )) is still a solution. Aubin and Talenti [], [46] gave the following variational characterization of W : let C N be the best constant in the Sobolev embedding u L C N u L. Then u L = C N u L, u 0, if and only if u = W θ0,x 0,λ 0 for some (θ 0, x 0, λ 0 ). Note that by the elliptic equation, W = W. Also, C N W = W L, so that 9 ( CN W = W ) N N. Hence, W = /C N N, and E(W ) = ( ) W = NCN N. Lemma. Assume that v < W and that E(v) ( δ 0 )E(W ), δ 0 > 0. Then there exists δ = δ(δ 0 ) so that: i) v ( δ) W. ii) v v δ v. iii) E(v) 0. Proof. Let f(y) = y C N y /, y = v. Note that f(0) = 0, f(y) > 0 for y near 0, y > 0, and that f (y) = C N y /, so that f (y) = 0 if and only if y = y c = C N = W. Also, f(y c ) = NC N = E(W ). Since 0 y < y c, f(y) ( δ 0 )f(y c ), f is non-negative and strictly increasing between 0 and y c, and f (y c ) 0, we have 0 f(y), y ( δ)y c = ( δ) W. This shows i).

0 For ii), note that v v = = ( v CN [ v CN v [ ( ) / v v ) /N ] C N ( δ) /N ( v [ ( δ) /N ], W ) /N ] which gives ii). Note from this that if u 0 < W, then E(u 0 ) 0, i.e., iii) holds. This static lemma immediately has dynamic consequences. Corollary 3 (Energy Trapping). Let u be a solution of (4) with maximal interval I, u 0 < W, E(u 0 ) < E(W ). Choose δ 0 > 0 such that E(u 0 ) ( δ 0 )E(W ). Then, for each t I, we have: i) u(t) ( δ) W, E(u(t)) 0. ii) u(t) u(t) δ u(t) ( coercivity ). iii) E(u(t)) u(t) u 0, with comparability constants which depend on δ 0 ( uniform bound ). Proof. The statements follow from continuity of the flow, conservation of energy and the previous Lemma. Note that iii) gives uniform bounds on u(t). However, this is a long way from giving Theorem. Remark 3. Let u 0 H, E(u 0 ) < E(W ), but u 0 > W. If we choose δ 0 so that E(u 0 ) ( δ 0 )E(W ), we can conclude, as in the proof of the Lemma, that u(t) ( + δ) W, t I. But then, u(t) u(t) = E(u 0 ) u N E(W ) N δ = (N )CN N < 0. C N N δ N Hence, if x u 0 (x) dx <, Glassey s proof shows that I cannot be infinite. If u 0 is radial, u 0 L, using a local virial identity (which we will see momentarily) one can see that the same result holds. C N N

Step : Concentration-compactness procedure. We now turn to the proof of i) in Theorem. By our variational estimates, if E(u 0 ) < E(W ), u 0 < W, if δ 0 is chosen so that E(u 0 ) ( δ 0 )E(W ), recall that E(u(t)) u(t) u 0, t I, with constants depending only on δ 0. Recall also that if u 0 < W, E(u 0 ) 0. It now follows from the local Cauchy theory that if u 0 < W and E(u 0 ) η 0, η 0 small, then I = (, + ) and u S(,+ ) <, so that u scatters. Consider now G = {E : 0 < E < E(W ) : if u 0 < W and E(u 0 ) < E, then u S(I) < } and E c = sup G. Then 0 < η 0 E c E(W ) and if u 0 < W, E(u 0 ) < E c, I = (, + ), u scatters and E c is optimal with this property. Theorem i) is the statement E c = E(W ). We now assume E c < E(W ) and will reach a contradiction. We now develop the concentration-compactness argument: Proposition. There exists u 0,c H, u 0,c < W, with E(u 0,c ) = E c, such that, for the corresponding solution u c, we have u c S(I) = +. Proposition. For any u c as in Proposition, with (say) u c S(I+) = +, I + = I [0, + ), there exist x(t), t I +, λ(t) R +, t I +, such that { ( ) } x x(t) K = v(x, t) = λ(t) u, t, t I N / + λ(t) has compact closure in H. The proof of Propositions and follows a general procedure which uses a profile decomposition, the variational estimates and the Perturbation Theorem. The idea of the decomposition is somehow a time-dependent version of the concentration-compactness method of P. L. Lions, when the local Cauchy theory is done in the critical space. It was introduced independently by Bahouri Gérard [] for the wave equation and by Merle Vega for the L critical NLS [35]. The version needed for Theorem is due to Keraani [7]. This is the evolution analog of the elliptic bubble decomposition, which goes back to work of Brézis Coron [5]. Theorem 5 (Keraani [7]). Let {v 0,n } H, with v 0,n H A. Assume that e it v S(,+ ) 0,n δ > 0. Then there exists a subsequence of {v 0,n } and a sequence {V 0,j } j= H and triples {(λ j,n, x j,n, t j,n )} R + R N R, with λ j,n + λ j,n + t j,n t j,n λ j,n λ j,n λ + x j,n x j,n j,n λ j,n, n for j j (we say that {(λ j,n, x j,n, t j,n )} is orthogonal), such that

i) V 0, H α 0 (A) > 0. ii) If Vj l(x, t) = eit V 0,j, then we have, for each J, ( J v 0,n = where lim n iii) v 0,n = E(v 0,n ) = e it wn J j= λ N / j,n V l j S(,+ ) J x x j,n, t j,n λ j,n λ j,n ) + w J n, 0, and for each J we have J V 0,j + wn J + o() as n and j= ( J E j= V l j ( t j,n λ j,n )) + E(w J n) + o() as n. Further general remarks: Remark 4. Because of the continuity of u(t), t I, in H, in Proposition we can construct λ(t), x(t) continuous in [0, T + (u 0 )), with λ(t) > 0. Remark 5. Because of scaling and the compactness of K above, if T + (u 0,c ) <, one always has that λ(t) C 0 (K)/(T + (u 0, c) t). Remark 6. If T + (u 0,c ) = +, we can always find another (possibly different) critical element v c with a corresponding λ so that λ(t) A 0 > 0 for t [0, T + (v 0,c )). (Again by compactness of K.) Remark 7. One can use the profile decomposition to also show that there exists a decreasing function g : (0, E c ] [0, + ) so that if u 0 < W and E(u 0 ) E c η, then u S(,+ ) g(η). Remark 8. In the profile decomposition, if all the v 0,n are radial, the V 0,j can be chosen radial and x j,n 0. We can repeat our procedure restricted to radial data and conclude the analog of Propositions and with x(t) 0. The final step in the proof is then: Step 3: Rigidity Theorem. Theorem 6 (Rigidity). Let u 0 H, E(u 0 ) < E(W ), u 0 < W. Let u be the solution of (4), with maximal interval I = ( T (u 0 ), T + (u 0 )). Assume that there exists λ(t) > 0, defined for t [0, T + (u 0 )), such that { ( ) } x K = v(x, t) = λ(t) u N / λ(t), t, t [0, T + (u 0 )) has compact closure in H. Assume also that, if T + (u 0 ) <, λ(t) C 0 (K)/(T + (u 0, c) t) and if T + (u 0 ) =, that λ(t) A 0 > 0 for t [0, + ). Then T + (u 0 ) = +, u 0 0.

3 To prove this, we split two cases: Case : T + (u 0 ) < +. (So that λ(t) + as t T + (u 0 ).) Fix φ radial, φ C0, φ on x, supp φ { x < }. Set φ R (x) = φ(x/r) and define y R (t) = u(x, t) φ R (x) dx. Then y r(t) = Im u u φ R, so that ( y R(t) C ) / ( ) u u / x C W, by Hardy s inequality and our variational estimates. Note that C is independent of R. Next, we note that, for each R > 0, u(x, t) dx = 0. lim t T +(u 0) x <R In fact, u(x, t) = λ(t) N / v(λ(t)x, t), so that u(x, t) dx = λ(t) x <R which is small with ɛ. = λ(t) + λ(t) = A + B. y <Rλ(t) y <ɛrλ(t) v(y, t) dy v(y, t) dy ɛrλ(t) y <Rλ(t) A λ(t) (ɛrλ(t)) v L Cɛ R W, B λ(t) (Rλ(t)) v L ( y ɛrλ(t)) t T +(u 0) v(y, t) dy (since λ(t) + as t T + (u 0 )) using the compactness of K. But then y R (0) CT + (u 0 ) W, by the fundamental theorem of calculus. Thus, letting R, we see that u 0 L, but then, using the conservation of the L norm, we see that u 0 L = u(t + (u 0 )) L = 0, so that u 0 0. Case : T + (u 0 ) = +. First note that the compactness of K, together with λ(t) A 0 > 0, gives that, given ɛ > 0, there exists R(ɛ) > 0 such that, for all t [0, + ), u + u + u x ɛ. x >R(ɛ) 0,

4 Pick δ 0 > 0 so that E(u 0 ) ( δ 0 )E(W ). Recall that, by our variational estimates, we have that u(t) u(t) C δ0 u 0 L. If u 0 L 0, using the smallness of tails, we see that, for R > R 0, u(t) u(t) C δ0 u 0 L. x <R Choose now ψ C0 radial with ψ(x) = x for x, supp ψ { x }. Define z R (t) = u(x, t) R ψ(x/r) dx. Similar computations to Glassey s blow-up proof give: z R(t) = R Im u u ψ(x/r) and z R(t) = 4 Re xl xj ψ(x/r) xl u xj u l,j R ψ(x/r) u 4 ψ(x/r) u. N Note that z R (t) C δ 0 R u 0, by Cauchy Schwartz, Hardy s inequality and our variational estimates. On the other hand, [ ] z R(t) u(t) u(t) x R ( ) C u(t) + u R x R x + u(t) C u 0, for R large. Integrating in t, we obtain z R (t) z R (0) Ct u 0, but z R(t) z R(0) CR u 0, which is a contradiction for t large, proving Theorem i). Remark 9. In the defocusing case, the proof is easier since the variational estimates are not needed. Remark 0. It is quite likely that for N = 3, examples similar to those by P. Raphäel [38] can be constructed, of radial data u 0 for which T + (u 0 ) < and u blows up exactly on a sphere.

5 We now turn to Theorem. We thus consider t u u = u 4/N u, (x, t) R N R, u t=0 = u 0 H (R n ), t u t=0 = u L (R n ), N 3. (7) Recall that W (x) = ( + x /N(N ) ) (N )/ is a static solution that does not scatter. The general scheme of the proof is similar to the one for Theorem. We start out with a brief review of the local Cauchy problem. We first consider the asociated linear problem, t w w = h, w t=0 = w 0 H (R n ), t w t=0 = w L (R N ). As is well known (see [4] for instance), the solution is given by ( w(x, t) = cos t ) ( w 0 + ( ) / sin t ) + t 0 ( ) / sin = S(t)((w 0, w )) + ( (t s) ) h(s) ds t 0 ( ) / sin w ( (t s) ) h(s) ds. The following are the relevant Strichartz estimates: for an interval I R, let (8) Then (see [4], [3]) f S(I) = f (N+)/N L, I Lx (N+)/N f W (I) = f (N+)/N L. I L (N+)/N x sup t (w(t), t w(t)) Ḣ L + D / w W (,+ ) + + t D / w W (,+ ) + w S(,+ ) + + w (N+)/N L t Lx (N+)/N { C (w 0, w ) Ḣ L + w (N+)/N+3 L t L (N+)/N+3 x Because of the appearance of D / in these estimates, we also need to use the following version of the chain rule for fractional derivatives (see [6]). }. (9)

6 Lemma. Assume F C, F (0) = F (0) = 0, and that for all a, b we have F (a + b) C { F (a) + F (b) } and F (a + b) C { F (a) + F (b) }. Then, for 0 < α <, p = p + p, p = r + r + r 3, we have i) D α F (u) L p C F (u) L p D α u L p, ii) D α (F (u) F (v)) L p C [ F (u) L p + F (v) L p ] D α (u v) L p +C [ F (u) L r + F (v) L r ] [ D α u L r + D α v L r ] u v L r 3. Using (9) and this Lemma, one can now use the same argument as for (4) to obtain: Theorem 7 ([4], [0], [4] and [3]). Assume that (u 0, u ) Ḣ L, (u 0, u ) Ḣ L A. Then, for 3 N 6, there exists δ = δ(a) > 0 such that if S(t)(u 0, u ) S(I) δ, 0 I, there exists a unique solution to (7) in R N I, with (u, t u) C(I; Ḣ L ) and D / u W (I) + t D / u W (I) <, u S(I) δ. Moreover, the mapping (u 0, u ) Ḣ L (u, t u) C(I; Ḣ L ) is Lipschitz. Remark. Again, using (9), if (u 0, u ) Ḣ L δ, the hypothesis of the Theorem is verified for I = (, + ). Moreover, given (u 0, u ) Ḣ L, we can find I 0 so that the hypothesis is verified on I. One can then define a maximal interval of existence I = ( T (u 0, u ), T + (u 0, u )), similarly to the case of (4). We also have the standard finite time blow-up criterion : if T + (u 0, u ) <, then u = +. Also, if T S(0,T+(u 0,u )) +(u 0, u ) = +, u scatters at + (i.e., (u + 0, u+ ) Ḣ L such that (u(t), t u(t)) S(t)(u + 0, u+ ) Ḣ L and only if u S(0,+ ) < +. Moreover, for t I, we have E((u 0, u )) = u 0 + u t + u 0 = E((u(t), t u(t))). 0) if It turns out that for (7) there is another very important conserved quantity in the energy space, namely momentum. This is crucial for us to be able to treat non-radial data. This says that, for t I, u(t) t u(t) = u 0 u. Finally, the analog of the Perturbation Theorem also holds in this context (see []). All the corollaries of the Perturbation Theorem also hold. Remark (Finite speed of propagation). Recall that if R(t) is the forward fundamental solution for the linear wave equation, the solution for (8) is given by (see [4]) w(t) = t R(t) w 0 + R(t) w t 0 R(t s) h(s) ds,

7 where stands for convolution in the x variable. The finite speed of propagation is the statement that supp R(, t), supp t R(, t) B(0, t). Thus, if supp w 0 C B(x 0, a), supp w C B(x 0, a), supp h C [ 0 t a B(x 0, a t) {t}], then w 0 on 0 t a B(x 0, a t) {t}. This has important consequences for solutions of (7). If (u 0, u ) (u 0, u ) on B(x 0, a), then the corresponding solutions agree on 0 t a B(x 0, a t) {t} R N (I I ). We now proceed with the proof of Theorem. As in the case of (4), the proof is broken up in three steps. Step: Variational estimates. Here these are immediate from the corresponding ones in (4). The summary is (we use the notation E(v) = v v ): Lemma 3. Let (u 0, u ) Ḣ L be such that E((u 0, u )) ( δ 0 )E((W, 0)), u 0 < W. Let u be the corresponding solution of (7), with maximal interval I. Then there exists δ = δ(δ 0 ) > 0 such that, for t I, we have i) u(t) ( δ) W. ii) u(t) u(t) δ u(t). iii) E(u(t)) 0 (and here E((u, t u)) 0). iv) E((u, t u)) (u(t), t u(t)) Ḣ L (u 0, u ) Ḣ L, with comparability constants depending only on δ 0. Remark 3. If E((u 0, u )) ( δ 0 )E((W, 0)), u 0 > W, then, for t I, u(t) ( + δ) W. This follows from the corresponding result for (4). We now turn to the proof of ii) in Theorem. We will do it for the case when u 0 L <. For the general case, see [3]. We know that, in the situation of ii), we have u(t) ( + δ) W, t I, Thus, E((W, 0)) E((u(t), t u)) + δ 0. u(t) ( t u(t)) + u(t) E((W, 0)) + δ 0, so that u(t) N ( t u(t)) + N N N u(t) E((W, 0)) + δ0. Let y(t) = u(t), so that y (t) = u(t) t u(t). A simple calculation gives y (t) = { ( t u) u(t) + u(t) }.

8 Thus, y (t) ( t u) + N ( t u) E((W, 0)) + N + δ0 + N u(t) u(t) = N 4(N ) = ( t u) + 4 u(t) N N 4 W + δ0 N 4(N ) ( t u) + δ0. N If I [0, + ) = [0, + ), there exists t 0 > 0 so that y (t 0 ) > 0, y (t) > 0, t > t 0. For t > t 0 we have ( ) y(t)y 4(N ) N (t) ( t u) u y (t), N N so that or y (t) y (t) ( ) N y (t) N y(t), y (t) C 0 y(t) (N )/(N ), for t > t 0. But, since N /N >, this leads to finite time blow-up, a contradiction. We next turn to the proof of i) in Theorem. Step : Concentration-compactness procedure. Here we proceed initially in an identical manner as in the case of (4), replacing the profile decomposition of Keraani [7] with the corresponding one for the wave equation, due to Bahouri Gérard []. Thus, arguing by contradiction, we find a number E c, with 0 < η 0 E c < E((W, 0)) with the property that if E((u 0, u )) < E c, u 0 < W, u S(I) < and E c is optimal with this property. We will see that this leads to a contradiction. As for (4), we have: Proposition 3. There exists (u 0,c, u,c ) Ḣ L, u 0,c < W, E((u 0,c, u,c )) = E c and such that for the corresponding solution u c on (7) we have u c S(I) = +. Proposition 4. For any u c as in Proposition 3, with (say) u c = +, I S(I+) + = I [0, + ), there exists x(t) R N, λ(t) R +, t I +, such that { ( ( ) ( )) } K = v(x, t) = u x x(t) λ(t) N / c λ(t), t, λ(t) N/ t u x x(t) c λ(t), t : t I + has compact closure in Ḣ L.

9 Remark 4. As in the case of (4), in Proposition 4 we can construct λ(t), x(t) continuous in [0, T + ((u 0,c, u,c ))). Moreover, by scaling and compactness of K, if T + ((u 0,c, u,c )) <, we have λ(t) C 0 (K)/(T + ((u 0,c, u,c )) t). Also, if T + ((u 0,c, u,c )) = +, we can always find another (possibly different) critical element v c, with a corresponding λ so that λ(t) A > 0, for t [0, T + ((v 0,c, v,c ))), using the compactness of K. We can also find g : (0, E c ] [0, + ) decreasing so that if u 0 < W and E((u 0,c, u,c )) E c η, then u S(,+ ) g(η). Up to here, we have used, in Step, only Step and general arguments. To proceed further we need to use specific features of (7) to establish further properties of critical elements. The first one is a consequence of the finite speed of propagation and the compactness of K. Lemma 4. Let u c be a critical element as in Proposition 4, with T + ((u 0,c, u,c )) < +. (We can assume, by scaling, that T + ((u 0,c, u,c )) =.) Then there exists x R N such that supp u c (, t), supp t u c (, t) B(x, t), 0 < t <. In order to prove this Lemma, we will need the following consequence of the finite speed of propagation: Remark 5. Let (u 0, u ) Ḣ L, (u 0, u ) Ḣ L A. If, for some M > 0 and 0 < ɛ < ɛ 0 (A), we have u 0 + u + u 0 x ɛ, x M then for 0 < t < T + (u 0, u ) we have u(t) + t u(t) + u(t) + u(t) x 3 M+t x Cɛ. Indeed, choose ψ M C, ψ M for x 3 M, with ψ M 0 for x M. Let u 0,M = u 0 ψ M, u,m = u ψ M. From our assumptions, we have (u 0,M, u,m ) Ḣ L Cɛ. If Cɛ 0 < δ, where δ is as in the local Cauchy theory, the corresponding solution u M of (7) has maximal interval (, + ) and sup t (,+ ) (u M (t), t u M (t)) Ḣ L Cɛ. But, by finite speed of propagation, u M u for x 3 M + t, t [0, T +(u 0, u )), which proves the Remark. We turn to the proof of the Lemma. Recall that λ(t) C 0 (K)/( t). We claim that, for any R 0 > 0, lim u c (x, t) t x+x(t)/λ(t) R0 + t u c (x, t) + u c(x, t) x = 0. )) Indeed, if v(x, t) = λ(t) N/ ( u c ( x x(t) λ(t) ), t x+x(t)/λ(t) R 0 u c (x, t) + t u c (x, t) = (, t u x x(t) c λ(t), t, y λ(t)r 0 v(x, t) dy t 0,

0 because of the compactness of K and the fact that λ(t) + as t. Because of this fact, using the Remark backward in time, we have, for each s [0, ), R 0 > 0, lim u c (x, s) + t u c (x, s) = 0. t x+x(t)/λ(t) 3 R0+(t s) We next show that x(t)/λ(t) M, 0 t <. If not, we can find t n so that x(t n )/λ(t n ) +. Then, for R > 0, { x R} { x + x(t n )/λ(t n ) 3 R + t n} for n large, so that, passing to the limit in n, for s = 0, we obtain u 0,c + u,c = 0, x R a contradiction. Finally, pick t n so that x(t n )/λ(t n ) x. Observe that, for every η 0 > 0, for n large enough, for all s [0, ), { x x + η 0 s} { x + x(t n )/λ(t n ) 3 R 0 + (t n s)}, for some R 0 = R 0 (η 0 ) > 0. From this we conclude that u(x, s) + s u(x, s) dx = 0, x x 0 +η 0 s which gives the claim. Note that, after translation, we can asume that x = 0. We next turn to a result which is fundamental for us to be able to treat non-radial data. Theorem 8. Let (u 0,c, u,c ) be as in Proposition 4, with λ(t), x(t) continuous. Assume that either T + (u 0,c, u,c ) < or T + (u 0,c, u,c ) = +, λ(t) A 0 > 0. Then u 0,c u,c = 0. In order to carry out the proof of this Theorem, a further linear estimate is needed: Lemma 5. Let w solve the linear wave equation t w w = h L t L x(r N+ ) w t=0 = w 0 H (R n ) Then, for a /4, we have ( ( sup x at w, t a x, t w t=0 = w L (R N ). t ax a ) ( x at, t w, a x, )) t ax a L (dx dx ) { } C w 0 H + w L + h L t L. x

The simple proof is omitted; see [3] for the details. Note that if u is a solution of (7), with maximal interval I and I I, u L (N+)/N I L(N+)/N x, and 4 since N + = N+ N, u 4/N u L I L x. Thus, the conclusion of the Lemma ( ) x applies, provided the integration is restricted to at, a x, R N I. t ax a Sketch of the proof of the Theorem. Assume first that T + (u 0,c, u,c ) =. Assume, to argue by contradiction, that (say) x (u 0,c )u,c = γ > 0. Recall that, in this situation, supp u c, t u c B(0, t), 0 < t <. For convenience, set u(x, t) = u c (x, + t), < t < 0, which is supported in B(0, t ). For 0 < a < /4, we consider the Lorentz transformation z a (x, x, t) = u ( x at a, x, ) t ax, a and we fix our attention on / t < 0. In that region, the previous Lemma and the comment following show, in conjunction with the support property of u, that z a is a solution in the energy space of (7). An easy calculation shows that supp z a (, t) B(0, t ), so that 0 is the final time of existence for z a. A lengthy calculation shows that E((z a (, /), t z a (, /))) E((u 0,c, u,c )) lim = γ a 0 a and that, for some t 0 [ /, /4], z a (t 0 ) < W, for a small (by integration in t 0 and a change of variables, together with the variational estimates for u c ). But, since E((u 0,c, u,c )) = E c, for a small this contradicts the definition of E c, since the final time of existence of z a is finite. In the case when T + (u 0,c, u,c ) = +, λ(t) A 0 > 0, the finiteness of the energy of z a is unclear, because of the lack of the support property. We instead do a renormalization. We first rescale u c and consider, for R large, u R (x, t) = R N / u c (Rx, Rt), and for a small, z a,r (x, x, t) = u R ( x at a, x, ) t ax. a We assume, as before, that x (u 0,c )u,c = γ > 0 and hope to obtain a contradiction. We prove, by integration in t 0 (, ), that if h(t 0 ) = θ(x)z a,r (x, x, t 0 ), with θ a fixed cut-off function, for some a small and R large, we have, for some t 0 (, ), that E((h(t 0 ), t h(t 0 ))) < E c γa and h(t 0 ) < W.

We then let v be the solution of (7) with data h(, t 0 ). By the properties of E c, we know that v S(,+ ) g( γa ), for R large. But, since u c S(0,+ ) = +, we have that u R (N+)/N L L. (N+)/N [0,] { x <} R But, by finite speed of propagation, we have that v = z a,r on a large set and, after a change of variables to undo the Lorentz transformation, we reach a contradiction from these two facts. From all this we see that, to prove Theorem, it suffices to show: Step 3: Rigidity Theorem. Theorem 9 (Rigidity). Assume that E((u 0, u )) < E((W, 0)), u 0 < W. Let u be the corresponding solution of (7), and let I + = [0, T + ((u 0, u ))). Suppose that: a) u 0 u = 0. b) There exist x(t), λ(t), t [0, T + ((u 0, u ))) such that { ( ( ) ( )) K = v(x, t) = u x x(t) λ(t) N / c λ(t), t, λ(t) N/ t u x x(t) c λ(t), t has compact closure in Ḣ L. : t I + } c) x(t), λ(t) are continuous, λ(t) > 0. If T + (u 0, u ) <, we have λ(t) C/(T + t), supp u, t u B(0, T + t), and if T + (u 0, u ) = +, we have x(0) = 0, λ(0) =, λ(t) A 0 > 0. Then T + (u 0, u ) = +, u 0. Clearly this Rigidity Theorem provides the contradiction that concludes the proof of Theorem. Proof of the Rigidity Theorem. For the proof we need some known identities (see [4], [3]). Lemma 6. Let r(r) = r(t, R) = x R } { u + t u + u + u x dx. Let u be a solution of (7), t I, φ R (x) = φ(x/r), ψ R (x) = xφ(x/r), where φ is in C0 (B ), φ on x. Then: ( ) i) t ψ R u t u = N ( t u) + N [ u u ] + O(r(R)).

3 ii) t ( ) φ R u t u = ( t u) u + u + O(r(R)). iii) t ( { ψ R u + ( tu) }) u = u t u + O(r(R)). We start out the proof of case, T + ((u 0, u )) = +, by observing that, if (u 0, u ) (0, 0) and E = E((u 0, u )), then, from our variational estimates, E > 0 and sup ( u(t), t u(t)) Ḣ L CE. t>0 We also have and u(t) u(t) C u(t), t > 0 ( t u(t)) + [ u(t) u(t) ] CE, t > 0. The compactness of K and the fact that λ(t) A 0 > 0 show that, given ɛ > 0, we can find R 0 (ɛ) > 0 so that, for all t > 0, we have x+ x(t) λ(t) R(ɛ) t u + u + u x + u ɛe. The proof of this case is accomplished through two lemmas. Lemma 7. There exist ɛ > 0, C > 0 such that, if 0 < ɛ < ɛ, if R > R 0 (ɛ), there exists t 0 = t 0 (R, ɛ) with 0 < t 0 CR, such that for 0 < t < t 0, we have < R R 0 (ɛ) and x(t) = R R 0 (ɛ). x(t) λ(t) λ(t) Note that in the radial case, since we can take x(t) 0, a contradiction follows directly from Lemma 7. This will be the analog of the local virial identity proof for the corresponding case of (4). For the non-radial case we also need: Lemma 8. There exist ɛ > 0, R (ɛ) > 0, C 0 > 0, so that if R > R (ɛ), for 0 < ɛ < ɛ, we have t 0 (R, ɛ) C 0 R/ɛ, where t 0 is as in Lemma 7. From Lemma 7 and Lemma 8 we have, for 0 < ɛ < ɛ, R > R 0 (ɛ), t 0 (R, ɛ) CR, while for 0 < ɛ < ɛ, R > R (ɛ), t 0 (R, ɛ) C 0 R/ɛ. This is clearly a contradiction for ɛ small. Proof of Lemma 7. Since x(0) = 0, λ(0) = ; if not, we have for all 0 < t < CR, with C large, that < R R 0 (ɛ). Let x(t) λ(t) z R (t) = ( N ψ R u t u + ) φ R u t u.

4 Then [ u u ] + O(r(R)). z R(t) = ( t u) But, for x > R, 0 < t < CR, we have x + x(t) R 0 (ɛ) so that r(r) CɛE. Thus, for ɛ small, z R (t) CE/. By our variational estimates, we also have z R (T ) C RE. Integrating in t we obtain CR CE/ C RE, which is a contradiction for C large. Proof of Lemma 8. For 0 t t 0, set { y R (t) = ψ R ( tu) + u } u. For x > R, x + x(t) R 0 (ɛ), so that, since u 0 u = 0 = u(t) t u(t), λ(t) y (R) = O(r(R)), and hence y R (t 0 ) y R (0) CɛEt 0. However, λ(t) y R (0) CR 0 (ɛ)e + O(Rr(R 0 (ɛ))) CE[R 0 (ɛ) + ɛr]. Also, y R (t 0 ) x+ x(t 0 ) λ(t 0 ) R 0(ɛ) λ(t 0 ) >R 0(ɛ) x+ x(t 0 ) ψ R { ( tu) + u u } ψ R { ( tu) + u u }. In the first integral, x R, so that ψ R (x) = x. The second integral is bounded by MRɛE. Thus, { y R (t 0 ) x ( tu) + u } u MRɛE. x+ x(t 0 ) λ(t 0 ) R 0(ɛ) The integral on the right equals x(t 0) λ(t 0 ) x+ x(t 0 ) λ(t 0 ) + x+ x(t 0 ) λ(t 0 ) R 0(ɛ) R 0(ɛ) ( x + x(t 0) λ(t 0 ) so that its absolute value is greater than or equal to } { ( tu) + u u + ) { ( tu) + u u (R 0 R 0 (ɛ))e C(R R 0 (ɛ))ɛe CR 0 (ɛ)e. },

5 Thus, y R (t 0 ) E(R R 0 (ɛ))[ Cɛ] CR 0 (ɛ)e MRɛE ER/4, for R large, ɛ small. But then ER/4 CE[R 0 (ɛ) + ɛr] CɛEt 0, which yields the Lemma for ɛ small, R large. We next turn to the case, T + ((u 0, u )) =, with supp u, t u B(0, t), λ(t) C/ t. For (7) we cannot use the conservation of the L norm as in the (4) case and a new approach is needed. The first step is: Lemma 9. Let u be as in the Rigidity Theorem, with T + ((u 0, u )) =. Then there exists C > 0 so that λ(t) C/ t. Proof. If not, we can find t n so that λ(t n )( t n ) +. Let ( N z(t) = x u t u + ) u t u, where we recall that z is well defined since supp u, t u B(0, t). Then, for 0 < t <, we have z (t) = ( t u) u u. By our variational estimates, E((u 0, u )) = E > 0 and sup (u(t), t u) Ḣ L CE 0<t< and z (t) CE, for 0 < t <. From the support properties of u, it is easy to see that lim t z(t) = 0, so that, integrating in t, we obtain z(t) CE( t), 0 t <. We will next show that z(t n )/( t n ) 0, yielding a contradiction. Because n u(t) t u(t) = 0, 0 < t <, we have ( z(t n ) (x + x(tn )/λ(t n )) u t u N = + t n t n ) u t u. t n Note that, for ɛ > 0 given, we have x + x(t n) λ(t n ) u(t n) t u(t n ) + u(t n ) t u(t n ) CɛE( t n ). x+ λ(tn) ɛ( t x(tn) n) Next we will show that x(t n )/λ(t n ) ( t n ). If not, B( x(t n )/λ(t n ), ( t n )) B(0, ( t n )) =, so that u(t n ) + t u(t n ) = 0, B( x(t n)/λ(t n),( t n))

6 while x+ x(tn) λ(tn) ( t n) ( ) u(t n ) + t u(t n ) = y u x(tn ), t n + y λ(t n)( t n) λ(t n ) ( ) + y x(tn ) dy tu, t n λ(t n ) λ(t n ) N 0, n which contradicts E > 0. Then t n x + x(t n) x+ λ(tn) ɛ( t x(tn) n) λ(t n ) u(t n) t u(t n ) 3 u(t n ) t u(t n ) = = 3 x+ λ(tn) ɛ( t x(tn) n) y ɛ( t n)λ(t n) 0 n ( y u x(tn ) λ(t n ) ) ( ) y x(tn ) dy, t n t u, t n λ(t n ) λ(t n ) N because of the compactness of K and the fact that λ(t n )( t n ). Arguing similarly for u tu t n, using Hardy s inequality (centered at x(t n )/λ(t n )), the proof is concluded. Proposition 5. Let u be as in the Rigidity Theorem, with T + ((u 0, u )) =, supp u, t u B(0, t). Then ( ) K = ( t) N / u(( t)x, t), ( t) N / t u(( t)x, t) is precompact in Ḣ (R N ) L (R N ). Proof. { } v(x, t) = ( t) N ( u(( t)(x x(t)), t), t u(( t)(x x(t)), t)), 0 t < has compact closure in L (R N ) N+, since we have c 0 ( t)λ(t) c and if K is compact in L (R N ) N+, } K = {λ N/ v(λx) : v K, c 0 λ c also has K compact. Let now ṽ(x, t) = ( t) N/ ( u(( t)x, t), t u(( t)x, t)), so that ṽ(x, t) = v(x+x(t), t). Since supp v(, t) {x : x x(t) } and E > 0, the fact that { v(, t)} is compact implies that x(t) M. But if K = { v(x + x 0, t) : x 0 M}, then K is compact, giving the Proposition.

7 At this point we introduce a new idea, inspired by the works of Giga Kohn [] in the parabolic case and Merle Zaag [36] in the hyperbolic case, who studied the equations ( t )u u p u = 0, for < p < 4 N +, in the radial case. In our case, p = 4 N + > 4 N +. We thus introduce self-similar variables. Thus, we set y = x/ t, s = log / t and define w(y, s; 0) = ( t) N / u(x, t) = e s(n )/ u(e s y, e s ), which is defined for 0 s < with supp w(, s; 0) { y }. We will also consider, for δ > 0, u δ (x, t) = u(x, t+δ) which also solves (7) and its corresponding w, which we will denote by w(y, s; δ). Thus, we set y = x/+δ t, s = log /+δ t and w(y, s; δ) = ( + δ t) N / u(x, t) = e s(n )/ u(e s y, + δ e s ). Here w(y, s; δ) is defined for 0 s < log δ and we have { supp w(, s; δ) y e s δ e s = t } + δ t δ. The w solve, where they are defined, the equation s w = ρ N(N ) div (ρ w ρ(y w)y) w + 4 + w 4/N w y s w (N ) s w, where ρ(y) = ( y ) /. Note that the elliptic part of this operator degenerates. In fact, ρ div (ρ w ρ(y w)y) = div (ρ(i y y) w), ρ which is elliptic with smooth coefficients for y <, but degenerates at y =. Here are some straightforward bounds on w( ; δ) (δ > 0): w H0 (B ) with w + s w + w C. B Moreover, by Hardy s inequality for H 0 (B ) functions [6], B w(y) ( y ) C. These bounds are uniform in δ > 0, 0 < s < log δ. Next, following [36], we introduce an energy, which will provide us with a Lyapunov functional for w. { Ẽ(w(s; δ)) = ( s w) + w (y w) } dy B ( y ) + / { N(N ) + w N } dy B 8 N w ( y ). /

8 Note that this is finite for δ > 0. We have: Lemma 0. For δ > 0, 0 < s < s < log /δ, s i) Ẽ(w(s )) Ẽ(w(s ( s w) )) = s B ds dy, so that Ẽ is increasing. ( y 3/ ) ii) [ ( s w) w + N ] s w dy B ( y ) / iii) s = s = Ẽ(w(s))ds + s w s N s B ds dy+ ( y / ) s + s B {( s w) + s wy w + sww y } y dy ( y ) /. lim Ẽ(w(s)) = E((u 0, u )) = E, so that, by i), Ẽ(w(s)) E for 0 s log /δ s < log /δ. The proof is computational; see [3]. Our first improvement over this is: Lemma. 0 B ( s w) dy ds C log /δ. y Proof. Notice that ( s w) y = d { [ ds ( sw) + ( w (y w) ) + (N )N + w N ] [ 8 N w log( y ) ] dy + [log( + y ) + ] y w s w log( y )( s w) ( s w). We next integrate in s, between 0 and, and drop the next to last term by sign. The proof is finished by using Cauchy Schwartz and the support property of w( ; δ). Corollary 4. a) w B 0 ( y ) / dy ds C(log /δ)/. b) Ẽ(w()) C(log /δ)/.

9 Proof. Part a) follows from ii), iii) above, Cauchy Schwartz and the previous Lemma. Note that we obtain the power / on the right hand side by Cauchy Schwartz. Part b) follows from i) and the fact that 0 Ẽ(w(s)) ds C(log /δ) /, which is a consequence of the definition of Ẽ and a). Our next improvement is: Lemma. log /δ B ( s w) ( y ) 3/ C(log /δ)/. Proof. Use i), iii) and the bound b) in the Corollary. Corollary 5. There exists s δ (, (log /δ) 3/4) such that sδ +(log /δ) /8 s δ B ( s w) ( y ) 3/ C (log /δ) /8. Proof. Split (, (log /δ) 3/4) into disjoint intervals of length (log /δ) /8. Their number is (log /δ) 5/8 and 5 8 8 =. Note that, in the Corollary, the length of the s interval tends to infinity, while the bound goes to zero. It is easy to see that if s δ (, (log /δ) 3/4), and s δ = log( + δ t δ ), then t δ + δ t Cδ/4, δ which goes to 0 with δ. From this and the compactness of K, we can find δ j 0, so that w(y, s δj + s; δ j ) converges, for s [0, S] to w (y, s) in C([0, S]; Ḣ 0 L ), and w solves our self-similar equation in B [0, S]. The previous Corollary shows that w must be independent of s. Also, the fact that E > 0 and our coercivity estimates show that w 0. (See [3] for the details.) Thus, w H 0 (B ) solves the (degenerate) elliptic equation ρ div (ρ w ρ(y w )y) N(N ) w + w 4/N w = 0, 4 ρ(y) = ( y ) /. We next point out that w satisfies the additional (crucial) estimates: w B [ w ( y ) + (y w ) ] <. / B ( y ) /

30 Indeed, for the first estimate it suffices to show that, uniformly in j large, we have sδj +δ s δj B w(y, s; δ j ) dy ds C, ( y / ) which follows from ii) above, together with the choice of s δj, by the Corollary, Cauchy Schwartz and iii). The proof of the second estimate follows from the first one, iii) and the formula for Ẽ. The conclusion of the proof is obtained by showing that a w in H 0 (B ), solving the degenerate elliptic equation with the additional bounds above, must be zero. This will follow from a unique continuation argument. Recall that, for y η 0, η 0 > 0, the linear operator is uniformly eliptic, with smooth coefficients and that the non-linearity is critical. An argument of Trudinger s [5] shows that w is bounded on { y η 0 } for each η 0 > 0. Thus, if we show that w 0 near y =, the standard Carleman unique continuation principle [9] will show that w 0. Near y =, our equation is modeled (in variables z R N, r R, r > 0, near r = 0) by r / r (r / r w ) + z w + cw + w 4/N w = 0. Our information on w translates into w H0 ((0, ] ( z < )) and our crucial additional estimates are: w dr (r, z) dz + r/ z w (r, z) dr dz <. r/ 0 z < 0 z < To conclude, we take advantage of the degeneracy of the equation. We desingularize the problem by letting r = a, setting v(a, z) = w (a, z), so that a v(a, z) = r / r w (r, z). Our equation becomes: av + z v + cv + v 4/N v = 0, 0 < a <, z <, v a=0 = 0, and our bounds give: z v(a, z) da dz = 0 0 z < z < a v(a, z) da a dz = 0 0 z < z < z w (r, z) dr dz <, r/ r w (r, z) dr dz <. Thus, v H0 ((0, ] B ), but in addition a v(a, z) a=0 0. We then extend v by 0 to a < 0 and see that the extension is an H solution to the same equation. By Trudinger s argument, it is bounded. But since it vanishes for a < 0, by Carleman s unique continuation theorem, v 0. Hence, w 0, giving our contradiction.

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