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A simple proof of scattering for the intercritical inhomogeneous NLS

We adapt the argument of Dodson-Murphy to give a simple proof of scattering below the ground state for the intercritical inhomogeneous nonlinear Schr\"odinger equation. The decaying factor in the nonlinearity obviates the need for a radial assumption.


Introduction
We revisit the problem of scattering below the ground state for the focusing, intercritical, inhomogeneous nonlinear Schrödinger equation (NLS). We restrict our attention to the case of a cubic nonlinearity in three dimensions, i.e.
with the parameter b chosen from the interval (0, 1 2 ). The scaling symmetry of (1.1) identifies the equation asḢ sc -critical, where s c = 1+b 2 ∈ ( 1 2 , 3 4 ). We call the equation intercritical because the critical regularity s c lies between the special values 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 simplify the presentation. In particular, the argument presented here should also apply to more general powers and dimensions d ≥ 3. The restriction on b, which arises from applications of Hardy's inequality, could perhaps be relaxed by modifying or refining the arguments given below.
Denoting by Q the ground state solution to ∆Q − Q + |x| −b Q 3 = 0 (1.2) and the conserved mass and energy of solutions by we will prove the following.
and ∇u 0 1+b Then the solution u to (1.1) with u| t=0 = u 0 is global in time and scatters, that is, there exist u ± ∈ H 1 such that lim t→±∞ u(t) − e it∆ u ± H 1 = 0. Theorem 1.1 was established first in the radial setting by [6], and subsequently in the non-radial setting by [2,9]. These works adopted the concentration-compactness approach to induction on energy pioneered in [8], reducing the problem of scattering to the problem of precluding a global non-scattering solution that is below the ground state threshold in the sense of (1.3) and (1.4) and has precompact orbit in H 1 . The preclusion of such a solution is achieved by using a localized virial argument. In non-radial problems, compact solutions are typically parametrized by some moving spatial center x(t). The key to passing from the radial to the nonradial case for (1.1) was the observation that the decaying factor in the nonlinearity already provides enough spatial localization to guarantee that x(t) ≡ 0, which in turn allows for a simple implementation of the virial argument. The basic idea is that if |x(t)| → ∞, then the solution would behave like an approximate solution to the linear Schrödinger 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. 1 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 evacuation condition for solutions below the ground state threshold. The radial assumption is used in both steps to derive quantitative decay estimates at large radii via the radial Sobolev embedding estimate of [11]. In both cases, however, this estimate is used only in controlling terms arising from the nonlinearity. Observing that the decaying factor in the nonlinearity of (1.1) already yields quantitative decay at large radii, we find that the simple argument of [4] suffices to treat (1.1), even in the non-radial case.
Without loss of generality, we consider scattering in the forward time direction only. After collecting a few preliminaries in Section 1.1, we will prove the scattering criterion in Section 2 and the virial/Morawetz estimate in Section 3. These two ingredients quickly imply Theorem 1.1.
1.1. Preliminaries. We will need a few results related to well-posedness and scattering for (1.1). We assume familiarity with the standard subcritical well-posedness theory for dispersive PDEs (e.g. the Duhamel formulation, Strichartz estimates, etc.). Otherwise, we refer the reader to [3] for a textbook treatment of nonlinear Schrödinger equations in general and to [7] for the specific case of the inhomogeneous NLS.
For any initial datum u 0 ∈ H 1 , there exists a unique maximal-lifespan solution to (1.1). Solutions conserve the mass and energy, and any solution that remains uniformly bounded in H 1 throughout its lifespan may be extended globally in time. For such solutions we have the following local estimate: for any Strichartz admissible pair (q, r). (1.5) We will also need the following small-data scattering result. 1 The paper [13] similarly adapted the arguments of the related work [1] to the 2d inhomogeneous NLS. However, the authors of [13] continued to work in the radial setting.
If ε is sufficiently small depending on E, then u scatters in H 1 as t → ∞.
x , where here and below all space-time norms are taken over (0, ∞) × R 3 . By interpolation, Sobolev embedding, and Strichartz estimates, we can deduce that e it∆ u 0 S E ε c for some c > 0. By Sobolev embedding, Strichartz estimates, Hölder's inequality, and Hardy's inequality, we can then estimate x . Thus for ε = ε(E) sufficiently small, we derive u S ε c . With this bound in hand, we may deduce that e −it∆ u(t) is Cauchy in H 1 as t → ∞ essentially by repeating the estimates above.
Next, we recall some properties of the ground state Q. For more details, we refer the reader to [5, Theorems 1.1 and 1.2].
The ground state Q arises as an optimizer for the Gagliardo-Nirenberg inequality Using Pohozaev identities (obtained by multiplying (1.2) by Q and x · ∇Q and integrating by parts), one can connect the sharp constant to norms of Q as follows: The proof of scattering is then connected to the following virial identity: which follows from (1.1) and integration by parts. In particular, the bound (1.8) and the sharp Gagliardo-Nirenberg inequality imply that the right-hand side of (1.9) is bounded 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 above identity, and accordingly we will need the following local form of coercivity. Lemma 1.3. Let u 0 ∈ H 1 satisfy the hypotheses of Theorem 1.1, and let u be the corresponding solution to (1.1). There exists δ ′ > 0 so that for all R sufficiently large, L 2 for some f ∈ H 1 and η ∈ (0, 1). Then (1.6) and (1.7) imply , which yields the following upon rearranging: (ii) Using (i), it suffices to to show that for R sufficiently large and some η > 0. As (1.8) holds and multiplication by χ R only decreases the L 2 -norm, it suffices to consider theḢ 1 -norm. For this, we use ). We conclude that (1.10) holds with η = 1 2 δ for all R sufficiently large.

Scattering criterion
The first ingredient for the proof of Theorem 1.1 is the following scattering criterion as in [12]. For the standard NLS, this criterion is only valid in the radial setting. As we will see, because of the decaying factor in the nonlinearity, this criterion is sufficient for (1.1) even in the non-radial setting.
then u scatters forward in time.
Proof. Throughout the proof, we allow implicit constants to depend on E. With ε > 0 and R > 1 to be chosen below, we first take T > ε −1 large enough that e it∆ u 0 where χ R is a smooth cutoff to {|x| ≤ R}. The goal is then to prove e i(t−T )∆ u(T ) < ε a for some a > 0, which (for ε sufficiently small) implies scattering via Lemma 1.2. To estimate this norm, we rewrite the Duhamel formula for u as follows: where for some c > 0 to be specified below. The linear term in (2.2) is controlled in acceptable manner by (2.1), and hence it suffices to estimate the two integral terms.
For the integral term in (2.2), we begin by using the triangle inequality, Sobolev embedding, and Strichartz estimates to obtain We estimate this term by interpolating between L 2 x andḢ 1 x . We first consider the estimate in L 2 x . We begin by extending the small mass condition at t = T in (2.1) to the interval I 1 . Using (1.1) to derive the identity we integrate by parts and use Cauchy-Schwarz to estimate d dt With R ≥ ε −2−c , this implies ε.
Recalling that b < 1 2 , we now choose an exponent r = r(b) satisfying 3 < r < 6 1+2b (2.5) and θ = θ(b) ∈ (0, 1) satisfying Writing r θ for the solution to 1 r = θ 2 + 1−θ r θ , we use the triangle inequality, Hölder's inequality, Hardy's inequality, and Sobolev embedding to estimate Here (2.5) and first constraint in (2.6) guarantee that we may apply Hardy's inequality, while the second constraint in (2.6) guarantees that the final norm is controlled by H 1 . Using Hölder's inequality, Sobolev embedding, and the local estimate (1.5), we therefore obtain We turn to theḢ 1 x estimate. This leads to two terms, which take the form The first term may be estimated exactly as above; we simply put ∇u in L 2 t L 6 x instead of u in (2.7). For the second term, we instead use Hölder's inequality, Hardy's inequality, Sobolev embedding, and (1.5) to estimate Here the application of Hardy's inequality requires b < 1 2 . Returning to (2.4), we obtain the following bound by interpolation: Here the estimate is the same as in [4]. We interpolate between the L 4 t L 3 x -norm and the L 4 t L ∞ x -norm, using the identity and Strichartz to obtain boundedness for the L 4 t L 3 x -norm. For the L 4 t L ∞ x -norm, we first use the dispersive estimate, Hardy's inequality, and Sobolev embedding to estimate , and hence we deduce the acceptable estimate I2 e i(t−s)∆ |x| −b |u| 2 u(s) ds

Virial/Morawetz estimate
In this section, we let u be a solution to (1.1) satisfying the hypotheses of Theorem 1.1. In particular, as discussed in Section 1.1, u is global, uniformly bounded in H 1 , and obeys (1.8). We will prove a virial/Morawetz estimate that implies the mass evacuation condition appearing in Proposition 2.1.
Proof. The proof is based off of the following identity, which follows from a direct computation using (1.1) and integration by parts: Given a smooth weight a : R 3 → R and defining A a (t) = 2 Im ūu j a j dx, where subscripts denote partial derivatives and repeated indices are summed, we have Inspired by [10], we choose a weight that interpolates between the standard virial and Morawetz weights. In particular, choosing R sufficiently large as in Lemma 1.3, we let a be a radial function satisfying a(x) = |x| 2 for |x| ≤ R 2 and a(x) = 2R|x| for |x| > R.
For R 2 < |x| ≤ R we impose ∂ r a ≥ 0, ∂ 2 r a ≥ 0, and |∂ α a(x)| α R|x| −|α|+1 for |α| ≥ 1, where ∂ r denotes radial derivative. We observe that the conditions above imply nonnegativity of the matrix a jk , and that we have the bound For |x| ≤ R 2 , we have a j = 2x j , a jk = 2δ jk , ∆a = 6, and ∆∆a = 0, while for |x| > R we have , ∆a = 4R |x| , ∆∆a = 0. Thus, by the identities above, where / ∇ denotes the angular part of the derivative. In (3.1), we insert χ 2 R and use the identity (1.11), Lemma 1.3, and uniform H 1 -boundedness of u to obtain For (3.2), the angular derivative term is nonnegative, while the nonlinear term is estimated by R −b . Similarly, in (3.3) the first term is nonnegative while the second term is estimated by R −b . Note that in contrast to [4], we do not use radial Sobolev embedding to obtain decay at large radii. Instead, the decay comes directly from the nonlinearity.
Applying the fundamental theorem of calculus on the interval [0, T ] now yields Proof of Theorem 1.1. Applying Proposition 3.1 with R ∼ T 1 b+1 and T sufficiently large, we may find a sequence of times t n → ∞ and radii R n → ∞ such that lim n→∞ |x|≤Rn |x| −b |u(t n , x)| 4 dx = 0.
We therefore derive scattering via Proposition 2.1.