A simple proof of scattering for the intercritical inhomogeneous NLS
aa r X i v : . [ m a t h . A P ] J a n A SIMPLE PROOF OF SCATTERING FOR THEINTERCRITICAL INHOMOGENEOUS NLS
JASON MURPHY
Abstract.
We adapt the argument of [4] to give a simple proof of scat-tering below the ground state for the intercritical inhomogeneous nonlinearSchr¨odinger equation. The decaying factor in the nonlinearity obviates theneed for a radial assumption. Introduction
We revisit the problem of scattering below the ground state for the focusing,intercritical, inhomogeneous nonlinear Schr¨odinger equation (NLS). We restrict ourattention to the case of a cubic nonlinearity in three dimensions, i.e.( i∂ t + ∆) u + | x | − b | u | u = 0 , ( t, x ) ∈ R × R , (1.1)with the parameter b chosen from the interval (0 , ). The scaling symmetry of(1.1) identifies the equation as ˙ H s c -critical, where s c = b ∈ ( , ). We callthe equation intercritical because the critical regularity s c lies between the specialvalues s c = 0 and s c = 1, corresponding to the mass- and energy-critical cases,respectively. Our restriction to the cubic nonlinearity serves primarily to simplifythe presentation. In particular, the argument presented here should also apply tomore general powers and dimensions d ≥
3. The restriction on b , which arisesfrom applications of Hardy’s inequality, could perhaps be relaxed by modifying orrefining the arguments given below.Denoting by Q the ground state solution to∆ Q − Q + | x | − b Q = 0 (1.2)and the conserved mass and energy of solutions by M ( u ) = Z | u | dx, E ( u ) = Z |∇ u | − | x | − b | u | dx, we will prove the following. Theorem 1.1.
Let < b < . Suppose u ∈ H ( R ) obeys E ( u ) b M ( u ) − b < E ( Q ) b M ( Q ) − b (1.3) and k∇ u k bL k u k − bL < k∇ Q k bL k Q k − bL . (1.4) Then the solution u to (1.1) with u | t =0 = u is global in time and scatters, that is,there exist u ± ∈ H such that lim t →±∞ k u ( t ) − e it ∆ u ± k H = 0 . Theorem 1.1 was established first in the radial setting by [6], and subsequently inthe non-radial setting by [2,9]. These works adopted the concentration-compactnessapproach to induction on energy pioneered in [8], reducing the problem of scatteringto the problem of precluding a global non-scattering solution that is below theground state threshold in the sense of (1.3) and (1.4) and has precompact orbitin H . The preclusion of such a solution is achieved by using a localized virialargument. In non-radial problems, compact solutions are typically parametrizedby some moving spatial center x ( t ). The key to passing from the radial to the non-radial case for (1.1) was the observation that the decaying factor in the nonlinearityalready provides enough spatial localization to guarantee that x ( t ) ≡
0, which inturn allows for a simple implementation of the virial argument. The basic idea isthat if | x ( t ) | → ∞ , then the solution would behave like an approximate solutionto the linear Schr¨odinger equation, contradicting the fact that it does not scatter.Roughly speaking, the non-radial problem may be treated as if it were radial.In this note we push this idea a bit further by showing that the argument of [4],which gives a simple proof of scattering for the radial NLS, may be adapted to(1.1) even in the non-radial case. The argument of [4] has two ingredients: (i)a scattering criterion as in [12] based on a ‘mass evacuation’ condition, and (ii)a hybrid virial/Morawetz estimate as in [10], which implies the mass evacuationcondition for solutions below the ground state threshold. The radial assumptionis used in both steps to derive quantitative decay estimates at large radii via theradial Sobolev embedding estimate of [11]. In both cases, however, this estimateis used only in controlling terms arising from the nonlinearity. Observing that thedecaying factor in the nonlinearity of (1.1) already yields quantitative decay atlarge radii, we find that the simple argument of [4] suffices to treat (1.1), even inthe non-radial case.Without loss of generality, we consider scattering in the forward time directiononly. After collecting a few preliminaries in Section 1.1, we will prove the scatteringcriterion in Section 2 and the virial/Morawetz estimate in Section 3. These twoingredients quickly imply Theorem 1.1.1.1.
Preliminaries.
We will need a few results related to well-posedness and scat-tering for (1.1). We assume familiarity with the standard subcritical well-posednesstheory for dispersive PDEs (e.g. the Duhamel formulation, Strichartz estimates,etc.). Otherwise, we refer the reader to [3] for a textbook treatment of nonlinearSchr¨odinger equations in general and to [7] for the specific case of the inhomoge-neous NLS.For any initial datum u ∈ H , there exists a unique maximal-lifespan solutionto (1.1). Solutions conserve the mass and energy, and any solution that remainsuniformly bounded in H throughout its lifespan may be extended globally in time.For such solutions we have the following local estimate: k u k L qt H ,rx ( I × R ) . (1 + | I | ) q for any Strichartz admissible pair ( q, r ) . (1.5)We will also need the following small-data scattering result. The paper [13] similarly adapted the arguments of the related work [1] to the 2 d inhomoge-neous NLS. However, the authors of [13] continued to work in the radial setting. NHOMOGENEOUS NLS 3
Lemma 1.2.
Let b ∈ (0 , ) . Suppose u is a forward global solution to (1.1) with u | t =0 = u ∈ H . Suppose further that k u k L ∞ t H x ((0 , ∞ ) × R ) = E and k e it ∆ u k L t L − bx ((0 , ∞ ) × R ) = ε. If ε is sufficiently small depending on E , then u scatters in H as t → ∞ .Sketch of proof. We choose a parameter ρ ∈ ( b , ∞ ) and set r = ρρ − ∈ (6 , − b ). Wethen define k u k S = k u k L t L rx + k u k L t L x , where here and below all space-time norms are taken over (0 , ∞ ) × R . By inter-polation, Sobolev embedding, and Strichartz estimates, we can deduce that k e it ∆ u k S . E ε c for some c >
0. By Sobolev embedding, Strichartz estimates, H¨older’s inequality,and Hardy’s inequality, we can then estimate k u k S . ε c + k| x | − b | u | u k L t H , x . ε c + X T ∈{ , ∇ , | x | − } k| x | − b u T u k L t L x . ε c + (cid:8) k| x | − b k L ρ ( | x | > + k| x | − b k L ( | x |≤ (cid:9) k u k S k u k L ∞ t H x . ε c + k u k S k u k L ∞ t H x . Thus for ε = ε ( E ) sufficiently small, we derive k u k S . ε c . With this bound in hand,we may deduce that e − it ∆ u ( t ) is Cauchy in H as t → ∞ essentially by repeatingthe estimates above. (cid:3) Next, we recall some properties of the ground state Q . For more details, we referthe reader to [5, Theorems 1.1 and 1.2].The ground state Q arises as an optimizer for the Gagliardo–Nirenberg inequality k| x | − b | u | k L ≤ C b k u k − bL k∇ u k bL . (1.6)Using Pohozaev identities (obtained by multiplying (1.2) by Q and x · ∇ Q andintegrating by parts), one can connect the sharp constant to norms of Q as follows: k∇ Q k bL k Q k − bL = b C − b and E ( Q ) b M ( Q ) − b = b ) b ( b ) b C − b . (1.7)Then, using (1.6), one can show that solutions obeying (1.3) and (1.4) are globalin time and uniformly bounded in H , withsup t ∈ R (cid:8) k∇ u ( t ) k bL k u ( t ) k − bL (cid:9) < (1 − δ ) k∇ Q k bL k Q k − bL for some δ > . (1.8)The proof of scattering is then connected to the following virial identity: ddt Z ¯ u ∇ u · x dx = 8 Z |∇ u | − b | x | − b | u | dx, (1.9)which follows from (1.1) and integration by parts. In particular, the bound (1.8) andthe sharp Gagliardo–Nirenberg inequality imply that the right-hand side of (1.9) isbounded below, yielding the monotonicity at the heart of scattering. In practice,the presence of the weight x in (1.9) necessitates spatial localization of the aboveidentity, and accordingly we will need the following local form of coercivity. JASON MURPHY
Lemma 1.3.
Let u ∈ H satisfy the hypotheses of Theorem 1.1, and let u be thecorresponding solution to (1.1) . There exists δ ′ > so that for all R sufficientlylarge, Z |∇ [ χ R u ( t, x )] | − b | x | − b | χ R u ( t, x ) | dx ≥ δ ′ Z | x | − b | χ R u ( t, x ) | dx uniformly over t ∈ R , where χ R is a smooth cutoff to | x | ≤ R .Proof. (i) First suppose k∇ f k bL k f k − bL < (1 − η ) k∇ Q k bL k Q k − bL for some f ∈ H and η ∈ (0 , k f k H − b k| x | − f k L ≥ η k f k H , which yields the following upon rearranging: k f k H − b k| x | − b f k L ≥ η − η b k| x | − b f k L . (ii) Using (i), it suffices to to show thatsup t ∈ R (cid:8) k∇ [ χ R u ( t )] k bL k χ R u ( t ) k − bL (cid:9) < (1 − η ) k∇ Q k bL k Q k − bL (1.10)for R sufficiently large and some η >
0. As (1.8) holds and multiplication by χ R only decreases the L -norm, it suffices to consider the ˙ H -norm. For this, we use Z χ R |∇ u | dx = Z |∇ [ χ R u ] | + χ R ∆( χ R ) | u | dx, (1.11)which implies k∇ [ χ R u ] k L ≤ k∇ u k L + O ( R − M ( u )) . We conclude that (1.10) holds with η = δ for all R sufficiently large. (cid:3) Scattering criterion
The first ingredient for the proof of Theorem 1.1 is the following scatteringcriterion as in [12]. For the standard NLS, this criterion is only valid in the radialsetting. As we will see, because of the decaying factor in the nonlinearity, thiscriterion is sufficient for (1.1) even in the non-radial setting.
Proposition 2.1.
Let b ∈ (0 , ) . Suppose u is a global solution to (1.1) obeying k u k L ∞ t H x ≤ E . Then there exist ε = ε ( E ) > and R = R ( E ) > so that if lim inf t →∞ Z | x |≤ R | u ( t, x ) | dx ≤ ε , then u scatters forward in time.Proof. Throughout the proof, we allow implicit constants to depend on E . With ε > R >
T > ε − large enough that k e it ∆ u k L t L − bx ([ T, ∞ ) × R ) < ε and Z χ R ( x ) | u ( T, x ) | dx ≤ ε , (2.1)where χ R is a smooth cutoff to {| x | ≤ R } . The goal is then to prove k e i ( t − T )∆ u ( T ) k L t L − bx ([ T, ∞ ) × R ) < ε a for some a > , NHOMOGENEOUS NLS 5 which (for ε sufficiently small) implies scattering via Lemma 1.2. To estimate thisnorm, we rewrite the Duhamel formula for u as follows: e i ( t − T )∆ u ( T ) = e it ∆ u + i Z I e i ( t − s )∆ | x | − b | u | u ( s ) ds (2.2)+ i Z I e i ( t − s )∆ | x | − b | u | u ( s ) ds, (2.3)where I = [ T − ε − c , T ] and I = [0 , T − ε − c ] for some c > (cid:13)(cid:13)(cid:13)(cid:13)Z I e i ( t − s )∆ | x | − b | u | u ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L t L − bx ([ T, ∞ ) × R ) . k|∇| b (cid:2) | x | − b | u | u (cid:3) k L t L x ( I × R ) . (2.4)We estimate this term by interpolating between L x and ˙ H x .We first consider the estimate in L x . We begin by extending the small masscondition at t = T in (2.1) to the interval I . Using (1.1) to derive the identity ∂ t | u | = − ∇ · Im(¯ u ∇ u ) , we integrate by parts and use Cauchy–Schwarz to estimate (cid:12)(cid:12)(cid:12)(cid:12) ddt Z χ R ( x ) | u ( t, x ) | dx (cid:12)(cid:12)(cid:12)(cid:12) . R − . With R ≥ ε − − c , this implies k χ R u k L ∞ t L x ( I × R ) . ε. Recalling that b < , we now choose an exponent r = r ( b ) satisfying3 < r < b (2.5)and θ = θ ( b ) ∈ (0 ,
1) satisfying θ < min { r − b ) , r − b − } . (2.6)Writing r θ for the solution to r = θ + − θr θ , we use the triangle inequality, H¨older’sinequality, Hardy’s inequality, and Sobolev embedding to estimate k| x | − b u k L rx . k| x | − b (1 − χ R ) u k L rx + k χ R u k θL x k| x | − b − θ u k − θL rθx . R − b + ε θ k|∇| b − θ + − rθ u k − θL x . ε θ uniformly over t ∈ I , where we have further imposed R ≥ ε − θb . Here (2.5) andfirst constraint in (2.6) guarantee that we may apply Hardy’s inequality, while thesecond constraint in (2.6) guarantees that the final norm is controlled by H . UsingH¨older’s inequality, Sobolev embedding, and the local estimate (1.5), we thereforeobtain k| x | − b | u | u k L t L x ( I × R ) . k x | − b u k L ∞ t L rx ( I × R ) k u k L t L rr − x ( I × R ) k u k L t L x ( I × R ) . ε θ | I | . ε θ − c . (2.7) JASON MURPHY
We turn to the ˙ H x estimate. This leads to two terms, which take the form | x | − b O ( u ∇ u ) and O ( | x | − b − u ) . The first term may be estimated exactly as above; we simply put ∇ u in L t L x instead of u in (2.7). For the second term, we instead use H¨older’s inequality,Hardy’s inequality, Sobolev embedding, and (1.5) to estimate k| x | − b − u k L t L x ( I × R ) . k|∇| b +13 u k L t L x ( I × R ) . k|∇| b +23 u k L t L x ( I × R ) . ε − c . Here the application of Hardy’s inequality requires b < .Returning to (2.4), we obtain the following bound by interpolation: (cid:13)(cid:13)(cid:13)(cid:13)Z I e i ( t − s )∆ | x | − b | u | u ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L t L − bx ([ T, ∞ ) × R ) . ε (1 − b ) θ − c , which (choosing c = c ( b ) sufficiently small) is acceptable.It remains to estimate (2.3) in L t L − b x on [ T, ∞ ) × R . Here the estimate is thesame as in [4]. We interpolate between the L t L x -norm and the L t L ∞ x -norm, usingthe identity i Z I e i ( t − s )∆ | x | − b | u | u ( s ) ds = e i ( t − T + ε − c )∆ [ u ( T − ε − c ) − u ]and Strichartz to obtain boundedness for the L t L x -norm. For the L t L ∞ x -norm,we first use the dispersive estimate, Hardy’s inequality, and Sobolev embedding toestimate (cid:13)(cid:13)(cid:13)(cid:13)Z I e i ( t − s )∆ | x | − b | u | u ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L ∞ x . Z I | t − s | − ds · k| x | − b u k L ∞ t L x . ( t − T + ε − c ) − k|∇| b + u k L ∞ t L x . Thus the L t L ∞ x -norm over [ T, ∞ ) is bounded by ε c , and hence we deduce theacceptable estimate (cid:13)(cid:13)(cid:13)(cid:13)Z I e i ( t − s )∆ | x | − b | u | u ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L t L − bx ([ T, ∞ ) × R ) . ε (1+ b ) c . (cid:3) Virial/Morawetz estimate
In this section, we let u be a solution to (1.1) satisfying the hypotheses of The-orem 1.1. In particular, as discussed in Section 1.1, u is global, uniformly boundedin H , and obeys (1.8). We will prove a virial/Morawetz estimate that implies themass evacuation condition appearing in Proposition 2.1. Proposition 3.1 (Virial/Morawetz estimate) . For any
T > and R > suffi-ciently large, T Z T Z | x |≤ R | x | − b | u ( t, x ) | dx dt . RT + R b . NHOMOGENEOUS NLS 7
Proof.
The proof is based off of the following identity, which follows from a directcomputation using (1.1) and integration by parts: Given a smooth weight a : R → R and defining A a ( t ) = 2 Im Z ¯ uu j a j dx, where subscripts denote partial derivatives and repeated indices are summed, wehave ddt A a = Z a jk ¯ u j u k − | u | a jjkk − | x | − b | u | a jj − b | x | − b − | u | x j a j dx. Inspired by [10], we choose a weight that interpolates between the standard virialand Morawetz weights. In particular, choosing R sufficiently large as in Lemma 1.3,we let a be a radial function satisfying a ( x ) = | x | for | x | ≤ R and a ( x ) = 2 R | x | for | x | > R. For R < | x | ≤ R we impose ∂ r a ≥ , ∂ r a ≥ , and | ∂ α a ( x ) | . α R | x | −| α | +1 for | α | ≥ , where ∂ r denotes radial derivative. We observe that the conditions above implynonnegativity of the matrix a jk , and that we have the boundsup t ∈ R | A a ( t ) | . R k u k L ∞ t H x . R. For | x | ≤ R , we have a j = 2 x j , a jk = 2 δ jk , ∆ a = 6 , and ∆∆ a = 0 , while for | x | > R we have a j = Rx j | x | , a jk = R | x | [ δ jk − x j x k | x | ] , ∆ a = R | x | , ∆∆ a = 0 . Thus, by the identities above, ddt A a = 8 Z | x |≤ R |∇ u | − b | x | − b | u | dx (3.1)+ Z | x | >R R | x | | / ∇ u | − R (2+ b ) | x | | x | − b | u | dx (3.2)+ Z R < | x |≤ R a jk ¯ u j u k + O ( R − b | u | + R − | u | ) dx, (3.3)where / ∇ denotes the angular part of the derivative.In (3.1), we insert χ R and use the identity (1.11), Lemma 1.3, and uniform H -boundedness of u to obtain(3.1) ≥ Z |∇ [ χ R u ] | − b | x | − b | χ R u | dx − O (cid:26) R − M ( u ) + Z [ χ R − χ R ] | x | − b | u | dx (cid:27) ≥ δ ′ Z | x | − b | χ R u | dx − O ( R − b ) . For (3.2), the angular derivative term is nonnegative, while the nonlinear term isestimated by R − b . Similarly, in (3.3) the first term is nonnegative while the secondterm is estimated by R − b . Note that in contrast to [4], we do not use radial Sobolev JASON MURPHY embedding to obtain decay at large radii. Instead, the decay comes directly fromthe nonlinearity.Applying the fundamental theorem of calculus on the interval [0 , T ] now yields Z T Z | x |≤ R | x | − b | u ( t, x ) | dx dt . R + T R − b . (cid:3) Proof of Theorem 1.1.
Applying Proposition 3.1 with R ∼ T b +1 and T sufficientlylarge, we may find a sequence of times t n → ∞ and radii R n → ∞ such thatlim n →∞ Z | x |≤ R n | x | − b | u ( t n , x ) | dx = 0 . Thus, given any
R >
0, we have by H¨older’s inequality that Z | x |≤ R | u ( t n , x ) | dx . R b (cid:18)Z | x |≤ R | x | − b | u ( t n , x ) | dx (cid:19) → n → ∞ . We therefore derive scattering via Proposition 2.1. (cid:3)
References [1] A. Arora, B. Dodson, and J. Murphy,
Scattering below the ground state for the 2d radialnonlinear Schr¨odinger equation.
Proc. Amer. Math. Soc. (2020), no. 4, 1653–1663.[2] M. Cardoso, L. G. Farah, C. M. Guzm´an, and J. Murphy,
Scattering below the ground statefor the intercritical non-radial inhomogeneous NLS.
Preprint arXiv:2007.06165 .[3] T. Cazenave,
Semilinear Schr¨odinger equations , Courant Lecture Notes in Mathematics, vol.10, New York University, Courant Institute of Mathematical Sciences, New York; AmericanMathematical Society, Providence, RI, 2003.[4] B. Dodson and J. Murphy,
A new proof of scattering below the ground state for the 3d radialfocusing cubic NLS.
Proc. Amer. Math. Soc. (2017), no. 11, 4859–4867.[5] L. G. Farah,
Global well-posedness and blow-up on the energy space for the inhomogeneousnonlinear Schr¨odinger equation.
J. Evol. Equ. (2016), no. 1, 193–208.[6] L. G. Farah and C. M. Guzm´an, Scattering for the radial 3D cubic focusing inhomogeneousnonlinear Schr¨odinger equation.
J. Differential Equations (2017), no. 8, 4175–4231[7] C. M. Guzm´an,
On well posedness for the inhomogeneous nonlinear Schr¨odinger equation .Nonlinear Anal. Real World Appl. (2017), 249–286.[8] C. E. Kenig and F. Merle, Global well-posedness, scattering and lbow-up for the energy-critical, focusing, non-linear Schr¨odinger equation in the radial case.
Invent. Math, (2006), no. 3, 645–675.[9] C. Miao, J. Murphy, and J. Zheng,
Scattering for the non-radial inhomogeneous NLS.
Preprint arXiv:1912.01318.
To appear in Math. Res. Lett.[10] T. Ogawa and Y. Tsutsumi,
Blow-up of H solution for the nonlinear Schr¨odinger equation. J. Differential Equations (1991), no. 2, 317–330.[11] W. A. Strauss, Existence of solitary waves in higher dimensions.
Comm. Math. Phys. (1977), no. 2, 149–162.[12] T. Tao, On the asymptotic behavior of large radial data for a focusing non- linearSchr¨odinger equation . Dyn. Partial Differ. Equ. (2004), no. 1, 1–48.[13] C. Xu and T. Zhao, A remark on the scattering theory for the 2d radial focusing INLS .Preprint arXiv:1908.00743.
Department of Mathematics & Statistics, Missouri S&T
Email address ::