BLOW-UP OF SOLUTIONS TO THE NONLINEAR SCHRO¨DINGER EQUATIONS ON STANDARD N-SPHERE 7 AND HYPERBOLIC N-SPACE 0 0 2 LI MA AND LIN ZHAO n a J Abstract. In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlin- 7 earSchro¨dingerequationsonsomeRiemannianmanifoldslikethestan- ] dard2-sphereS2 andthehyperbolic2-spaceH2(−1). Usingthesimilar A idea, we establish such blow-up results on higher dimensional standard C sphere and hyperbolic n-space. Extensions to n-dimensional Riemann- . ian warped product manifolds with n≥2 are also given. h Keywords: Schro¨dingerequation,blow-up,Riemannianman- t a ifold m AMS Classification: Primary 35J [ 1 v 0 1. Introduction 0 2 In this paper, we partially settle down the long standing open prob- 1 lemofthefinitetimeblow-uppropertyaboutthenonlinearSchro¨dinger 0 equations on some Riemannian manifolds like the standard 2-sphere S2 7 0 and the hyperbolic 2-space H2( 1). The nonlinear Schro¨dinger equa- / − h tions of the following form t a (1) iu = ∆u+F( u 2)u m t | | : play an important role in many areas of applied physics, such as non- v relativistic quantum mechanics, laser beam propagation, Bose-Einstein i X condensates and so on (see [18]). The initial value problems (IVP) or ar the initial-boundary value problems (IBVP) of (1) on Rn have been ex- tensively studied in the last two decades (see [8, 13, 21, 10, 19, 20] ). In particular, the blow-up properties in finite time for IVP or IBVP have caught sufficient attention (see [11, 12, 16, 14, 15]). However, much less results have been known on bounded domains in Rn or on compact manifolds (M,g), with the notable exception of the works of H.Bre´zis and T.Gallouet [5], J.Bourgain [1, 2, 3] (In [1], the case M = R2/Z2 was discussed in detail), and N.Burq, P.Ge´rard and N.Tzvetkov [6, 7]. In particular, the blow-up in finite time of Schro¨dinger equations (1) The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP20060003002. 1 2 LIMAANDLINZHAO posed on an arbitrary Riemannian manifold (M,g) is a long widely known open problem. To our knowledge, the only examples of such blow-up phenomena on Riemannian manifolds are given by the follow- ing result, attributed to Ogawa-Tsutsuni [17] if the dimension n of M equals to 1 and generalized to the case n = 2 by N.Burq, P.Ge´rard and N.Tzvetkov [6]. Theorem 1. Let (M,g) be a compact Riemannian manifold of dimen- sion n = 1 or n = 2. Assume there exist x0 M and a system of ∈ coordinates near x0 in which d g = dx2. j j=1 X Then there exist smooth solutions u C ([0,T) M) of ∞ ∈ × iu = ∆u+ u 4/nu t | | such that as t T, → u(t,x) 2 ⇀ Q 2 δ(x x0), | | k kL2(Rn) − where Q is the ground state solution on Rn of ∆Q+Q1+4/n = Q. Even though the condition of Theorem 1 that the manifold near x0 is flat is a very strong restriction, the result is also impressive. In this paper, we concentrate onthe analysis ofthe blow-up phenom- enaforIVPorBIVPoftheSchro¨dingerequationsposedonRiemannian manifolds. To be precise, the IVP and BIVP are of the following forms respectively iu = ∆u+F( u 2)u, on M, t | | (2) IVP u(0,x) = u (x), 0  ∂M = ;  ∅  iu = ∆u+F( u 2)u, on M, t | | (3) BIVP u R ∂M = 0,  | × u(0,x) = u (x),  0 where F is a real-valued smooth function on the n-dimensional Rie-  mannian manifold (M,g) and F satisfies F(s) C(1 + s(p 1)/2) for − ≤ some p > 1 on [0, ). Here ∆ is the Laplacian operator of the metric g with the sign ∆u∞= u′′ on the real line R. Noticing that (3) reduces to (2) when ∂M = , it’s convenient to establish our blow-up results ∅ in the context of (3). The local wellposedness in Hs(M) for s > n/2 to (3) with the interesting case F( u 2)u = u p 1u (p > 1) is a classical − | | | | BLOW-UP 3 consequence of energy estimates, and therefore, it’s relaxed to assume that the solution u(t) to (3) satisfies (4) u C1([0,T),L2) C([0,T),H2 H1 Lp+1), ∈ ∩ ∩ 0 ∩ where T is the maximal existence time for the solution u(t). Before stating our main results, let’s introduce some exact notations concerning Riemannian manifolds used below. Let (M,g) be a com- plete Riemannian manifold of dimension n with boundary ∂M or not. Wedenoteby D theLevi-Civita connection, andbyTM = T M, x M x where T M is the tangent space at x M. It’s well known∈that the x ∈ S smooth sections of TM are just vector fields. For f C1(M), its ∈ gradient is defined as the unique vector field f such that ∇ x M, ξ TM, g( f(x),ξ(x)) = (ξf)(x). ∀ ∈ ∀ ∈ ∇ Thedivergence divXofa smoothvector field Xisdefined astheunique smooth function on M such that f C (M), fdivX = Xf. ∀ ∈ 0∞ − Z Z The Laplace-Beltrami operator ∆ on M is the second order differential operator defined by f C2(M), ∆f = div( f). ∀ ∈ ∇ Corresponding to our analysis, we need to extend g to be defined on complex valued vector fields. For complex valued vectors X + iX , 1 2 Y +iY , where X , X , Y , Y are real, we define 1 2 1 2 1 2 g(X +iX ,Y +iY ) 1 2 1 2 = g(X ,X )+ig(X ,Y )+ig(X ,Y )+g(Y ,Y ). 1 2 1 1 2 1 1 2 It’s easy to see that g defined in such a way is bilinear in the field C and accordingly f = f +i f, ∆f = ∆ f +i∆ f. ∇ ∇ℜ ∇ℑ ℜ ℑ When M has a nonempty boundary ∂M, we denote by v the outer unit normal vector along ∂M. Forthesakeofsimplicity, weomitthespatialintegralvariablex M ∈ and omit the integral region when it’s the whole space M, and we abbreviateLq(M),Hk(M)toLq,Hk respectively. Wewritetheintegral dV and dV as and respectively, and the norm of Lq as M M ∂M ∂M . We denote by Sn the standard sphere of dimension n and by q kR · k R R H Hn( 1) the hyperbolic n-space respectively. We denote by N the north − pole of Sn and by dist(N,x) the distance between N and x Sn N . ∈ \{ } Our main results in 2-dimensions are the following two Theorems. 4 LIMAANDLINZHAO Theorem 2. Consider the Schro¨dinger equation iu = ∆u+ u p 1u, on S2, t − | | u(0) = u H1. 0 (cid:26) ∈ For p 5, if u (x) = u (r), where r = dist(N,x) and u (r) is an 0 0 0 ≥ asymmetric function at r = π with u(π) = 0 and E < 0, then the 2 2 0 asymmetric solution satisfying (4) blows up in finite time. Theorem 3. Consider the Schro¨dinger equations (2) on M = H2( 1). − Assume there exists a constant κ 3 such that ≥ sF(s) κG(s), s 0. ≥ ∀ ≥ Then any solution satisfying (4) with E < 0 blows up in finite time. 0 For higher dimensional results, please see Theorem 12 and 13. Here, we just want to point out that the range of the exponent p for blow-up of the solutions to the Schro¨dinger equation (1) with F(s) = s(p 1)/2 − on Rn is p 1+ 4, on Sn is p 5 and on Hn( 1) is p 1+ 4 . ≥ n ≥ − ≥ n 1 Wetrytopresent very elementary proofsofourblowupresults−start- ing from 2-sphere. This paper is organized as follows. In section 2, we establish some new invariant quantities for the Schro¨dinger equations on general Riemannian manifolds, which generalize the corresponding classical results on Rn. In section 3, we construct blow-up solutions on the unit sphere S2. In section 4, we establish the blow-up results on a class of noncompact manifolds. We discuss the blow-up properties on n-dim manifolds with n 3 in section 5. ≥ 2. Preliminary lemma The following lemma is a generalization of the identities obtained by Glassey [11] (see also [12]). We define u G(u) = F(s)ds. Z0 Lemma 4. Suppose that (M,g) is a complete Riemannian manifold of dimension n with boundary ∂M or not, and v is the outer unit normal vector along ∂M. Let u be a solution of (3) satisfying (4), ρ be an arbitrary smooth function on M, and X be a real smooth vector field on M. Define J(t) := ρ u 2. Then we have | | (A). u(t) = u , 2 0 2 k k k k R (B). (g( u, u¯) G( u 2)) const := E , 0 ∇ ∇ − | | ≡ (C). J (t) = 2 g( ρ, u)u¯, ′ R − ℑ ∇ ∇ R BLOW-UP 5 (D). d g(X, u)u¯ dtℑ ∇ Z 1 = 2 DX( u, u¯)+ (∆divX) u 2 − ∇ ∇ 2 | | Z Z + (divX)(F( u 2) u 2 G( u 2)) | | | | − | | Z + g( u, u¯)g(X,v). ∇ ∇ I Proof. The facts that ρ u 2 and g(X, u)u¯ are of C1[0,T) are | | ℑ ∇ straightforward and the reader can refer to [12] for details. R R For (A), multiply both sides of (3) by 2u¯ and take the imaginary part to obtain ∂ (5) u 2 = 2 (u¯ u). ∂t| | ∇·ℑ ∇ Integrating it over M we get (A). For (B), multiply (3) by 2u¯ , integrate, and take the real part of the t resulting expression. For (C), multiply (5) by ρ and integrate by parts over M. The derivation of (D) is a bit involved. We first multiply (3) by 2DXu¯ to obtain (6) 2i(DXu¯)ut = 2(DXu¯)∆u+2(DXu¯)F( u 2)u | | := I +I . 1 2 Then, we take the real part of the left-hand side (LHS) of (6) to get (LHS) = i((DXu¯)ut (DXu)u¯t) ℜ − = i((uDXu¯)t DX(uu¯t)) − = (i(uDXu¯)t) (iDX(uu¯t)) ℜ −ℜ d = (g(X, u)u¯) (iDX(uu¯t)). dtℑ ∇ −ℜ Integrating this identity over M yields d (7) LHS = g(X, u)u¯ iDX(uu¯t). ℜ dtℑ ∇ −ℜ Z Z Z 6 LIMAANDLINZHAO Using integration by parts we have iDX(uu¯t) ℜ Z = (divX)(iuu¯ ) t −ℜ Z = (divX)( u∆u¯ F( u 2) u 2) −ℜ − − | | | | Z = (g( (divX),u u¯)+(divX)g( u, u¯))+ (divX)F( u 2) u 2 −ℜ ∇ ∇ ∇ ∇ | | | | Z Z 1 = g( (divX), u 2) (divX)g( u, u¯) −2 ∇ ∇| | − ∇ ∇ Z Z + (divX)F( u 2) u 2 | | | | Z 1 = (∆divX) u 2 (divX)g( u, u¯)+ (divX)F( u 2) u 2. 2 | | − ∇ ∇ | | | | Z Z Z Inserting this into (7) we obtain that d 1 (8) LHS = g(X, u)u¯ (∆divX) u 2 ℜ dtℑ ∇ − 2 | | Z Z Z + (divX)g( u, u¯) (divX)F( u 2) u 2. ∇ ∇ − | | | | Z Z To handle the right-hand side of (6), we use integration by parts again to obtain I1 = 2 (DXu¯)∆u ℜ ℜ Z Z = 2 g( DXu¯, u)+2 g((DXu¯) u,v) − ℜ ∇ ∇ ℜ ∇ Z I 1 = 2 (DX( u, u¯)+ DXg( u, u¯)) − ℜ ∇ ∇ 2 ∇ ∇ Z +2 g((DXu¯) u,v) ℜ ∇ I = 2 DX( u, u¯) DXg( u, u¯) − ∇ ∇ − ∇ ∇ Z Z +2 g( u, u¯)g(X,v). ∇ ∇ I The last ”=” in the above expression follows from the fact that u = ∂M 0, which implies u = g( u,v)v. | ∇ ∇ BLOW-UP 7 Noticing that DXg( u, u¯) = (divX)g( u, u¯)+ g( u, u¯)g(X,v), ∇ ∇ − ∇ ∇ ∇ ∇ Z Z I we then get (9) I = 2 DX( u, u¯)+ (divX)g( u, u¯) 1 ℜ − ∇ ∇ ∇ ∇ Z Z Z + g( u, u¯)g(X,v). ∇ ∇ I For I we have 2 (10) I2 = 2 (DXu¯)F( u 2)u = (DX u 2)F( u 2) ℜ ℜ | | | | | | Z Z Z = DXG( u 2) = (divX)G( u 2). | | − | | Z Z Combining (8)-(10) with (6) we get (D), and the proof of the lemma (cid:3) is concluded. Remark 5. If we choose X = ρ in lemma 2 (D), we then arrive at ∇ (11) J′′(t) = 4 D2ρ( u, u¯) (∆2ρ) u 2 ∇ ∇ − | | Z Z 2 (∆ρ)(F( u 2) u 2 G( u 2)) − | | | | − | | Z 2 g( u, u¯)g( ρ,v). − ∇ ∇ ∇ I This identity will play a vital role in our analysis. In particular, when ρ = v, we have ∇ g( u, u¯)g( ρ,v) = g( u, u¯) 0, ∇ ∇ ∇ ∇ ∇ ≥ which is an important fact in our proof. 3. Blow-up phenomena on S2 To investigate the blow-up nature of the solution u, one method is to observe the long time behavior of J(t) := ρ u 2. If there exists ρ 0 | | ≥ such that J(t) becomes negative after some finite time T due to the R conservation of the L2 norm and the energy E , then u must blow up 0 beforethetime T. It’sclassical thatρ(x) = x 2 when M = Rd. But for | | an arbitrary manifold, the sharp ρ adopted to the blow-up properties is unknown explicitly. It seems that ∆ x 2 = 2n is a nice property for us to use (11) on M = Rn. For nonco|m|pact manifolds and compact manifolds with boundary, it’s possible to find ρ such that ∆ρ = const. 8 LIMAANDLINZHAO We put this idea in practice on the half sphere S2 := S2 x1 0 + ∩ { ≥ } and the hyperbolic space M = H2( 1). We now state the result for − S2. + Theorem 6. Consider the Schro¨dinger equations (3) on M = S2. + Assume there exists a constant κ 3 such that ≥ sF(s) κG(s), s 0. ≥ ∀ ≥ Then any solution satisfying (4) with E < 0 blows up in finite time. 0 Proof. In R3, S2 = (x1)2 + (x2)2 +(x3)2 = 1 x1 0 . We want + { }∩{ ≥ } to construct a function ρ such that (a). ρ C4(S2); ∈ + (b). ρ > 0 on S2 N ; + \{ } (c). ∆ρ = 1 on S2; + (d). D2ρ( u, u¯) g( u, u¯), u C1(S2). ∇ ∇ ≤ ∇ ∇ ∀ ∈ + We now give the form of ρ exactly. To make the calculations clear, we recall the expressions of f, ∆f and D2f for f C2(M) in local ∇ ∈ coordinates. We use Einstein’s convention. Let g = g dxidxj, G = ij det(g ) and (gij) = (g ) 1. Then ij ij − (12) f = gijf ∂ , i j ∇ 1 (13) ∆f = ∂ (gij√Gf ), i j √G (14) D2f = (f Γkf )dxi dxj, ij − ij k ⊗ where 1 ∂g ∂g ∂g Γk = gkl( il + lj ij). ij 2 ∂xj ∂xi − ∂xl In our situation, we use the geodesic polar coordinate (r,θ) at the north pole N for S2, i.e., + x1 = cosr, x2 = sinrcosθ,  x3 = sinrsinθ,  where r (0,π/2], θ [0,2π). In this coordinate, ∈ ∈  dx1 = sinrdr, − dx2 = cosrcosθdr sinrsinθdθ,  − dx3 = cosrsinθdr +sinrcosθdθ,  and hence  3 g = (dxi)2 = dr2 +sin2rdθ2, i=1 X BLOW-UP 9 Γ1 = Γ1 = Γ1 = 0, Γ1 = sinrcosr. 11 12 21 22 − By (13) we have for ̺ = ̺(r) C2(0,π/2] that ∈ 1 ′′ ∆̺(r) = (sinr̺(r)) = ̺ (r)+̺(r)cotr. ′ ′ ′ sinr Solving the ODE ′′ ̺ (r)+̺(r)cotr = 1, 0 < r π/2, ′ ≤ ̺(r) > 0, 0 < r π/2,  ≤ ̺(π/2) = 1,  ′ we get a solution ̺(r) = 2logcos(r/2). We then define ρ(r) as  − ̺(r), 0 < r π/2, ρ(r) = ≤ 0, r = 0. (cid:26) It’s then easy to see that ρ(r) C4[0,π/2], i.e, ρ C4(S2). ∈ ∈ + From (12) and (14), we have for any u C1(S2), ∈ + 1 g( u, u¯) = u 2 + u 2 ∇ ∇ | r| sin2r| θ| and cosr D2ρ( u, u¯) = ρ′′(r) u 2 +ρ(r) u 2 ∇ ∇ | r| ′ sin3r| θ| 1 cosr (1 cosr)cosr = − u 2 + − u 2. sin2r | r| sin4r | θ| It’s obvious that D2ρ( u, u¯) g( u, u¯) provided 0 < r π/2. ∇ ∇ ≤ ∇ ∇ ≤ By a standard approximation process we get D2ρ( u, u¯) g( u, u¯), u H1(S2). ∇ ∇ ≤ ∇ ∇ ∀ ∈ + ZS+2 ZS+2 Notice that ρ = v on the boundary ∂S2. Then by (11) we have ∇ + (15) J′′(t) 4 D2ρ( u, u¯) 2 (F( u 2) u 2 G( u 2)) ≤ ∇ ∇ − | | | | − | | ZS+2 ZS+2 2 g( u, u¯)g( ρ,v) − ∇ ∇ ∇ I∂S+2 4 g( u, u¯) 2(κ 1) G( u 2). ≤ ∇ ∇ − − | | ZS+2 ZS+2 Combining Lemma 2 (B) with (15) we obtain J′′(t) 4E +(6 2κ) G( u 2) 4E < 0, 0 0 ≤ − | | ≤ ZS+2 10 LIMAANDLINZHAO which implies that J(t) becomes negative after some finite time T, i.e., the solution u satisfying E < 0 blows up in finite time. (cid:3) 0 Remark7. Here weonlyestablishedtheblow-up resultforthe Schro¨dinger equations on S2. The method above can’t be applied to the whole sphere + S2 or some other compact manifolds because of the nonexistence of sub- harmonic functions on compact manifolds. But this result allows us to construct blow-up solutions for the Schro¨dinger equations posed on S2. See the proof of Theorem 2. Remark 8. If F(s) = sp−21 where p > 1, then G(s) = 2 sp+21, and p+1 the condition sF(s) κG(s) on [0,+ ) for some κ 3 is equivalent ≥ ∞ ≥ to p 5. It’s already known that when 1 < p < 3, the Schro¨dinger ≥ equation iu = ∆u+ u p 1u with u Hs (s 1) on S2 has a unique t − 0 global solution u C| (|R,Hs) (see [7∈]). We≥point out when p 5, ∈ ≥ blow-up phenomena may occur by our Theorem 2. Proof of Theorem 2: If u (r) is asymmetric in r, then by the symme- 0 try of the Schro¨dinger equation, the solution u(t,x) is also asymmetric, that is, u(t,r) is asymmetric with respected to r, which implies that u(t,π/2) 0. We then cut the sphere S2 into two parts S2 and S2, ≡ + where − S2 = (x1)2 +(x2)2 +(x3)2 = 1 x1 0 , + { }∩{ ≥ } S2 = (x1)2 +(x2)2 +(x3)2 = 1 x1 0 . { }∩{ ≤ } − We write by ρ the function obtained in the proof of Theorem 6 on + S2, and define + ρ (x) := ρ ( x) for x S2. + We denote by v and−v the oute−r normal v∈ecto−r along ∂S2 and ∂S2. + + Then v = v , and −ρ = v , ρ = v . − + + + − − ∇ ∇ − − We now define J(t) := ρ u 2 + ρ u 2 := J +J . + 1 2 ZS+2 | | ZS−2 −| | Since u = 0 on ∂S2 = ∂S2, we can use (11) directly for both J and + 1 J to get that − 2 (16) J′′(t) 4 D2ρ ( u, u¯) 2 (F( u 2) u 2 G( u 2)) 1 ≤ + ∇ ∇ − | | | | − | | ZS+2 ZS+2 2 g( u, u¯)g( ρ ,v ) + + − ∇ ∇ ∇ I∂S+2 4 g( u, u¯) 2(κ 1) G( u 2), ≤ ∇ ∇ − − | | ZS+2 ZS+2