Scattering theory for the defocusing energy-supercritical nonlinear wave equation with a convolution

Nonlinear Analysis 152 (2017) 220–249

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Nonlinear Analysis www.elsevier.com/locate/na

Scattering theory for the defocusing energy-supercritical nonlinear wave equation with a convolution Wei Han Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, PR China

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Article history: Received 12 October 2016 Accepted 11 January 2017 Communicated by Enzo Mitidieri Keywords: Scattering theory Strichartz estimate Concentration compactness

abstract In this paper, we study the global well-posedness and scattering problem for the nonlinear wave equation with a convolution utt − ∆u + (|x|−γ ∗ |u|2 )u = 0 in dimensions d ≥ 6. We show that if the solution u is apriorily bounded in the critical ˙ sc −1 ), with sc := γ−2 > 1, ˙ sc homogeneous Sobolev space, that is, u ∈ L∞ t (Hx × Hx 2 then u is global and it scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schr¨ odinger equation. Our analysis derived from the concentration compactness method to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios: the finite time blow-up solution, the soliton-like solution and the low-to-high frequency cascade. We note that, authors preclude the finite time blow-up solution to wave equation usually by the property of the finite speed of propagation in the previous literature, however, the finite speed of propagation is broken in the nonlocal nonlinear wave equation with a convolution nonlinearity. For this, we will initially establish the low regularity results for almost periodic solutions including the finite time blow-up solutions and initially preclude the finite time blow-up solutions by lowering the regularity. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we shall consider the initial value problem for the defocusing nonlinear wave equation with a Convolution in dimensions d ≥ 6:  utt − ∆u + (|x|−γ ∗ |u|2 )u = 0, d ≥ 6, (NLWH) (u, ut )|t=0 = (u0 , u1 ) ∈ H˙ xsc × H˙ xsc −1 (Rd ), where the nonlinearity F (u) = (|x|−γ ∗ |u|2 )u is energy-supercritical, that is, 4 < γ < d; and u(t, x) is a real-valued function on I × Rd and 0 ∈ I ⊂ R is a time interval, sc = γ−2 2 . E-mail address: sh [email protected]. http://dx.doi.org/10.1016/j.na.2017.01.008 0362-546X/© 2017 Elsevier Ltd. All rights reserved.

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Formally, the solution u of (NLWH) conserves the energy     1 1 |u(t, x)|2 |u(t, y)|2 1 dxdy E(u(t), ut (t)) = |∇u(t)|2 + |ut (t)|2 dx + 2 2 4 |x − y|γ Rd ×Rd Rd ≡ E(u0 , u1 ). The class of solutions to wave equation utt − ∆u + (|x|−γ ∗ |u|2 )u = 0 is left invariant by the scaling u(t, x) → uλ (t, x) = λ

2+d−γ 2

u(λt, λx) ∀λ > 0.

Moreover, it leaves the Sobolev norm H˙ xsc (Rd ) with sc = γ−2 2 invariant, this defines a notion of criticality. In view of the cubic convolution nonlinear term, 4 < γ < d implies sc > 1, and therefore we say that 4 < γ < d is the energy-supercritical regime for (NLWH). We study (NLWH) in the energy-supercritical regime sc > 1, in dimensions d ≥ 6 in this paper. We consider solutions to (NLWH), that is, functions u : I × Rd → R 2(1+d) d+2−γ such that for every K ⊂ I compact, (u, ut ) ∈ Ct (K; H˙ xsc × H˙ xsc −1 ), u ∈ Lt,x (K × Rd ), and satisfying the Duhamel formula  t sin((t − t′ )|∇|) u(t) = W(t)(u0 , u1 ) + F (u(t′ ))dt′ |∇| 0

for every t ∈ I, where 0 ∈ I ⊂ R is a time interval and the wave propagator W(t)(u0 , u1 ) = cos(t|∇|)u0 +

sin(t|∇|) u1 |∇|

is the solution to the linear wave equation with initial data (u0 , u1 ). We refer to I as the interval of existence of u, and we say that I is the maximal interval of existence if u cannot be extended to any larger time interval. We say that u is a global solution if I = R, and that u is a blow-up solution if ∥u∥ 2(1+d) = ∞. d+2−γ Lt,x (I×Rd )

Now we state our main result: 4d d Theorem 1.1. Assume that d ≥ 6, sc = γ−2 2 and d−1 < γ < d. Let u : I × R → R be a solution to (NLWH) with maximal interval of existence I ⊂ R satisfying

˙ sc −1 ). ˙ sc (u, ut ) ∈ L∞ t (I; Hx × Hx

(1.1)

Then u is global and  

2(1+d)

|u(t, x)| d+2−γ dxdt ≤ C, R

Rd

for some constant C = C(∥(u, ut )∥L∞ (I;H˙ xsc ×H˙ xsc −1 ) ). t

± ˙ sc ˙ sc −1 such that Moreover, u scatters in the sense that there exist unique (u± 0 , u1 ) ∈ Hx × Hx ± ± ± lim ∥(u(t), ut (t)) − (W(t)(u± ˙ xsc ×H ˙ xsc −1 = 0. 0 , u1 ), ∂t W(t)(u0 , u1 ))∥H

t→±∞

Remark 1.1. The restriction

4d d−1

< γ in Theorem 1.1 stems from Lemmas 4.2 and 4.3. More precisely, we

2d need to find an exponent q = q(d) simultaneously satisfies that q(d) ∈ ( 2(d−1) d−3 , d+2−γ ) and 2d(d−1) d2 +d−γd+γ ,

when d ≥ 6. These conditions on q(d) impose the restriction Section 4 for more details.

4d d−1

4d 3d−2γ

< q(d) <

< γ < d. One may refer to

The highlight of the paper is that we establish the low regularity results Theorem 4.1 for almost periodic solutions including the finite time blow-up solutions. In Section 5, we preclude the finite time blow-up solution by the low regularity results, while the previous papers in the literature, they use the finite speed

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of propagation to rule out the finite time blow-up solution, but in our paper, we consider the Wave-Hartree equation, there is no finite speed of propagation for the problem. On the one hand, for the energy-supercritical case, Kenig and Merle [8] first studied the 3-dimensional 4 (d is the space dimensions), they obtained the semilinear wave equation with nonlinearity |u|p u, p > d−2 global existence and scattering theory for radial initial data, and Killip and Visan [13] obtained the global existence and scattering theory for general data in d = 3 with even values of p and also in [14] for d ≥ 3 and radial initial data with a specified range of p. Aynur Bulut [2] obtained the global well-posedness and scattering theorems for the cubic nonlinearity F (u) = |u|2 u in the case of higher dimensions d ≥ 6 with no radial assumption on the initial data and in the energy-supercritical regime. On the other hand, the scattering theory for the nonlinear dispersive equation with Hartree nonlinearity has been studied by a series of works [15,16,18,17,19,20]. In the case sc = 1 of nonlinear wave equation with the energy critical defocusing cubic convolution nonlinearity (|x|−4 ∗ |u|2 )u, global well-posedness and scattering theory in the defocusing case was obtained by Miao, Zhang and Zheng [19]. The main difficulties of their paper are that the nonlocal nonlinear term, i.e., the Hartree term lacks the classical finite speed of propagation (i.e. the monotonic local energy estimate on the light cone), to compensate it, they resort to the extended causality and utilize the strategy derived from concentration compactness ideas. The another contribution of their paper is that they establish the lower bound of the potential energy in the sense of average which is independent of the nonlinear term. Since for the Hartree nonlinear term, the potential energy does not have the pointwise lower bound, to overcome this difficulty, they have proved that the potential energy have a lower bound in the sense of average. By using this property, they can preclude the soliton-like solution and low-to-high cascade solution. Their paper provides a method of dealing with the nonlocal nonlinear term. However, to the best of our knowledge, there are no global well-posedness and scattering theory results for the corresponding energy-supercritical case in the literature. This paper attempts to fill this gap in the literature. Killip and Visan [11,12] used the double Duhamel technique to deal with the higher dimensional scattering problem. We also used this principle to deal with the higher dimensional (d ≥ 6) Wave-Hartree equation. But for the case of low dimensional case d = 4, 5, this method is not suitable. In the present context, the restriction to dimensions d ≥ 6 appears as a consequence of our use of this technique; see the discussion in Section 4 for a more detailed account. The next we will deal with the energy-supercritical problem in the case of d = 4, 5. 1.1. Outline of the proof of Theorem 1.1 The proof of Theorem 1.1 is an argument by contradiction. Failure of Theorem 1.1 implies the existence of very special types of counterexamples. Such counterexamples are then shown to have so many properties, in fact, they cannot exist. While we will make some further reductions later, the main property of the special counterexamples is almost periodicity modulo symmetries: Definition 1.1. A solution u to (NLWH) with time interval I is said to be almost periodic modulo symmetries if (u, ut ) is bounded in H˙ xsc × H˙ xsc −1 and there exist functions N : I → R+ , x : I → Rd and C : R+ → R+ such that for all t ∈ I and η > 0,  (| |∇|sc u(t, x)|2 + | |∇|sc −1 ut (t, x)|2 )dx ≤ η, |x−x(t)|≥C(η)/N (t)

and  |ξ|≥C(η)N (t)

(|ξ|2sc |ˆ u(t, ξ)|2 + |ξ|2(sc −1) |ˆ ut (t, ξ)|2 )dξ ≤ η.

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We refer to the function N (t) as the frequency scale function for the solution u, to x(t) as the spatial center function, and to C(η) as the compactness modules function. Remark 1.2 (See [12,13]). If u is an almost periodic solution with modulo symmetries, then for each η > 0 there exist constants c1 (η), c2 (η) > 0 such that for all t ∈ I,   2d 2d |u(t, x)| d+2−γ dx + |ut (t, x)| d+4−γ dx ≤ η, |x−x(t)|≥c1 (η)/N (t)

|x−x(t)|≥c1 (η)/N (t)

and also 

u(t, ξ)|2 + |ξ|2(sc −1) |ˆ ut (t, ξ)|2 dξ ≤ η. |ξ|2sc |ˆ

(1.2)

|ξ|≤c2 (η)N (t)

We are now ready to state the first major milestone in the proof of Theorem 1.1. Theorem 1.2 (The Reduction to Almost Periodic Solutions, [8]). Assume Theorem 1.1 fails. Then there ˙ sc ˙ sc −1 ), and exists a maximal-lifespan solution u : I × Rd → R to (NLWH) such that (u, ut ) ∈ L∞ t (I; Hx × Hx ∥u∥ 2(1+d) = ∞, and u is a minimal blow-up solution in the following sense: for any solution v with d+2−γ Lt,x (I×Rd )

maximal interval of existence J such that ∥v∥

2(1+d) d+2−γ Lt,x (J×Rd )

= ∞, we have supt∈I ∥(u(t), ut (t))∥H˙ xsc ×H˙ xsc −1 ≤

supt∈J ∥(v(t), vt (t))∥H˙ xsc ×H˙ xsc −1 . Moreover, there exist N : I → R+ and x : I → Rd such that the set       1 x 1 x K= t, x(t) + , :t∈I , t, x(t) + 2+d−γ u 4+d−γ ut N (t) N (t) N (t) 2 N (t) 2

(1.3)

is precompact in H˙ xsc × H˙ xsc −1 (Rd ). The reduction to almost periodic solutions is now a standard technique in the analysis of dispersive equations at critical regularity by the linear profile decomposition from [1] together with the stability (Theorem 3.2). Their existence was first proved in the pioneering work by Keraani [9] for the mass-critical NLS. Kenig and Merle [7] adapted the argument to the energy-critical NLS, and first applied this to study the wellposedness problem. For the semilinear wave equations utt − ∆u + |u|2 u = 0, Kenig and Merle [8] proved such theorem in three dimensions with radial initial data, however, as Kenig [6,5] pointed out, when a satisfactory local theory is present the proof is independent of the dimension and the assumption of radial symmetry. The method of establishing the above theorem is similar as Kenig and Merle [8] by using a concentration compactness result in the form of a profile decomposition theorem for solutions of the linear wave equation. We will also need the following further refinement of Theorem 1.2. This part of the argument is quite standard and was achieved before in [13]. Here we just record the result without proving it. Theorem 1.3. Assume that Theorem 1.1 fails, then there exists a solution u : I × Rd → R to (NLWH) with maximal interval of existence I such that u is almost periodic modulo symmetries, ˙ sc ˙ sc −1 ), (u, ut ) ∈ L∞ t (I; Hx × Hx

and

∥u∥

2(1+d)

= ∞,

d+2−γ Lt,x (I×Rd )

and u satisfies one of the following: • (finite time blow-up solution) either sup I < ∞ or inf I > −∞, and inf t∈I N (t) ≥ 1. • (soliton-like solution) I = R and N (t) = 1 for all t ∈ R. • (low-to-high frequency cascade solution) I = R, inf N (t) ≥ 1,

t∈R

and

lim sup N (t) = ∞. t→∞

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Based on this important Theorem, the proof of Theorem 1.1 is reduced to show that each of the scenarios identified in Theorem 1.3 cannot occur. We study the three scenarios, the finite time blow-up solution, the soliton-like solution and the low-to-high frequency cascade. Not as in [2,12,13], in all these cases, we prove that the solutions including the finite time blow-up solution possess an additional decay property: for almost periodic solutions with the function N (t) bounded away from zero, the a priori bound ˙ sc ˙ sc −1 ) allows us to obtain the bound (u, ut ) ∈ L∞ ˙ 1−ϵ × H˙ x−ϵ ) for some ϵ > 0 (see (u, ut ) ∈ L∞ t (Hx × Hx t (Hx Theorem 4.1 for further details). We should point out that our low regularity results do not need the global solution hypothesis. For the proof of the additional decay property, we will introduce the following Duhamel formula. The argument is similar as in [13]. Lemma 1.4 ([13,25]). Let u : I × Rd → R be a solution to (NLWH) with maximal interval of existence I which is almost periodic modulo symmetries. Then for all t ∈ I,    T T sin((t − t′ )|∇|) ′ ′ ′ ′ ′ F (u(t ))dt , cos((t − t )|∇|)F (u(t ))dt ⇀ (u(t), ut (t)), (1.4) T →sup I |∇| t t and   −

t

T

sin((t − t′ )|∇|) F (u(t′ ))dt′ , − |∇|





t

cos((t − t′ )|∇|)F (u(t′ ))dt′

T

⇀ (u(t), ut (t))

T →inf I

(1.5)

weakly in H˙ xsc × H˙ xsc −1 . The rest of the paper is arranged as follows. We introduce some notations and present some preliminaries in Section 2. Section 3 is devoted to the study of the local well-posedness and stability. In Section 4, we prove an additional decay result for the finite-time blow-up, soliton-like and low-to-high frequency cascade scenarios. This result is then used to rule out these three cases in Sections 5–7 respectively. 2. Notation and useful lemmas We write X . Y to indicate that there exists a constant C > 0 such that X ≤ CY . We use the symbol ∇ for the derivative operator in the space variable. We write Lqt Lrx to denote the Banach space with the spacetime norm    q  q1 |u(t, x)|r dx

∥u∥Lqt Lrx = R

r

dt

Rd

with the usual modifications when q or r is equal to infinity, or when the domain R × Rd is replaced by a smaller region of spacetime such as I × Rd . When q = r we abbreviate Lqt Lqx as Lqt,x . Next, we define the Fourier transform of f (x) on Rd by  fˆ(ξ) := (2π)−d/2 e−ix·ξ f (x)dx. Rd

We also define the homogeneous Sobolev space H˙ xs (Rd ), s ∈ R via the norm ∥f ∥H˙ xs := ∥ |∇|s f ∥L2x , where the s f (ξ) := |ξ|s fˆ(ξ).  fractional differentiation operator is given by |∇| We first introduce the following standard dispersive estimate of the linear wave propagator, that is, Lemma 2.1 (Dispersive Estimate, See for Examples [2,22]). For any d ≥ 2, 2 ≤ p < ∞ and t ̸= 0 we have     eit|∇|   d+1 d−1 d−1 2     f  . |t|− 2 (1− p ) |∇| 2 − p f  p′ . (2.1)   |∇|  p Lx Lx

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In particular,    sin(t|∇|)    f   |∇| 

. |t|−

   cos(t|∇|)    g    |∇|2

. |t|−

(d−1) 2

  d+1 d−1  (1− p2 )  |∇| 2 − p f 

(2.2)

′

d Lp x (R )

d Lp x (R )

and (d−1) 2

1 p′

+

1 p

′

d Lp x (R )

d Lp x (R )

for all f, g ∈ S(Rd ), where

  d+1  d−3 (1− p2 )  |∇| 2 − p g 

,

= 1.

Definition 2.1 (Wave Admissible Pairs). For s ≥ 0, we say that a pair of exponents (q, r) is H˙ xs -wave admissible if q, r ≥ 2, r < ∞ and it satisfies 1 d−1 d−1 + ≤ , q 2r 4

and

1 d d + = − s. q r 2

Lemma 2.2 (Strichartz Estimates, See for Examples [2,23]). Assume u : I × Rd → R with time interval 0 ∈ I ⊂ R is a solution to the nonlinear wave equation  utt − ∆u + F = 0 (u, ut )|t=0 = (u0 , u1 ) ∈ H˙ xµ × H˙ xµ−1 (Rd ), µ ∈ R. Then s˜ µ−1 2 + ∥ |∇| 2 . ∥(u0 , u1 )∥ ˙ µ ∥ |∇|s u∥Lqt Lrx + ∥ |∇|s−1 ut ∥Lqt Lrx + ∥ |∇|µ u∥L∞ ut ∥L∞ ˙ xµ−1 + ∥ |∇| F ∥Lq˜′ Lr˜′ H x ×H t Lx t Lx x

t

(2.3) q , r˜) is H˙ x1+˜s−µ -wave admissible. for s ≥ 0, where the pair (q, r) is H˙ xµ−s -wave admissible and the pair (˜ For convenience, we define the following Strichartz norms. For each I ⊂ R and s ≥ 0, we set ∥u∥Ss (I) = ∥u∥Ns (I) =

sup ˙ s -wave admissible (q,r) H x

inf

˙ s -wave admissible (q,r) H x

∥u∥Lqt Lrx (I×Rd ) , ∥u∥Lq′ Lr′ (I×Rd ) . x

t

Taking the supremum over (q, r) H˙ xµ−s -wave admissible and the infimum over (˜ q , r˜)H˙ x1+˜s−µ -wave admissible pairs in (2.3), we also have, ∥ |∇|s u∥Sµ−s (I) + ∥ |∇|s−1 ut ∥Sµ−s (I) . ∥(u0 , u1 )∥H˙ xµ ×H˙ xµ−1 + ∥ |∇|s˜F ∥N1+˜s−µ (I) . Now we give a few basic estimates. Lemma 2.3 (Product Rule for Fractional Derivatives, See for Example [3]). For all s ≥ 0 we have ∥ |∇|s (f g)∥Lpx ≤ ∥ |∇|s f ∥Lpx1 ∥g∥Lpx2 + ∥f ∥Lpx3 ∥ |∇|s g∥Lpx4 , where 1 < p1 , p4 < ∞ and 1 < p, p2 , p3 ≤ ∞ satisfy

1 p

=

1 p1

+

1 p2

=

1 p3

+

1 p4 .

This together with Hardy–Littlewood–Sobolev inequality yields the following nonlinear estimates that will help us control the nonlinear term in establishing the local well-posedness and perturbation theory. Lemma 2.4. For d ≥ 6, we have     γ−3 γ−3     . |∇| 2 f  |∇| 2 ((|x|−γ ∗ g)f )

∥g∥

N1

S1

2

2

d+1 d+2−γ Lt,x

  γ−3   + |∇| 2 f 

S1 2

  γ−3   |∇| 2 g 

d+1

Ltd+2−γ Lx2d

2d(d+1) 2 +d−γd−3+γ

.

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1

2(d+1) ˙2 Proof. Note that ( 2γ−d−3 , d22d(d+1) +d−2γ+2 ) is an Hx wave admissible pair. By Lemma 2.3 and Sobolev’s inequality, we get,     γ−3 γ−3     . |∇| 2 ((|x|−γ ∗ g)f ) 2(d+1) 2d(d+1) |∇| 2 ((|x|−γ ∗ g)f ) N1

Lt3d+5−2γ Lxd

2

  γ−3   . |∇| 2 f 

2d

L2t Lxd−2

+ ∥f ∥

2d L2t Lxd+1−γ

  γ−3   . |∇| 2 f 

∥ |x|−γ ∥ d L∗γ

2d L2t Lxd−2

2d

L2t Lxd−2

+ ∥f ∥

2d L2t Lxd+1−γ

2d(d+1) 2 +d+γ−γd−3

  γ−3   |∇| 2 g 

2d(d+1) 2 +d+γ−γd−3

d+1

Ltd+2−γ Lx2d

d+1

  γ−3   |∇| 2 g 

d+1

Ltd+2−γ Lx2d

∥g∥

d+1 d+2−γ Lt,x

2

d+1

d+2−γ Lt,x

2d L2t Lxd−2

S1

d+1

d+2−γ Lt,x

Ltd+2−γ Lx2d

d+1 d+2−γ Lt,x

∥g∥

∥g∥

  γ−3   |∇| 2 g 

∥g∥

  γ−3   + |∇| 2 f    γ−3   . |∇| 2 f 

d

L∗γ

∥ |x|−γ ∥

  γ−3   . |∇| 2 f    γ−3   . |∇| 2 f 

2d(d+1)

d+1

Ltd+2−γ Lxγd−d+2γ−3+γ

2d

2d L2t Lxd+1−γ

d(d+1)

d+1

Ltd+2−γ Lx d+γ

   γ−3  −γ   |x| ∗ |∇| 2 g 

L2t Lxd−2

+ ∥f ∥

∥ |x|−γ ∗ g∥

2 +d+2γ−2

  γ−3   + |∇| 2 f 

2d(d+1) 2 +d+γ−γd−3

  γ−3   |∇| 2 g 

S1

d+1

Ltd+2−γ Lx2d

2

2d(d+1) 2 +d−γd−3+γ

1

2d ) is an H˙ x2 admissible pair, which gives the right hand side of the desired where we used the fact that (2, d−2 inequality. 

Similarly, we have the following Lemma 2.5. For d ≥ 6, we have     γ−3 γ−3     . |∇| 2 f  |∇| 2 ((|x|−γ ∗ g)f )

∥g∥

N1

S1

2

2

Lemma 2.6. For d ≥ 6, we have     γ−3 γ−3     . |∇| 2 f  |∇| 2 (|x|−γ ∗ (f g))

d+1 d+2−γ Lt,x

∥g∥

N1

S1

2

2

d+1 d+2−γ Lt,x

+ ∥f ∥

2(d+1) d+2−γ Lt,x

  γ−3   |∇| 2 g 

  γ−3   + |∇| 2 f 

S1

2(d+1)

  γ−3   |∇| 2 g 

N1

  γ−3   . |∇| 2 f 

2d

  γ−3   . |∇| 2 f 

∥f ∥2

L2t Lxd−2

2

S1

Proof. The proof is as in the proof of Lemma 2.4, we omit it here. We next state some basic facts from Littlewood–Paley theory:

∥f ∥2

2(d+1)

d+2−γ Lt,x

2(d+1)

d+2−γ Lt,x

2



d+1

Ltd+2−γ Lx2d

2

We also need the following estimate, that is, Lemma 2.7. For d ≥ 6, we have   γ−3   |∇| 2 ((|x|−γ ∗ |f |2 )f )

2d(d+1) 2 −γd+d−2

.

Lt2d−γ+3 Lx2d

.

2d(d+1) 2 +d−γd−3+γ

.

W. Han / Nonlinear Analysis 152 (2017) 220–249

227

Lemma 2.8 (Littlewood–Paley Theory). Let φ(ξ) be a real valued radially symmetric bump function supported in the ball {ξ ∈ Rd : |ξ| ≤ 2} which equals 1 on the ball {ξ ∈ Rd : |ξ| ≤ 1}. For any dyadic number N = 2k , k ∈ Z, we define the following Littlewood–Paley operators: ˆ P ≤N f (ξ) = φ(ξ/N )f (ξ), ˆ P >N f (ξ) = (1 − φ(ξ/N ))f (ξ), ˆ P N f (ξ) = (φ(ξ/N ) − φ(2ξ/N ))f (ξ). Similarly, we define P
and also PM <·≤N := P≤N

PN + PN ,  − P≤M = M
These operators commute with one another, with derivative operators, the free propagator, and the conjugation operation. Moreover, they are bounded on Lpx for 1 ≤ p ≤ ∞ and obey the following Bernstein inequalities, d

d

∥PN f ∥Lqx . N p − q ∥PN f ∥Lpx , ∥ |∇|±s PN f ∥Lpx ∼ N ±s ∥PN f ∥Lpx , d

d

∥P≤N f ∥Lqx . N p − q ∥P≤N f ∥Lpx , with s ≥ 0 and 1 ≤ p ≤ q ≤ ∞. We will introduce the following Morawetz estimate for the wave-Hartree equation considered in this paper, this method for the Morawetz estimate is motivated by Miao, Zhang and Zheng [19] and is similar to the corresponding part of [19]. However, we still provide the whole proof of the Morawetz estimate to the reader for the completeness of the paper. Theorem 2.9 (Morawetz Estimate). Assume u : I × Rd → R is a solution to (NLWH). Then we have   ∗ |u(t, x)|2 dxdt ≤ CE(u, ut ), |x| I Rd where 2∗ =

2d d−2 .

Proof. Let ψ = ur + 

(d−1)u 2|x| ,

then we have

    1 (d − 1) (∂tt − ∆)u + (V ∗ |u|2 )u ψ = ∂t (uψ) ˙ + ∇ · −(∇uψ) + θℓ(u) + |u|2 ∇ 2 2|x| |uθ |2 (d − 1)(d − 3)|u|2 1 + + − θ · ∇(V ∗ |u|2 )|u|2 , r 4r3 2

where V (x) = |x|−γ , r = |x|,

θ=

ℓ(u) = x , |x|

 1 −|u| ˙ 2 + |∇u|2 + (V ∗ |u|2 )|u|2 , 2 ur = θ · ∇u,

uθ = ∇u − θur .

    Integrating the above equality with respect to (t, x) over W = (t, x) 0 < a ≤ t ≤ b, x ∈ Rd , we obtain that         |uθ |2 (d − 1)(d − 3)|u|2 1 2 x t=b  2 ¯  . + − |u| · ∇(V ∗ |u| ) dxdt ≤ C u ˙ ψdx (2.4)  t=a   Rd r 4r3 2 |x| W

W. Han / Nonlinear Analysis 152 (2017) 220–249

228

Since −

x |x| |y| − x · y  1 y  1  ∇V (x − y) = γ ≥ 0, − + |x| |y| |x − y|γ+2 |x| |y|

we have 

b



x y  ∇V (x − y)|u(x)|2 |u(y)|2 dydxdt ≥ 0. − |y| Rd ×Rd |x|

− a

So we have −

1 2



|u|2

W

x · ∇(V ∗ |u|2 )dxdt ≥ 0. |x|

(2.5)

Substituting (2.5) into (2.4) and using the Hardy inequality, we obtain that      t=b  |uθ |2 (d − 1)(d − 3)|u|2    + ˙ 22 + ∥∇u∥22 ) ≤ CE. dxdt ≤ C  uψdx ˙  ≤ C(∥u∥ t=a   Rn r 4r3 W On the other hand, we have   ∞   ∞ ∗ ∗ ∗ |u|2 dx = r−1 |u(rθ)|2 dθrd−1 dr = rd−2 ∥u∥2L2∗ (S d−1 ) dr. θ r Rd 0 S d−1 0

(2.6)

(2.7)

Note that the following interpolation and the Sobolev embedding ∗

[H 1 (S d−1 ), Lβ (S d−1 )] d2 = Hqσ (S d−1 ) ↩→ L2 (S d−1 ), where β =

2(d−1) d−2 ,

σ=

d−2 1 d , q

=

σ d−1

+

1 2∗

∗

 Rd

|u|2 dx ≤ r

= (1 − d2 ) 12 +  0

2 dβ ,

(2.8)

it follows that

∞

4

rd−2 ∥u∥2H 1 (S d−1 ) ∥u∥ d−2 β

Lθ (S d−1 )

θ

dr.

(2.9)

We introduce a Sobolev embedding for the polar coordinates established in [21]. Lemma 2.10 ([21], Proposition 3.7). Let 1 ≤ p < d. Then

where β =

(d−1)p d−p ,

ν=

d−p p

and L∞,ν r

H˙ p1 ↩→ Lβθ L∞,ν ↩→ L∞,ν Lβθ , r r   = u : ∥u∥L∞,ν = ∥rν u(r)∥L∞ <∞ . r r

(2.10)

Applying this lemma with p = 2 to (2.9), we have   ∞ ∗ 4  d−2  d−2 |u|2  2  dx ≤ r u ∞ β r−3 ∥u∥2H 1 (S d−1 ) rd−1 dr θ r Lr Lθ 0 Rd    4 |u|2 |uθ |2 ≤ ∥∇u∥Ld−2 + dx. (2.11) 2 3 r r Rd b 2∗ Integrating (2.11) with respect to time t and using (2.6), we get that a Rd |u|r dxdt ≤ C(E). This implies Theorem 2.9.  Lemma 2.11 (Gronwall’s Inequality, See [13,2] for Instance). Given β, β ′ , C, η > 0, let {xk }k≥0 be a bounded non-negative sequence obeying xk ≤ C2−βk + η

k−1  l=0

If η ≤

1 4

′

2−β(k−l) xl + η

∞ 

2−β

′

|k−l|

xl ,

for all k ≥ 0.

l=k

min{1 − 2−β , 1 − 2−β , 1 − 2ρ−β } for some ρ ∈ (0, β), then xk ≤ (4C + ∥x∥l∞ )2−ρk .

W. Han / Nonlinear Analysis 152 (2017) 220–249

229

We now recall that the wave propagator W(t) is weakly continuous for all t ∈ R. Lemma 2.12 (Weak Continuity of the Wave Propagator). Suppose {(fn , gn )} ⊂ H˙ x1 × L2x is a sequence such that for some (f, g) ∈ H˙ x1 × L2x , we have (fn , gn ) ⇀ (f, g) weakly in H˙ x1 × L2x .

(2.12)

2d

Then for every τ ∈ R, W(τ )(fn , gn ) ⇀ W(τ )(f, g) weakly in Lxd−2 . Proof. This is exactly Proposition A.2 in Bulut [2].



3. Local well-posedness and stability In this section, we will give a standard local well-posedness theory and a stability result for (NLWH). All results follow from the Strichartz inequalities (2.3), the Hardy–Littlewood- Sobolev inequality and the usual contraction mapping arguments. We make use of the convolution nature of the nonlinearity F (u) = (|x|−γ ∗ |u|2 )u. The following properties for the nonlinear Wave-Hartree equation are analogous to the nonlinear Schr¨ odinger equation in [4]. For convenience, we state the theorems and sketch their proofs. Theorem 3.1 (Local Well-Posedness). Let d ≥ 6 and sc = γ−2 2 . Then for all A > 0, there exists δ0 = δ0 (d, A) > 0 such that for every 0 < δ ≤ δ0 , 0 ∈ I ⊂ R, and (u0 , u1 ) ∈ H˙ xsc × H˙ xsc −1 (Rd ) with ∥(u0 , u1 )∥H˙ xsc ×H˙ xsc −1 ≤ A, the condition ∥W(t)(u0 , u1 )∥ I × Rd with ∥u∥

2(d+1)

(3.1)

≤ δ implies that there exists a unique solution u to (NLWH) on

2(d+1) d+2−γ Lt,x (I×Rd )

≤ 2δ, and

d+2−γ Lt,x

  γ−3   |∇| 2 u

S 1 (I)

  γ−3   + |∇| 2 −1 ut 

< ∞.

S 1 (I)

2

2

Proof. We use a contraction mapping argument. By the Duhamel representation for the solution to (NLWH), we have  t  sin((t − s)|∇|)  u(t) = W(t)(u0 , u1 ) + (|x|−γ ∗ |u(s)|2 )u(s) ds. |∇| 0 For all a, b > 0, we define the contraction space  Ba,b :=

v : ∥v∥

2(d+1)

d+2−γ Lt,x

  γ−3   ≤ a, |∇| 2 v 

S1 2



  γ−3   + |∇| 2 −1 vt 

≤b ,

S1 2

and the map  Φ(v)(t) := W(t)(u0 , u1 ) + 0

t

 sin((t − s)|∇|)  (|x|−γ ∗ |v(s)|2 )v(s) ds. |∇|

We will show that for suitably chosen a and b, we have Φ(Ba,b ) ⊂ Ba,b and the mapping Φ : Ba,b → Ba,b is a contraction. Noticing the assumption (3.1), the Strichartz inequality, Lemma 2.7 and by using Minkowski’s inequality, we get for v ∈ Ba,b ,

W. Han / Nonlinear Analysis 152 (2017) 220–249

230

  γ−3   |∇| 2 Φ(v)

  γ−3   + |∇| 2 −1 ∂t Φ(v)

S1

S1

2

  γ−3   ≤ |∇| 2 W(t)(u0 , u1 )

S1 2

2    t   γ−3 sin((t − s)|∇|)   −γ 2 + |∇| 2 [(|x| ∗ |v| )v(s)]ds   |∇| 0

S1

2    t   γ−3   −γ 2 −1 +  |∇| 2 cos((t − s)|∇|)[(|x| ∗ |v| )v(s)]ds   0

  γ−3   + |∇| 2 −1 ∂t W(t)(u0 , u1 )

S1

S1

2

2

. ∥(u0 , u1 )∥H˙ xsc ×H˙ xsc −1

  γ−3   + |∇| 2 [(|x|−γ ∗ |v|2 )v(s)]

N1 2

  γ−3   ≤ CA + C ′ |∇| 2 v 

∥v∥2

S1

2(d+1)

d+2−γ Lt,x

2

   γ−3   ≤ CA + Ca |∇| 2 v  2

S1



  γ−3   +  |∇| 2 −1 ∂t v 

≤ CA + Ca2 b.

(3.2)

S1

2

2

Similarly, using Minkowski’s inequality together with the assumption (3.1), we estimate    t sin((t − s)|∇|)    −γ 2 ∥Φ(v)∥ 2(d+1) ≤ ∥W(t)(u0 , u1 )∥ 2(d+1) +  ((|x| ∗ |v| )v(s))ds 2(d+1) d+2−γ d+2−γ   d+2−γ |∇| 0 Lt,x Lt,x Lt,x   γ−3   −γ 2 ≤ δ + C |∇| 2 ((|x| ∗ |v| )v(s)) N1 2

  γ−3   ≤ δ + C |∇| 2 v 

2

∥v∥

2(d+1) d+2−γ Lt,x

S1 2

≤ δ + Ca2 b.

Choosing b = 2AC and a such that Ca2 ≤ 21 , we obtain   γ−3   |∇| 2 Φ(v)

≤ b.

(3.3)

S1 2

If we also fix δ =

a 2

and a small enough such that Ca b ≤ a2 , we have 2

∥Φ(v)∥

≤ a.

2(d+1)

(3.4)

d+2−γ Lt,x

Combining (3.3) and (3.4) with the above choices of a, b and δ, we have the desired inclusion Φ(Ba,b ) ⊂ Ba,b . We now prove that the mapping Φ is a contraction for suitable a, b and δ. Let a, b and δ be as chosen above. Note that by the Strichartz inequality, Lemma 2.4 and Minkowski’s inequality, we get   γ−3 γ−3   |∇| 2 [Φ(u) − Φ(v)] + ∥ |∇| 2 −1 ∂t [Φ(u) − Φ(v)]∥S 1 + ∥Φ(u) − Φ(v)∥ 2(d+1) S1

d+2−γ Lt,x

2

2   γ−3   −γ 2 . |∇| [((|x| ∗ |v|2 )v(s)) − ((|x|−γ ∗ |u|2 )u(s))]

N1

2   γ−3   −γ 2 −γ 2 = |∇| [(|x| ∗ |u| )(u − v) − (|x| ∗ (u + v)(u − v))v(s)]

N1

  γ−3   ≤ |∇| 2 [(|x|−γ ∗ |u|2 )(u − v)]

N1

2   γ−3   + |∇| 2 [(|x|−γ ∗ (u + v)(u − v))v(s)]

N1

2

  γ−3   ≤ |∇| 2 (u − v)

∥u2 ∥

S1 2

  γ−3   + |∇| 2 v 

d+1 d+2−γ Lt,x

∥(u2 − v 2 )∥

S1 2

2

  γ−3   + |∇| 2 (u − v)

d+1 d+2−γ Lt,x

S1

  γ−3   |∇| 2 (u2 )

+ ∥v∥

2(d+1) d+2−γ Lt,x

d+1

2d(d+1) 2 +d−γd−3+γ

Ltd+2−γ Lx2d

2

  γ−3   |∇| 2 (u2 − v 2 )

2(d+1)

Lt2d−γ+3 Lx2d

2d(d+1) 2 −γd+d−2

W. Han / Nonlinear Analysis 152 (2017) 220–249

  γ−3   ≤ |∇| 2 (u − v)

∥u∥2

2(d+1) d+2−γ Lt,x

Ssc −α

  γ−3   + |∇| 2 v 

∥(u − v)∥

S1

2(d+1)

∥(u + v)∥

∥u∥2

2(d+1) d+2−γ Lt,x

S1 2

  γ−3   + |∇| 2 v 

∥(u − v)∥

S1

2d

Ssc −α

∥(u + v)∥

2(d+1)

2d

L2t Lxd−2

 2

. ∥v − u∥Ba,b ∥u∥

2(d+1)

d+2−γ Lt,x

∥u∥

2(d+1) d+2−γ Lt,x

2(d+1) d+2−γ Lt,x

∥u∥

2(d+1)

2(d+1)

d+2−γ Lt,x

2

d+2−γ Lt,x

2(d+1)

d+2−γ Lt,x

S1

+ ∥v∥

2(d+1)

d+2−γ Lt,x

  γ−3   + |∇| 2 (u + v)

 2d

L2t Lxd−2

  γ−3   |∇| 2 u

∥u∥

S1

∥(u − v)∥

2(d+1)

d+2−γ Lt,x

2(d+1)

d+2−γ Lt,x

2

2(d+1)

∥(u + v)∥

  γ−3   + |∇| 2 u

 + ∥v∥

∥u∥

d+2−γ Lt,x

   γ−3   |∇| 2 (u − v)

d+2−γ Lt,x

2(d+1)

d+2−γ Lt,x

  γ−3   + |∇| 2 (u − v)

2(d+1)

2d(d+1) 2 −d−3+γ

2(d+1)

∥(u + v)∥

d+2−γ Lt,x

2

2(d+1)

Ltd+2−γ Lxd

d+2−γ Lt,x

L2t Lxd−2

  γ−3   ≤ |∇| 2 (u − v)

  γ−3   |∇| 2 u

2

   γ−3   |∇| 2 (u − v)

d+2−γ Lt,x

+ ∥v∥

S1

d+2−γ Lt,x

2

+ ∥v∥

2(d+1)

  γ−3   + |∇| 2 (u − v)

231

  γ−3   + |∇| 2 (u + v)

∥(u − v)∥

2(d+1)

d+2−γ Lt,x



 ∥u∥

S1

2(d+1)

d+2−γ Lt,x

2

S1

2d

L2t Lxd−2

  γ−3   + |∇| 2 v 

  γ−3   + |∇| 2 u



+ ∥v∥

2(d+1) d+2−γ Lt,x



  γ−3   + |∇| 2 v 

S1

2

2

2

. ∥u − v∥Ba,b (a + ab),

(3.5) 1 2

2(d+1) ˙ where we note that ( d+2−γ , d22d(d+1) −d−3+γ ) is an Hx admissible pair, which gives the right hand side of the desired inequality, and we use H¨ older’s inequality and Lemma 2.3 to obtain (3.5). Thus, if a is chosen such that C(a2 + ab) < 1 we conclude that Φ is a contraction as desired. 

Theorem 3.2 (Stability). Let d ≥ 6 and sc = γ−2 ˜ be a 2 . Assume 0 ∈ I ⊂ R is a compact interval and let u function on I × Rd which is a near-solution to (NLWH) in the sense that u ˜tt − ∆˜ u + (|x|−γ ∗ |˜ u|2 )˜ u = e, for some function e. Then for every E, L > 0, there exists ϵ1 = ϵ1 (E, L) > 0 such that for each 0 < ϵ < ϵ1 , the conditions sup ∥(˜ u(t), u ˜t (t))∥H˙ xsc ×H˙ xsc −1 (Rd ) ≤ E, t∈I

∥(u0 − u ˜(0), u1 − u ˜t (0))∥H˙ xsc ×H˙ xsc −1 (Rd ) ≤ ϵ,   γ−3   ≤ ϵ, and ∥˜ u∥ 2(d+1) ≤ L |∇| 2 e N 1 (I)

d+2−γ Lt,x

2

imply that there exists a unique solution u : I × Rd → R to (NLWH) with initial data (u0 , u1 ) such that   γ−3   ˜ ) ≤ C(E, L)ϵ, (3.6) |∇| 2 (u − u S 1 (I) 2

∥˜ u − u∥

2(d+1) d+2−γ Lt,x

  γ−3   |∇| 2 u

≤ C(E, L)ϵ,

(3.7)

≤ C(E, L).

(3.8)

S 1 (I) 2

Proof. The first step of the proof is to obtain a bound on ∥ |∇|

γ−3 2

u ˜∥S 1 (I) , for this purpose, we fix ϵ1 , η > 0 2

(to be determined later in the argument) and partition I into J0 = J0 (L, η) subintervals Ij = [tj , tj+1 ] such

W. Han / Nonlinear Analysis 152 (2017) 220–249

232

that for each j = 1, . . . , J0 , ∥˜ u∥

≤ η. Noting the Strichartz inequality and Lemma 2.7, we

2(d+1) d+2−γ Lt,x (Ij ×Rd )

obtain   γ−3   ˜ |∇| 2 u

S 1 (Ij )

  γ−3   . ∥(˜ u(tj ), u ˜t (tj ))∥H˙ xsc ×H˙ xsc −1 + |∇| 2 e

N 1 (Ij )

2

  γ−3   + |∇| 2 F (˜ u(s))

N 1 (Ij )

2

2

  γ−3   . E + ϵ + |∇| 2 [(|x|−γ ∗ |˜ u|2 )˜ u]

N 1 (Ij ) 2

  γ−3   ˜ |∇| 2 u

2

. E + ϵ + ∥˜ u∥

2(d+1) d+2−γ Lt,x

S 1 (Ij )

  γ−3   . E + ϵ1 + η 2 |∇| 2 u ˜

S 1 (Ij )

2

2

for each ϵ < ϵ1 . Choosing η > 0 sufficiently small and ϵ1 < E, we obtain ∥|∇|

γ−3 2

u ˜∥S 1 (Ij ) . E. Summing up 2

about j, we get that   γ−3   ˜ |∇| 2 u

S 1 (I)

. C(E, L).

(3.9)

2

2(d+1) Next, fixing ϵ1 ≤ E and δ > 0 (to be determined later in the argument), noting that ( d+2−γ , d22d(d+1) −d−3+γ ) is 1 an H˙ x2 -wave admissible pair, and using (3.9), we can divide I into J1 = J1 (E, L, δ) subintervals Ij = [tj , tj+1 ]

such that for each j = 1, . . . , J1 , we have ∥|∇|

γ−3 2

u ˜∥

2(d+1) Ltd+2−γ

≤ δ, and ∥ |∇|

2d(d+1) 2 Lxd −d−3+γ γ−3 2

w =u−u ˜, and define, for t ∈ I and j = 1, . . . , J1 , γj (t) := ∥ |∇|

γ−3 2

u ˜∥

2d

L2t Lxd−2

[F (˜ u + w) − F (˜ u)]∥N 1 ([tj ,t]) . 2

For each j ∈ {1, . . . , J1 }, we now estimate γj (t). Noticing that F (u) − F (v) = (|x|−γ ∗ |u|2 )u − (|x|−γ ∗ |v|2 )v = (|x|−γ ∗ |u|2 )(u − v) + (|x|−γ ∗ ((u − v)(u + v)))v, and by using Lemma 2.4, Minkowski’s and H¨ older’s inequalities, we get, for each j ∈ {1, . . . , J1 },   γ−3   u + w|2 )w + (|x|−γ ∗ (w(2˜ u + w)))˜ u] γj (t) = |∇| 2 [(|x|−γ ∗ |˜ N 1 ([tj ,t])

  γ−3   ≤ |∇| 2 w

∥ |˜ u + w|2 ∥

d+1 d+2−γ Lt,x

S1 2

  γ−3   + |∇| 2 u ˜

2d

L2t Lxd−2

  γ−3   + |∇| 2 u ˜

2d L2t Lxd−2

S1

∥w(2˜ u + w)∥

d+1

d+2−γ Lt,x

  γ−3   u + w)) |∇| 2 (w(2˜

d+1

Ltd+2−γ Lx2d

2d(d+1) 2 +d−γd−3+γ

 2

∥˜ u ∥

d+1

+ ∥˜ uw∥

d+2−γ Lt,x

S1 2

  γ−3   + |∇| 2 w

S1

  γ−3   u + w) |∇| 2 (˜

2(d+1)

2d(d+1) 2 −d−3+γ

2(d+1)

 2d

2d

L2t Lxd−2

d+2−γ Lt,x

d+1

d+2−γ Lt,x

∥˜ u + w∥

∥w˜ u∥

2(1+d)

d+2−γ Lt,x

Ltd+2−γ Lxd

L2t Lxd−2

  γ−3   + |∇| 2 u ˜

+ ∥w ∥

d+2−γ Lt,x

2

  γ−3   + |∇| 2 u ˜

2

d+1

 2

+ ∥w ∥

d+1 d+2−γ Lt,x

   γ−3   |∇| 2 w

  γ−3   u + w) |∇| 2 (2˜

d+1

d+2−γ Lt,x

2(d+1)

Ltd+2−γ Lxd

2d(d+1) 2 −d−3+γ

∥2˜ u + w∥

 2(d+1)

Ltd+2−γ Lxd

d+1

Ltd+2−γ Lx2d

2



  γ−3   . |∇| 2 w

+ ∥w∥

2   γ−3   u + w)2  |∇| 2 (˜

  γ−3   + |∇| 2 w

2d(d+1) 2 −d−3+γ

2(d+1)

d+2−γ Lt,x

2d(d+1) 2 +d−γd−3+γ

≤ δ. Let

W. Han / Nonlinear Analysis 152 (2017) 220–249

233



  γ−3   . |∇| 2 w

∥w∥2

+ ∥˜ u∥2

2(d+1) d+2−γ Lt,x

S1 2

  γ−3   ˜ + |∇| 2 u

2(d+1) Ltd+2−γ

  γ−3   + |∇| 2 w

2(d+1)

+ ∥˜ u∥

∥˜ u∥

∥w∥

2(d+1)

d+2−γ Lt,x

  γ−3   + |∇| 2 u ˜

2(d+1) d+2−γ Lt,x

2d(d+1) 2 −d−3+γ

2(d+1)

d+2−γ Lt,x

∥˜ u∥

2d(d+1) 2 Lxd −d−3+γ

Ltd+2−γ Lxd

2(d+1)

d+2−γ Lt,x

2(d+1)

∥w∥

2d(d+1) 2 −d−3+γ



  γ−3   + |∇| 2 w

2(d+1)

d+2−γ Lt,x

2(d+1)

d+2−γ Lt,x

Ltd+2−γ Lxd

2(d+1)

Ltd+2−γ Lxd

2d(d+1) 2 −d−3+γ

∥w∥

2(d+1) d+2−γ Lt,x



  γ−3   ˜ + |∇| 2 u

∥w∥

2d

L2t Lxd−2

  γ−3   + |∇| 2 w

∥˜ u∥

S1

 3 γ−3   . |∇| 2 w

2(d+1)

d+2−γ Lt,x

2

S 1 (Ij )

2(d+1)

∥˜ u∥

d+2−γ Lt,x

2(d+1)

+ ∥w∥2

d+2−γ Lt,x

  γ−3   + |∇| 2 w

∥w∥

S1

S 1 (Ij )

2

2(d+1)

+ ∥w∥

d+2−γ Lt,x

2

 2 γ−3   + δ |∇| 2 w

2(d+1)

d+2−γ Lt,x

2(d+1) d+2−γ Lt,x

  γ−3   ˜ |∇| 2 u

 2(d+1)

Ltd+2−γ Lxd

  γ−3   + δ 2 |∇| 2 w

2d(d+1) 2 −d−3+γ

(3.10)

S 1 (Ij )

2

2

where we have used Lemma 2.3 and Sobolev inequality to obtain the last inequality. We next show by induction that for every j = 1, . . . , J1 , there exists a constant C(j, d) > 0 such that γj (t) ≤ C(j, d)ϵ.

(3.11)

In order to prove the inequality (3.11). Firstly, we let ϵ ∈ R be arbitrary such that ϵ < ϵ1 and we note that without loss of generality we may assume t1 = 0. Secondly, we notice that when j = 1, by Strichartz inequality, for every t ∈ I1 ,       γ−3 γ−3 γ−3       . ∥(w(t1 ), wt (t1 ))∥H˙ xsc ×H˙ xsc −1 + |∇| 2 [F (˜ + |∇| 2 e u) − F (u)] |∇| 2 w S 1 ([t1 ,t])

N 1 ([t1 ,t])

2

N 1 (I1 )

2

2

. ∥(w(0), wt (0))∥H˙ xsc ×H˙ xsc −1 + γ1 (t) + ϵ . ϵ + γ1 (t) + ϵ.

(3.12)

Noting (3.10) and (3.12), we get γ1 (t) . (γ1 (t) + ϵ)3 + δ(γ1 (t) + ϵ)2 + δ 2 (γ1 (t) + ϵ). By bootstrap argument, we get for δ and ϵ sufficiently small, γ1 (t) . ϵ for all t ∈ I1 , this completes the induction base. Next we assume that for all j ≤ j0 there exists C(j, d, δ) > 0 such that γj (t) ≤ C(j, d)ϵ for all t ∈ Ij . We then will prove (3.11) for j = j0 + 1 in the following. For every t ∈ Ij0 +1 , by the Strichartz inequality, we have     γ−3 γ−3     . ∥(w(tj0 +1 ), wt (tj0 +1 ))∥H˙ xsc ×H˙ xsc −1 + |∇| 2 [F (˜ u) − F (u)] |∇| 2 w S 1 ([tj0 +1 ,t])

N 1 ([tj0 +1 ,t])

2   γ−3   + |∇| 2 e

2

N 1 (Ij0 +1 ) 2

. ∥(w(tj0 +1 ), wt (tj0 +1 ))∥H˙ xsc ×H˙ xsc −1 + γj0 +1 (t) + ϵ   γ−3   . ∥(w(0), wt (0))∥H˙ xsc ×H˙ xsc −1 + |∇| 2 [F (˜ u) − F (u)]

N 1 ([0,tj0 +1 ]) 2

. 3ϵ + γj0 +1 (t) +

j0  k=1

Noting

j0

k=1

 γk (tk+1 ) .

3+

j0 

  γ−3   + |∇| 2 e

N 1 ([0,tj0 +1 ])

+ γj0 +1 (t) + ϵ

2

 C(k, d) ϵ + γj0 +1 (t).

k=1

C(k, d) . C(j0 , d), (3.10) and (3.13), we obtain γj0 +1 (t) . (γj0 +1 (t) + ϵ)3 + δ(γj0 +1 (t) + ϵ)2 + δ 2 (γj0 +1 (t) + ϵ).

(3.13)

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Therefore by bootstrap argument, we get, for δ and ϵ1 sufficiently small, γj0 +1 (t) . ϵ for all t ∈ Ij0 +1 . This completes the induction step and obtains the desired estimates (3.11). Thus we have   γ−3   u)] |∇| 2 [F (u) − F (˜

N 1 (I) 2

.

J1 

γj (tj+1 ) . C(E, L)ϵ

(3.14)

j=1

where J1 = J1 (E, L). Next we will show the desired bounds (3.7)–(3.8). Firstly, for (3.6), by the Strichartz inequality, the γ−3 γ−3 definition of the S 21 norm, and (3.14), we have ∥|∇| 2 (˜ u −u)∥S 1 . ϵ+∥ |∇| 2 F (˜ u)−F (u)∥N 1 . C(E, L)ϵ, 2

2

therefore we get (3.6). Secondly, for (3.7), by the Sobolev embedding, we have     γ−3 γ−3     u − u) 2(d+1) 2d(d+1) . |∇| 2 (˜ u − u) ∥˜ u − u∥ 2(d+1) . |∇| 2 (˜

.

S1

2 −d−3+γ

d+2−γ Lt,x

Ltd+2−γ Lxd

2

We can directly obtain (3.7) from (3.6). Thirdly, for (3.8), by Minkowski’s inequality and (3.9), we have         γ−3  γ−3 γ−3  γ−3       ˜) + |∇| 2 u ˜ . |∇| 2 (˜ u − u) + C(E, L). |∇| 2 u ≤ |∇| 2 (u − u S1

S1

S1

S1

2

2

2

2



We also can get (3.8) by the expression (3.6). 4. Additional decay

In this section, we will prove that the finite time blow-up solution, soliton-like and frequency cascade solutions identified in Theorem 1.3 satisfy an additional decay property. More precisely, for d ≥ 6 we show ˙ 1−ϵ × H˙ x−ϵ ) for some ϵ = ϵ(d) > 0, where I = (tmin , tmax ) is the maximal-lifespan time that (u, ut ) ∈ L∞ t (I; Hx 2 ˙1 interval, tmin < 0 < tmax . In particular, we obtain that such solutions belong to L∞ t (Hx × Lx ). Throughout this section, we assume that inf N (t) ≥ 1.

t∈I

(4.1)

The main result of this section is: 4d < γ < d and that u : I × Rd → R is an almost periodic solution to Theorem 4.1. Assume d ≥ 6, d−1 ˙ sc −1 ), where I = (tmin , tmax ) is the maximal-lifespan time interval, ˙ sc (NLWH) with (u, ut ) ∈ L∞ t (I; Hx × Hx tmin < 0 < tmax . Then under the assumption (4.1),

˙ 1−ϵ × H˙ x−ϵ ) (u, ut ) ∈ L∞ t (I; Hx

(4.2)

2 ˙1 for some ϵ = ϵ(d) > 0. In particular, (u, ut ) ∈ L∞ t (I; Hx × Lx ).

Remark 4.1. In Theorem 4.1, the maximal-lifespan time interval I may be the finite time interval, or be the semi-infinite time axis, or be the entire time axis R. Compared to the previous results in the literature, for example, [2, Theorem 7.1], here we do not need the global solution hypothesis, we can establish the low regularity results also for the finite time blow-up solutions. ˙ sc In order to obtain Theorem 4.1, we need to employ the following two lemmas. First of all, u ∈ L∞ t (Hx ). 2d

d+2−γ From Sobolev embedding, one has u ∈ L∞ ). Our first goal here is to show the additional integrability: t (Lx

Lemma 4.2. Suppose d ≥ 6 and that u : I × Rd → R is an almost periodic solution to (NLWH) with ˙ sc ˙ sc −1 ), where I = (tmin , tmax ) is the maximal-lifespan time interval, tmin < 0 < tmax . (u, ut ) ∈ L∞ t (I; Hx × Hx 2(d−1) 2d q0 Then under the assumption (4.1), we have u ∈ L∞ t Lx for every q0 ∈ ( d−3 , d+2−γ ].

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235

The second goal is to perform a double Duhamel technique [4,24] to improve this decay to (u, ut ) ∈ × H˙ xsc −1−s0 ) for some s0 = s0 (d, q0 ) > 0. In fact, including this situation, we will prove the following general Lemma 4.3. If this is done, then Theorem 4.1 can be obtained by iterating the second step finitely many times. ˙ sc −s0 L∞ t (Hx

Lemma 4.3. Suppose d ≥ 6 and that u : I × Rd → R is an almost periodic solution to (NLWH) with ˙ sc ˙ sc −1 ), where I = (tmin , tmax ) is the maximal-lifespan time interval, tmin < 0 < tmax , (u, ut ) ∈ L∞ t (I; Hx × Hx 4d and satisfies the assumption (4.1). Moreover, assume that there exist 3d−2γ < q1 < d22d(d−1) +d−γd+γ and s ∈ [1, sc ] q1 s ∞ 2 such that u ∈ L∞ L and |∇| u ∈ L L . Then t x t x ˙ s−s0 × H˙ s−1−s0 ), (u, ut ) ∈ L∞ t (I; Hx x

(4.3)

for some s0 = s0 (d, q1 , γ) > 0. We will prove Lemmas 4.2 and 4.3 in detail in the rest subsections; however, with these two lemmas in hand, we can immediately complete the proof of the main theorem of this section. Proof of Theorem 4.1. We begin by choosing a suitable exponent to be able to apply Lemmas 4.2 and 4.3. To this end, we only need to find an exponent q = q(d) simultaneously satisfies that when d ≥ 6, 2d(d−1) 2d 4d 4d q(d) ∈ ( 2(d−1) d−3 , d+2−γ ) and 3d−2γ < q(d) < d2 +d−γd+γ , when d−1 < γ < d, this can be achieved. Fix s0 = s0 (d, q(d)) as in Lemma 4.3. By induction, we now prove that for each k ∈ N with ˙ sc −ks0 × H˙ xsc −1−ks0 ). We first note that for k = 0 the sc − (k − 1)s0 ≥ 1, we have (u, ut ) ∈ L∞ t (I; Hx ˙ sc ˙ sc −1 ). For the induction step, we assume that result follows from the hypothesis (u, ut ) ∈ L∞ t (I; Hx × Hx ˙ sc −(k−1)s0 , so that if k the result holds for some k − 1 ∈ N with sc − (k − 2)s0 ≥ 1. We then have u ∈ L∞ t Hx ∞ s −ks also satisfies sc − (k − 1)s0 ≥ 1, then by Lemma 4.3, we have (u, ut ) ∈ Lt (I; H˙ xc 0 × H˙ xsc −1−ks0 ). This completes the induction step. Taking k ∈ N as the largest integer such that sc − (k − 1)s0 ≥ 1, therefore we obtain the desired result (4.2) with ϵ = 1 − (sc − ks0 ).  In the following subsections, we will turn our attention to the proofs of Lemmas 4.2 and 4.3. We start with, 4.1. Proof of Lemma 4.2 Let η > 0 be a small constant to be chosen later. Assume u is a solution to (NLWH) as stated in Lemma 4.2. Then almost periodicity together with the condition (4.1) implies that we may find a dyadic number N0 such that 2 ≤ η. ∥ |∇|sc u≤N0 ∥L∞ t Lx

(4.4) d

d−γ

6d 2d R −1− 2 ∥u R for each dyadic Let us now fix R ∈ ( 2(d−1) N ∥L∞ d−3 , min{ 3d−2−γ , d−4 }) and define S(N ) = N t Lx n number N ∈ {2 : n ∈ Z}. α R . N To prove Lemma 4.2, it is enough to show ∥uN ∥L∞ for some α > 0 and N sufficiently small t Lx depending on u, d and R (see the argument at the end of this section). This bound will follow from the following decay estimate.

Lemma 4.4 (Decay Estimate). For all dyadic numbers N ≤ 8N0 , if tmax = +∞, or assume in addition, N ≤ min{8N0 , t−1 max }, if tmax < +∞, we have S(N ) .



N N0

d− Rd −3− d−γ 2

+η

N0   N1 = 2N 8

N N1

d− Rd −3− d−γ 2

     d − d +2 N1 R 2 S(N1 ) + η S(N1 ) . (4.5) N N1 ≤ N 8

W. Han / Nonlinear Analysis 152 (2017) 220–249

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In particular, S(N ) . N

γ−4 2

(4.6)

for every N ≤ 8N0 , if tmax = +∞; assume in addition, N ≤ min{8N0 , t−1 max }, if tmax < +∞. ˙ sc Proof. By Bernstein’s inequality, the Sobolev embedding and u ∈ L∞ t Hx , we have S(N ) . N

γ−2 2

sc 2 < ∞. 2 . ∥ |∇| uN ∥L∞ ∥uN ∥L∞ t Lx t Lx

In order to prove (4.5), we first note that using the time translation symmetry, it suffices to prove the result when t = 0. In two cases, that is, for every N ≤ 8N0 , if tmax = +∞; and assume N ≤ min{8N0 , t−1 max }, if tmax < +∞, we will prove that d

N R −1−

d−γ 2

d

. N d− R −3− ∥uN (0)∥LR x

d−γ 2

′. ∥PN F (u)∥L∞ LR x t

In fact, we discuss two cases respectively in the following: Case 1. If tmax = +∞, let N ≤ 8N0 . Then, by the Duhamel formula (1.4) and Minkowski’s inequality, we obtain N

d−γ d R −1− 2



∥uN (0)∥LR .N x

∞

+ N −1

N −1



d−γ d R −1− 2

0

   sin(−t′ |∇|) ′  P F (u(t ))   N |∇|

LR x

dt′

 ′ dt . R

   sin(−t′ |∇|) ′  P F (u(t ))   N |∇|

(4.7)

Lx

Using Bernstein’s inequality on the first term and the dispersive inequality (2.2) on the second term, we get  −1   N ′ d d d  |∇|) ′  −1− d−γ R 2 (4.7) . N P F (u(t )) N 2 − R  sin(−t  dt′ N |∇| L2x

0



∞

1 ′ −(d−1)( 21 − R )

|t |

+ d

d−γ 2



N −1

∞

+ N −1

.N .N

γ 2

−3

d

dt

d

  d+1 d−1 1 1   |t′ |−(d−1)( 2 − R ) |∇| 2 − R PN F (u(t′ ))

′ LR x

2 + N ∥PN F (u)∥L∞ t Lx

d d− R −3− d−γ 2



N 2 − R ∥ |∇|−1 PN F (u(t′ ))∥L2x dt′

0



′

′ LR x

N −1

. N R −1−

  d+1 d−1   |∇| 2 − R PN F (u(t′ ))

d d− R −3− d−γ 2

dt′



′ ∥PN F (u)∥L∞ LR x t

′ ∥PN F (u)∥L∞ LR x

(4.8)

t

where in passing from the first line to the third we use (2.2) once more and in passing from the fourth line to the fifth line, we used the fact that (d − 1)( 21 − R1 ) > 1 to observe the finiteness of the integral. d

Collecting (4.7) and (4.8), we obtain N R −1−

d−γ 2

min{8N0 , t−1 max },

Case 2. If tmax < +∞, let N ≤ and Minkowski’s inequality, we obtain d

N R −1−

d−γ 2

d

∥uN (0)∥LR . N R −1− x

d

∥uN (0)∥LR . N d− R −3− x then tmax ≤ N

d−γ 2

 0

tmax

−1

d−γ 2

′. ∥PN F (u)∥L∞ LR x t

. Then, by the Duhamel formula (1.4)

   sin(−t′ |∇|) ′  P F (u(t ))   N |∇|

LR x

dt′ .

(4.9)

W. Han / Nonlinear Analysis 152 (2017) 220–249

237

We then use Bernstein’s inequality and the dispersive inequality (2.2) to obtain  tmax   ′ d d d   |∇|) −1− d−γ R 2 (4.9) . N N 2 − R  sin(−t PN F (u(t′ )) 2 dt′ |∇| Lx 0 tmax   d−γ d d d . N R −1− 2 N 2 − R |∇|−1 PN F (u(t′ ))L2 dt′ x 0  tmax γ 2 dt′ . N 2 −2 ∥PN F (u)∥L∞ t Lx 0  N −1 γ γ −2 2 2 . N 2 ∥PN F (u)∥L∞ dt′ = N 2 −3 ∥PN F (u)∥L∞ t Lx t Lx 0

d

. N d− R −3−

d−γ 2

′. ∥PN F (u)∥L∞ LR x

(4.10)

t

d

Collecting (4.9) and (4.10), we obtain N R −1−

d−γ 2

d

∥uN (0)∥LR . N d− R −3− x

d−γ 2

′. ∥PN F (u)∥L∞ LR x t

′ . We start by decomposing u as In order to establish (4.5), it remains to estimate the term ∥PN F (u)∥L∞ LR x t

u = u≤ N + u N <·≤N0 + u>N0 =: u1 + u2 + u3 . 8

8

By this decomposition, we have ′ ∥PN F (u)∥L∞ LR x t

   3 2   3        −γ 2 −γ       ′ = = ∥PN [(|x| ∗ |u| )u]∥L∞ LR |x| ∗ ui ui PN  x t   ∞ R′ i=1 i=1 Lt Lx    3   3        |x|−γ ∗ ui uj uk  = PN   i,j=1 k=1 R′ L∞ t Lx       3  −γ    = P |x| ∗ (u u ) u N i j k   i,j,k=1  ∞ R′ Lt Lx       −γ 2 . PN (|x| ∗ u1 )u1  ∞ R′ + PN (|x|−γ ∗ u22 )u2  ∞ R′ Lt Lx

+

3  i,j=1

Lt Lx

       −γ PN |x| ∗ u u uj  3 i  

+

R′ L∞ t Lx

2   i=1

 PN 



    |x|−γ ∗ u2 u1 ui  

.

R′ L∞ t Lx

Using this inequality and the boundedness of PN , we obtain d

N R −1−

d−γ 2

d

∥uN (0)∥LR . N d− R −3− x

d−γ 2

′ ∥PN F (u)∥L∞ LR x t       d−γ d PN (|x|−γ ∗ u21 )u1  ∞ R′ +  (|x|−γ ∗ u22 )u2  ∞ R′ ≤ N d− R −3− 2 L L L L t

+

x

t

3 2    −γ  −γ      |x| ∗ (u3 ui ) uj  ∞ R′ +  |x| ∗ (u1 u2 ) ui  ∞ R′ L L L L t

i,j=1 d

= N d− R −3−

d−γ 2

x

i=1

  (I) + (II) + (III)i,j + (IV )i .

t

x



x

(4.11)

We now estimate each of the above terms (I), (II), (III)i,j and (IV )i separately. Term (I): Noting the support of the Fourier transform of (|x|−γ ∗ u1 (t)2 )u1 (t), we have PN [(|x|−γ ∗ u1 (t)2 )u1 (t)] ≡ 0, thus (I) = 0.

(4.12)

W. Han / Nonlinear Analysis 152 (2017) 220–249

238

Term (II): Using H¨ older’s inequality, the Sobolev embedding, the boundedness of P> N and Bernstein’s 8 inequality, we obtain ′ ≤ ∥u2 ∥ ∥(|x|−γ ∗ u22 )u2 ∥L∞ LR x

2d

d+2−γ L∞ t Lx

t

. ∥ |x|−γ ∥

d

L∗γ

∥ |x|−γ ∗ u22 ∥

∥u22 ∥

2Rd Rd−2d−2R+Rγ L∞ t Lx

2Rd

3Rd−2d−2R−Rγ L∞ t Lx

. ∥u2 ∥2

4Rd

3Rd−2d−2R−Rγ L∞ t Lx

∥u2 ∥

∥u2 ∥

2d d+2−γ L∞ t Lx

2d

d+2−γ L∞ t Lx



 

2 . ∥ |∇|sc u≤N0 ∥L∞ t Lx 2N 8

∥uN1 ∥

 2N 8

×

− Rd−2d−6R+Rγ 4R N2

sc

2 . ∥ |∇| u≤N0 ∥L∞ t Lx

6d−3Rd+2R+Rγ 4R



2 . ∥ |∇|sc u≤N0 ∥L∞ t Lx

4Rd

3Rd−2d−2R−Rγ L∞ t Lx

≤N1 ≤N2 ≤N0

N1

∥uN2 ∥

4Rd

3Rd−2d−2R−Rγ L∞ t Lx

R ∥uN1 ∥L∞ t Lx

≤N1 ≤N2 ≤N0



  Rd−2d−6R+Rγ   4R uN2  |∇|

4Rd

3Rd−2d−2R−Rγ L∞ t Lx

  N0 

6d−3Rd+2R+Rγ 4R

N1

R ∥uN1 ∥L∞ t Lx

N1 = 2N 8

 ×

N0 

− Rd−2d−6R+Rγ 4R 2 N2 ∥ |∇|sc uN2 ∥L∞ t Lx



N2 =N1 sc

. ∥ |∇| u≤N0 ∥

2 L∞ t Lx

  N0 

6d−3Rd+2R+Rγ 4R

N1

R ∥uN1 ∥L∞ t Lx

N1 = 2N 8

 ×

− Rd−2d−6R+Rγ 4R N1 ∥(u, ut )∥L∞ (R;H˙ xsc ×H˙ xsc −1 ) t

where to obtain the fifth inequality we note that R <



6d 3d−2−γ .

Therefore, using (4.4) in the last inequality above, we obtain (II) . η

N0 

2d

N1R

−d+2

R ∥uN1 ∥L∞ t Lx

N1 = 2N 8

=η

N0 

2d

N1R

−d+2

d+2−γ 2

N1

d −R

d

N1R

− d+2−γ 2

R ∥uN1 ∥L∞ t Lx

N1 = 2N 8

=η

N0 

d

N1R

−d+3+ d−γ 2

S(N1 ).

(4.13)

N1 = 2N 8

Term (III)i,j : Fix i, j ∈ {1, 2, 3}. Using H¨ older’s inequality, the Bernstein and Sobolev inequalities, we get  −γ  −γ      |x| ∗ (u3 ui ) uj  ∞ R′ =  |x| ∗ (u>N0 ui ) uj  ∞ R′ Lt Lx Lt Lx  −γ   |x| ∗ (u>N0 ui ) ≤ ∥uj ∥ 2d 2Rd d+2−γ L∞ t Lx

≤ ∥uj ∥

2d d+2−γ L∞ t Lx

Rd−2d−2R+Rγ L∞ t Lx

 −γ   |x| 

d

L∗γ

∥u>N0 ui ∥

2Rd

3Rd−2d−2R−Rγ L∞ t Lx

W. Han / Nonlinear Analysis 152 (2017) 220–249

≤ ∥uj ∥

2d d+2−γ L∞ t Lx

≤ ∥u∥2

2d

d+2−γ L∞ t Lx

∥ui ∥

2d

d+2−γ L∞ t Lx

∥u>N0 ∥

∥u>N0 ∥

239

Rd

Rd−2R−d L∞ t Lx

Rd

Rd−2R−d L∞ t Lx

6R+2d−Rd−γR 2R

  Rd−6R−2d+γR   2R u>N0   |∇|

. N0

Rd

Rd−2R−d L∞ t Lx

6R+2d−Rd−γR 2R

2 2 ∥u∥ ∥ |∇|sc u>N0 ∥L∞ t Lx

. N0

∥u∥2

2d

d+2−γ L∞ t Lx

2d

d+2−γ L∞ t Lx

γ d 3+ R −d 2−2

sc 2 ∥ |∇| u∥2L∞ ∥ |∇|sc u>N0 ∥L∞ 2 t Lx t Lx

. N0

γ d 3+ R −d 2−2

. N0

(4.14)

2d where in passing from the sixth line to the seventh line, we use R < d−4 , and in the last inequality we sc sc 2 s s −1 observed that ∥|∇| u>N0 ∥L∞ L2x ≤ ∥ |∇| u∥L∞ ≤ ∥(u, u )∥ ∞ c c t ˙ ˙ L (I;Hx ×Hx ) < ∞. t Lx t

t

older’s inequality, the Sobolev and Bernstein inequalities, we have Term(IV )i : Fix i ∈ {1, 2}. By H¨     −γ (|x|−γ ∗ (u1 u2 ))ui  ∞ R′ ≤ ∥ui ∥  |x| ∗ (u1 u2 ) 2d 2Rd L L d+2−γ Rd−2R−2d+Rγ t

L∞ t Lx

x

L∞ t Lx

 −γ   |x| 

∥u1 u2 ∥ 2Rd 3Rd−2R−2d−Rγ L∞ t Lx         N N  ≤ ∥ui ∥ 2d 2Rd u  u <·≤N ≤ 0 ∞ L2 d+2−γ 2Rd−2d−2R−Rγ 8 8 ∞ L Lt Lx L∞ x t t Lx         2 . P> N P≤N0 u ∞ u≤ N  ∥ |∇|sc u∥L∞ 2Rd t Lx 2Rd−2R−2d−Rγ 8 8 Lt L2x L∞ L x t    −sc    ∥uN1 ∥ . N8 2Rd  |∇|sc u N <·≤N0  ∞ 2Rd−2R−2d−Rγ ≤ ∥ui ∥

2d

d+2−γ L∞ t Lx

d

L∗γ

Lt L2x

8

γ

. N 1− 2 η



∥uN1 ∥

γ



(4.15)

2Rd

2Rd−2R−2d−Rγ L∞ t Lx

N1 ≤ N 8

. N 1− 2 η

L∞ t Lx

N1 ≤ N 8

d

N1R

− 2Rd−2R−2d−Rγ 2R

R ∥uN1 ∥L∞ t Lx

(4.16)

N1 ≤ N 8 γ

= N 1− 2 η



2d

N1R

−d+1+ γ2

R ∥uN1 ∥L∞ t Lx

N1 ≤ N 8 γ

= N 1− 2 η



2d

N1R

−d+1+ γ2

d+2−γ 2

N1

d −R

N1 ≤ N 8 d

= N R −d+

d−γ 2

+3

γ

S(N1 ) = N 1− 2 η



d

N1R

−d 2 +2

S(N1 )

N1 ≤ N 8

η

   d − d +2 N1 R 2 N

S(N1 ),

(4.17)

N1 ≤ N 8

where to obtain (4.15) we note that N ≤ 8N0 and to obtain (4.16) we used R <

6d 3d−2−γ .

Collecting the estimates (4.11)–(4.14) and (4.17), we obtain the desired inequality (4.5). To obtain (4.6), we define xk = S(2−k N0 ), k ∈ N. Noting (4.5), and applying Gronwall’s inequality d Lemma 2.11, we have xk . 2−kρ for each ρ ∈ (0, d − R − 3 − d−γ 2 ) (see for instance [2]). Thus, for each N = −k −k 2 N0 ≤ 8N0 , if tmax = +∞, or assume in addition, N = 2 N0 ≤ min{8N0 , t−1 max }, if tmax < +∞, we obtain d−γ d S(N ) = S(2−k N0 ) . (2−k )ρ ∼ N ρ . Taking ρ = γ−4 ∈ (0, d− −3− ) gives the desired bound (4.6).  2 R 2 With this lemma in hand, we are now ready to prove Lemma 4.2:

W. Han / Nonlinear Analysis 152 (2017) 220–249

240

Proof of Lemma 4.2. Noticing the definition of S(N ), (4.6) shows that for all N ≤ 8N0 , if tmax = +∞; or assume in addition, N ≤ min{8N0 , t−1 max }, if tmax < +∞, d+2−γ 2

∥uN ∥L∞ LR . N

d −R

x

t

N

γ−4 2

d

d

= N 2 − R −1 .

(4.18)

Then, by using (4.18) and the Bernstein inequalities, we obtain R ≤ ∥u≤N ∥L∞ LR + ∥u>N ∥L∞ LR ∥u∥L∞ 0 0 x x t Lx t t d d  2 R + ∥uN ∥L∞ N 2 − R ∥uN ∥L∞ . t Lx t Lx

N ≤N0

.



N >N0

N

d d 2 − R −1

+

N ≤N0

=





d

d

N 2−R N−

γ−2 2

∥ |∇|sc u∥L∞ L2x t

N >N0 d

d

N 2 − R −1 +

N ≤N0



N

d−γ 2

d −R +1

∥ |∇|sc u∥L∞ L2 . 1, t

x

N >N0

d d where we note that our hypotheses on d and R ensure that d2 − R − 1 > 0 and 1 − R + d−γ 2 < 0. Thus we 2(d−1) 6d 2d ∞ R obtain that u ∈ Lt Lx . Since R is arbitrary, we obtain the lemma for every q0 ∈ ( d−3 , min{ 3d−2−γ , d−4 }). 2d

2d ∞ d+2−γ We note that the lemma holds for every q0 ∈ ( 2(d−1) d−3 , d+2−γ ] by using interpolation with the Lt Lx ∞ ˙ sc bound which results from combining the a priori bound u ∈ Lt Hx with the Sobolev embedding. 

4.2. Proof of Lemma 4.3 1 ). By using the Bernstein Let u, q1 and s be given as stated in the lemma and choose s0 ∈ (0, 2q1 +q1 γ+2d−dq q1 inequalities, we have  s−1−s0 s−1−s0 2 + ∥ |∇| 2 ≤ 2 + ∥ |∇| 2 ∥ |∇|s−s0 u∥L∞ ut ∥L∞ ∥ |∇|s−s0 uN ∥L∞ ∂t uN ∥L∞ t Lx t Lx t Lx t Lx

N ≤1

+



s−1−s0 2 + ∥ |∇| 2 ∥ |∇|s−s0 uN ∥L∞ ∂t uN ∥L∞ t Lx t Lx

N >1

.



  s−1 ∞ 2 2 N −s0 ∥ |∇|s uN ∥L∞ + ∥ |∇| ∂ u ∥ t N Lt Lx t Lx

N ≤1

+



  sc −1 ∞ 2 2 N s−s0 −sc ∥ |∇|sc uN ∥L∞ + ∥ |∇| ∂ u ∥ t N Lt Lx t Lx

N >1

.



   s−1 ∞ 2 2 N −s0 ∥ |∇|s uN ∥L∞ + ∥ |∇| ∂ u ∥ N s−s0 −sc t N Lt Lx + t Lx

N ≤1

.



N >1

  −s0 s s−1 ∞ ∞ N ∥ |∇| uN ∥Lt L2x + ∥ |∇| ∂t uN ∥Lt L2x + 1

(4.19)

N ≤1

where we note ∥(u, ut )∥L∞ (H˙ xsc ×H˙ xsc −1 ) ≤ C to obtain the third inequality followed by t for s − s0 − sc < 0 to obtain the fourth inequality.



N >1

N s−s0 −sc < ∞

s−1 2 + ∥ |∇| 2 in In order to obtain (4.3), it thus remains to estimate the term ∥ |∇|s uN ∥L∞ ∂t uN ∥L∞ t Lx t Lx (4.19). We begin by noting that the unitary property of the linear propagator W(·) implies that for every t1 , t2 ∈ R and g, h ∈ L2 ,   2 |∇|) 1 |∇|) |∇| sin(t|∇| g, −|∇| sin(t|∇| h + ⟨cos(t1 |∇|)g, − cos(t2 |∇|)h⟩ = ⟨g, − cos((t1 − t2 )|∇|)h⟩ .

W. Han / Nonlinear Analysis 152 (2017) 220–249

241

Next, without loss of generality we take t = 0, and by using the above observation and Lemma 1.4 we write   T sin(−t′ |∇|) s 2 s−1 2 PN |∇|s−1 F (u(t′ ))dt′ , ∥ |∇| uN (0)∥L2x + ∥ |∇| ∂t uN (0)∥L2x = lim lim |∇| |∇| ′ T →tmax T →tmin

 − |∇| 

0

0

sin(−τ ′ |∇|) PN |∇|s−1 F (u(τ ′ ))dτ ′ |∇|

T′



T

cos(−t′ |∇|)PN |∇|s−1 F (u(t′ ))dt′ ,

+ 0 0



′

−

′

s−1

′



cos(−τ |∇|)PN |∇| F (u(τ ))dτ tmax  0  ⟨PN |∇|s−1 F (u(t′ )), ≤  tmin 0   ′ ′ s−1 ′ − cos((t − τ )|∇|)PN |∇| F (u(τ ))⟩dτ ′ dt′ . T′



Setting r =

(4.20)

2dq1 2q1 γ+4d−dq1

and using H¨ older’s inequality, Lemma 2.1 and Bernstein’s inequalities, we obtain     cos((t′ −τ ′ )|∇|) s ′ P |∇| F (u(τ ))   PN |∇|s F (u(t′ )), 2 N |∇|   ′ ′   −τ )|∇|) . ∥PN |∇|s F (u(t′ ))∥Lrx  cos((t|∇| PN |∇|s F (u(τ ′ )) r′ 2 Lx   d+1 d−3  − s ′  s ′ 1 r′ P r |∇| 2 |∇| F (u(τ )) . ∥P |∇| F (u(t ))∥  r N N L 1 1 x (d−1)( − ′ ) Lx 2 r |t′ −τ ′ | .

N

d+1 d−3 − 2 r′ (d−1)

|t′ −τ ′ |

( 12 − r1′ )

∥PN |∇|s F (u(t′ ))∥2L∞ r. t Lx

(4.21)

On the other hand, using the Cauchy–Schwarz inequality, Lemma 2.1(with p = 2) and Bernstein’s inequality, we get     ′     −τ ′ )|∇|) cos((t′ −τ ′ )|∇|) PN |∇|s F (u(τ ′ ))  . ∥PN |∇|s F (u(t′ ))∥L2x  cos((t|∇| PN |∇|s F (u(τ ′ ))  PN |∇|s F (u(t′ )), 2 |∇|2 2 Lx

−2

s

. ∥PN |∇| F (u)∥L2x ∥ |∇| .N

−2

s

∥PN |∇| 2d

. N −2+ r 2d d−γ

where we recall that r < 2. (We have used that q1 < 2d d−γ

=

−d

s

PN |∇| F (u)∥L2x

F (u)∥2L2x

∥ |∇|s F (u)∥2L∞ r, t Lx

2d(d−1) d2 −γd−d+γ ,

since q1 <

(4.22)

2d(d−1) d2 −γd+d+γ

<

2dq1 2q1 γ+4d−dq1

2d(d−1) d2 −γd−d+γ

=

=⇒ r = < 2.) Invoking the bounds (4.21) and (4.22) in (4.20) using Lemma 2.3 and Hardy–Littlewood–Sobolev inequality, we obtain ∥ |∇|s uN (0)∥2L2x + ∥ |∇|s−1 ∂t uN (0)∥2L2x   tmax  0 s 2 ≤ ∥ |∇| F (u)∥L∞ min r t Lx 0

≤ ∥ |∇|s F (u)∥2L∞ r t Lx ≤ ∥ |∇|

s



∞



tmin 0

N

(d−1)

|t′ −τ ′ |



N

min 0

4 u∥2L∞ 2 ∥u∥ ∞ q1 Lt Lx t Lx

−∞  ∞

0



−∞

2d

4 = N −2+d− r′ ∥ |∇|s u∥2L∞ 2 ∥u∥ ∞ q1 L Lx t Lx t

( 21 − r1′ )

(d−1)

min 0

( 12 − r1′ )

d+1 d−3 − 2 r′

|t′ −τ ′ |



d+1 d−3 − 2 r′

 0

2d

, N −2+d− r′

d+1 d−3 − 2 r′





dt′ dτ ′

dt′ dτ ′

−2+d− 2d r′



,N dt′ dτ ′ ( 12 − r1′ )  −(d−1)  12 − 1′ r min |tN′ −τ ′ |d−1 , 1 dt′ dτ ′ .

N

(d−1)

|t′ −τ ′ | ∞ 0 −∞

,N

−2+d− 2d r′

W. Han / Nonlinear Analysis 152 (2017) 220–249

242

We conclude the proof by estimating the above integral. To this end, we use the bound  ∞ 0  −(d−1)  12 − 1′ r min |tN′ −τ ′ |d−1 , 1 dt′ dτ ′ . N −2 , 0

(4.23)

−∞

which follows from the assumption q1 < d22d(d−1) +d−γd+γ and a straightforward computation. q1 s ∞ 2 Invoking this bound in (4.19) and using the hypotheses u ∈ L∞ t Lx and |∇| u ∈ Lt Lx , we get    d −d 1− 2q1 γ+4d−dq1 −2−s d 0 s−s0 −1 −s0 −2+ d 2dq1 2 − r′ + 1 = ∞ 2 2 + ∥ |∇| N N ∥ |∇|s−s0 u∥L∞ u ∥ . +1 N2 t Lt Lx t Lx N ≤1

N ≤1

=



N

2q γ+4d−2dq1 −s0 +2+ 1 2q 1

+ 1 < +∞.

N ≤1 1 > 0, so that the desired bound (4.3) holds. Noticing our choice of s0 , we have −s0 + 2 + 2q1 γ+4d−2dq 2q1



5. Finite time blow-up solution In this section, we preclude the existence of finite time blow-up solution in the sense of Theorem 1.3. The method we deal with the finite time blow-up solution is different with the previous methods. For the previous papers in the literature, the authors preclude the finite time blow-up solution by the finite speed of propagation, while the Hartree nonlinear term in our paper is the nonlocal nonlinear term, it does not have the property of the finite speed of propagation, we will employ the low regularity results we established in Section 4 for almost periodic solutions including the finite time blow-up solution to preclude the finite time blow-up solution. We point out that for the finite time blow-up solution to (NLWH), Theorem 4.1, Lemmas 4.2 and 4.3 still hold. Additionally, the method here we rule out the finite time blow-up solution in the sense of Theorem 1.3 is similar to the way we preclude the low-to-high cascade identified in Theorem 1.3. We will prove that the solution must have zero energy, contradicting the fact that the solution blows up. More precisely, we have Theorem 5.1 (No Finite Time Blowup). There is no solution u : I × Rd → R to (NLWH) with maximal interval of existence I satisfying the properties of a finite time blow-up solution in the sense of Theorem 1.3. Proof. Suppose for contradiction there exists such a solution u. By the time-reversal we may assume that sup I < ∞. As a consequence, we may choose a sequence {tn } ⊂ I with tn → sup I, such that N (tn ) → ∞ as n → ∞ (see for example [2,11,25]). ˙ 1−ϵ × H˙ −ϵ ) for some ϵ > 0 (Theorem 4.1) Using (1.2) followed by H¨ older’s inequality with u ∈ L∞ t (I; Hx x we have, for all n ∈ N and η > 0,  |ξ|2 |ˆ u(tn , ξ)|2 + |ˆ ut (tn , ξ)|2 dξ |ξ|≤c(η)N (tn )



c

|ξ|2sc |ˆ u(tn , ξ)|2 dξ

. |ξ|≤c(η)N (tn )

2(sc −1)

|ξ|

+

ϵ

∥(u, ut )∥

. η ϵ+sc −1 .

2

 ϵ+sϵ −1  c

sc −1  ϵ+s −1 c

−2ϵ

|ξ|

|ˆ ut (tn , ξ)| dξ

|ξ|≤c(η)N (tn ) ϵ ϵ+sc −1

c

|ξ|2(1−ϵ) |ˆ u(tn , ξ)|2 dξ

|ξ|≤c(η)N (tn )



.η

sc −1  ϵ+s −1

 ϵ+sϵ −1 

2

|ˆ ut (tn , ξ)| dξ

|ξ|≤c(η)N (tn )

2(sc −1) ϵ+sc −1 ˙ x1−ϵ ×H ˙ x−ϵ ) L∞ (I;H

(5.1)

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On the other hand, by Chebyshev’s inequality 

|ξ|2 |ˆ u(tn , ξ)|2 + |ˆ ut (tn , ξ)|2 dξ ≤ [c(η)N (tn )]−2(sc −1)



|ξ|2sc |ˆ u(tn , ξ)|2 + |ξ|2(sc −1) |ˆ ut (tn , ξ)|2 dξ

Rd

|ξ|≥c(η)N (tn )

. [c(η)N (tn )]−2(sc −1) ∥(u, ut )∥2L∞ (I;H˙ sc ×H˙ sc −1 ) x

−2(sc −1)

. [c(η)N (tn )]

x

,

(5.2)

for all η > 0 and n ∈ N. To continue, we now estimate the nonlinear term in the energy. Note that using Sobolev’s inequality ˙ sc followed by interpolation with u ∈ L∞ t Hx ,  |u(tn , x)|2 |u(tn , y)|2 dxdy ≤ ∥u(tn , ·)∥4 4d ≤ ∥∇u(tn )∥2L2x ∥ |∇|sc u∥2L∞ 2 t Lx |x − y|γ Lx2d−γ Rd ×Rd = ∥∇u(tn )∥2L2x ∥u∥2L∞ H˙ sc . ∥∇u(tn )∥2L2x . t

x

(5.3)

Combining (5.1), (5.2) and invoking Plancherel’s theorem in (5.3), we estimate the energy as    2 2 2 E(u(tn ), ut (tn )) . |ξ| |ˆ u(tn )| dξ + |ˆ ut (tn )| dξ + |ξ|2 |ˆ u(tn )|2 dξ Rd Rd Rd   . |ξ|2 |ˆ u(tn )|2 dξ + |ξ|2 |ˆ u(tn )|2 dξ |ξ|≤c(η)N (tn ) |ξ|≥c(η)N (tn )   2 + |ˆ ut (tn )| dξ + |ˆ ut (tn )|2 dξ |ξ|≤c(η)N (tn )

|ξ|≥c(η)N (tn )

ϵ

. η ϵ+sc −1 + [c(η)N (tn )]−2(sc −1) ,

(5.4)

for all η > 0 and n ∈ N. Letting n → ∞ in (5.4) and using the conservation of energy, now N (tn ) → ∞ yields for all η > 0, ϵ

E(u(0), ut (0)) . η ϵ+sc −1 . Taking η → 0, we obtain E(u(0), ut (0)) = 0. Thus u ≡ 0 contradicting our assumption that u is a finite time blow-up solution. Thus such a solution cannot exist.  6. The soliton In this section, we will use the Morawetz estimate to preclude the existence of soliton-like solution described in Theorem 1.3. Our approach to obtain the desired contradiction is to get an upper and lower bound on the quantity   ∗ |u(t, x)|2 dxdt, (6.1) |x| I Rd 2d where 2∗ = d−2 , with a time interval I ⊂ R. Indeed, the Morawetz estimate (Theorem 2.9) and the additional decay property given in Theorem 4.1 immediately imply that (6.1) is bounded from above independent of I. The contradiction will then follow once we obtain a lower bound on (6.1) which grows to infinity as |I| → ∞. The original method was given in [12,13]. We obtain the lower bound in two steps: The first step is to get an estimate on the growth of x(t) via the standard argument. For this purpose, one can adopt the proof of Lemma 5.18 in [10] to prove a similar result for wave equation that the almost periodic solutions satisfy the following local constancy property. We now state the following estimate for x(t) which will play an important role in precluding the finite time blow up solutions.

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Lemma 6.1 (Local Constancy of N (t), x(t)). Let u be a non-zero maximal-lifespan solution to (NLWH) with lifespan I that is almost periodic modulo symmetries with parameters N (t), x(t). Then there exists a small number δ, depending on u, such that for every t0 ∈ I we have [t0 − δN (t0 )−1 , t0 + δN (t0 )−1 ] ⊂ I, and N (t) ∼u N (t0 ), |x(t) − x(t0 )| .u N (t0 )−1 whenever |t − t0 | ≤ δN (t0 )−1 . Especially, noting that N (t) ≡ 1, and the maximal-lifespan of the solution I = R in the soliton-like solution case stated in Theorem 1.3, then for every t ≥ 0 we have, |x(t) − x(0)| . t. This result follows from standard arguments; one can see for instance [10, Lemma 5.18]. ∗ The second step in obtaining the lower bound on (6.1) is then to show that the L2t,x norm of u over unit time intervals and localized in space near x(t) is bounded away from zero. This result is included in the following lemma which employs the almost periodicity as well as the dispersive estimate. Lemma 6.2. Suppose that u : R × Rd → R is a solution to (NLWH) which satisfies the properties of a soliton-like solution stated in Theorem 1.3. Then there exist R > 0 and c > 0 such that for every s ∈ R,  s+1  ∗ |u(t, x)|2 dxdt ≥ c, (6.2) s ∗

where 2 =

|x−x(t)|≤R

2d d−2 .

Proof. Firstly, we claim that there exists C1 > 0 such that for every s ∈ R,      2d   |u(t)| d−2 dx ≥ C1  ≥ C1 .  t ∈ [s, s + 1] :   d R

(6.3)

For this purpose, suppose to the contrary that the claim failed. Then there exists a sequence of times {sn } ⊂ R such that for every n ∈ N,     2d 1  1  d−2 |u(sn + τ )| dx ≥  τ ∈ [0, 1] : < .   n n Rd This in turn implies that the sequence gn : [0, 1] → R defined by  2d gn (τ ) = |u(sn + τ )| d−2 dx Rd

converges to zero in measure as n → ∞. We next extract a subsequence (still labeled sn ) such that  2d |u(sn + τ )| d−2 dx → 0 for a.e. τ ∈ [0, 1] as n → ∞.

(6.4)

Rd

To continue, using the hypothesis that u is a soliton-like solution and the almost periodicity of u, we choose a further subsequence (still labeled sn ) and a pair (f, g) ∈ H˙ xsc × H˙ xsc −1 such that (u(sn , x(sn ) + ·), ut (sn , x(sn ) + ·) → (f, g)

in H˙ xsc × H˙ xsc −1 .

(6.5)

Moreover, using the additional decay property (Theorem 4.1) we observe that the sequence {(u(sn , x(sn )+·), ut (sn , x(sn ) + ·))} is bounded in H˙ x1 × L2x , and we therefore pass to another subsequence to find (f ′ , g ′ ) ∈

W. Han / Nonlinear Analysis 152 (2017) 220–249

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H˙ x1 × L2x such that weakly in H˙ x1 × L2x .

(u(sn , x(sn ) + ·), ut (sn , x(sn ) + ·)) ⇀ (f ′ , g ′ )

(6.6)

Next, we will claim that W(τ )(f ′ , g ′ )(x) = 0,

for a.e. τ ∈ [0, 1] and a.e. x ∈ Rd .

(6.7)

Indeed, we first note that (6.6) yields W(τ )(u(sn ), x(sn ) + ·), ut (sn , x(sn ) + ·)) ⇀ W(τ )(f ′ , g ′ ) weakly in 2d

Lxd−2 for every τ ∈ R. By weak lower semicontinuity of the norm, we have, for every τ ∈ R, ∥W(τ )(f ′ , g ′ )∥

2d

Lxd−2

≤ lim ∥W(τ )(u(sn ), ut (sn ))∥ n→∞

2d

Lxd−2

.

(6.8)

Fix τ ∈ [0, 1]. Using the Duhamel formula, the dispersive estimate, Lemma 2.3, and the Sobolev embedding, we obtain for all n ∈ N, ∥W(τ )(u(sn ), ut (sn ))∥ ≤ ∥u(sn + τ )∥

2d

. ∥u(sn + τ )∥

2d

Lxd−2

Lxd−2

2d

Lxd−2  sn +τ

   sin((sn + τ − τ ′ )|∇|)   −γ 2 ′   + (|x| ∗ |u| )u(τ )  2d dτ ′  |∇| sn Lxd−2  sn +τ   d−1  1   ′ + |sn + τ − τ ′ |− d |∇| d (|x|−γ ∗ |u|2 )u(τ ′ )  d+2 2d dτ sn sn +τ

 . ∥u(sn + τ )∥

2d Lxd−2

+

2d Lxd−2

2d Lxd−2

d−1 d

∥u(τ ′ )2 ∥ Lx2d

sn +τ

+

|sn + τ − τ ′ |−

d−1 d

∥u(τ ′ )∥2

sn

 . ∥u(sn + τ )∥

|sn + τ − τ ′ |−

sn

 . ∥u(sn + τ )∥

Lx

Lx2d

sn +τ

+

|sn + τ − τ ′ |−

d−1 d

2d2 2 −γd−2

4d2 2 −γd−2

∥u(τ ′ )∥2

4d2 2 Lx2d −γd−2

sn

  1   |∇| d u(τ ′ )

2d2 2 +2d+2−γd

dτ ′

Lxd

∥u(τ ′ )∥H˙ xsc dτ ′ dτ ′ .

(6.9)

We estimate the above integral as follows: Using interpolation, we deduce  sn +τ d−1 |sn + τ − τ ′ |− d ∥u(τ ′ )∥2 4d2 dτ ′ 2 −γd−2

sn

Lx2d



τ

=

|τ − τ ′ |−

d−1 d

∥u(sn + τ ′ )∥2

|τ − τ ′ |−

d−1 d

∥u(sn + τ ′ )∥2θ 2d ∥u(sn + τ ′ )∥

0



τ

. τ

|τ − τ ′ |−

.

d−1 d

.

2(1−θ) 2d Lxd+2−γ

τ

|τ − τ ′ |−

d−1 d

dτ ′

2(1−θ)

∥u(sn + τ ′ )∥2θ 2d ∥u∥L∞ H˙ sc dτ ′ Lxd−2

0



dτ ′

Lxd−2

0



4d2 2 Lx2d −γd−2

t

x

∥u(sn + τ ′ )∥2θ 2d dτ ′

(6.10)

Lxd−2

0

for some θ ∈ (0, 1). Then, by virtue of Theorem 4.1 and (6.4), the dominated convergence theorem yields  τ d−1 |τ − τ ′ |− d ∥u(sn + τ ′ )∥2θ 2d dτ ′ → 0. (6.11) Lxd−2

0

Thus appealing to (6.4) once again, together with (6.11), we use (6.9) to obtain ∥W(s)(u(sn ), ut (sn ))∥

2d

Lxd−2

→0

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which in turn gives the claim (6.7). Based on this claim, we can assert that (f ′ (x), g ′ (x)) = (0, 0) for a.e. x ∈ Rd . In fact, by using (6.7) and W(τ )(f ′ , g ′ ) ∈ Cτ0 (H˙ x1 ) ∩ Cτ1 (L2x ), we get ∥f ′ ∥

2d

Lxd−2

. ∥f ′ ∥H˙ x1 = lim ∥W(τ )(f ′ , g ′ )∥H˙ x1 = 0, τ →0

as well as   1  ′ ′ ′ ′  ∥g ′ ∥L2x = lim ∥∂τ W(τ )(f ′ , g ′ )∥L2x = lim lim  [W(τ + h)(f , g ) − W(τ )(f , g )]   2 = 0. τ →0 τ →0 h→0 h L x

′

′

Thus f (x) = g (x) = 0 a.e. as we asserted. Now, by combining (6.5) and (6.6) with the Sobolev embedding and uniqueness of weak limits in Lpx spaces, we get (f (x), g(x)) = (f ′ (x), g ′ (x)) for a.e. x ∈ Rd . Thus, using (6.5) with f (x) = g(x) = 0 for a.e. x ∈ Rd , we may choose n so that ∥(u(sn , x(sn ) + ·), ut (sn , x(sn ) + ·))∥H˙ xsc ×H˙ xsc −1 is arbitrarily small. By local theory, we obtain ∥u∥ 2(d+1) < ∞, contradicting our hypothesis that u is a blow-up solution. Thus (6.3) d+2−γ Lt,x

holds. Our second step is to adjust the domain of integration in (6.3). To this end, let C1 be as in (6.3). Fix η > 0 to be determined later in the argument and let s ∈ R be given. Then, by the almost periodicity of u, we may choose C2 (η) > 0 such that ∥u(t)∥

2d Lxd+2−γ

≤η

d+2−γ 2d

.

(|x−x(t)|≥C2 (η))

Let ϵ > 0 be as in Theorem 4.1. Using interpolation followed by the Sobolev embedding, we have ∥u(t)∥

2d

Lxd−2 (|x−x(t)|≥C2 (η))

≤ ∥u(t)∥δ

∥u(t)∥1−δ

2d

2d d−2(1−ϵ)

Lxd+2−γ (|x−x(t)|≥C2 (η))

Lx

δ

≤ C∥u(t)∥

2d

Lxd+2−γ (|x−x(t)|≥C2 (η))

≤ Cη

(Rd )

1−δ ∥u(t)∥H ˙ x1−ϵ (Rd )

(d+2−γ)δ 2d

for some δ ∈ (0, 1), where we note that d ≥ 6 yields

(6.12)

2d d−2(1−ϵ)

<

2d d−2

<

2d d+2−γ .

 (d+2−γ)δ 2d 2d Choose η small enough so that (Cη 2d ) d−2 < C21 . Then for all t ∈ [s, s + 1], Rd |u(t, x)| d−2 ≥ C1 implies    2d 2d 2d |u(t, x)| d−2 dx = |u(t, x)| d−2 dx − |u(t, x)| d−2 dx |x−x(t)|≤C2 (η)

Rd

|x−x(t)|≥C2 (η)

C1 . ≥ 2 Thus, we obtain from (6.3) that for all s ∈ R      2d C   1 |u(t, x)| d−2 dx ≥  t ∈ [s, s + 1] :  ≥ C1  2  |x−x(t)|≤C2 (η)

(6.13)

from which we settle the second step. We use (6.13) to obtain the desired estimate (6.2). Similarly as in (6.12), we have, for all s ∈ R  s+1  s+1  2d 2d ∥u(t)∥ d−22d dt |u(t, x)| d−2 dxdt = s

|x−x(t)|≤C2 (η)

s

 ≥ C1 ·

C1 2



Lxd−2 (|x−x(t)|≤C2 (η)) C12

=

2

.

Since C1 , C2 and C are independent of s, this completes the proof of the estimate (6.2).



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We finally preclude the existence of soliton-like solution by using the Morawetz estimate and the above two lemmas obtained in this Section. Again, we will follow the idea in [2,13]. We will prove that Theorem 6.3 (No Soliton). There is no solution u : R × Rd → R such that u solves (NLWH) and satisfies the properties of a soliton-like solution in the sense of Theorem 1.3. Proof. Suppose for a contradiction that there exists such a solution u. Fix T > 0 and choose R, c as in Lemma 6.2, and noting Lemma 6.1, we then write, 2d 2d  T ⌊T ⌋−1  i+1   |u(t, x)| d−2 |u(t, x)| d−2 dxdt ≥ dxdt. (6.14) |x| |x| 0 Rd i |x−x(t)|≤R i=0 Note that for all i ∈ {0, . . . , ⌊T ⌋ − 1}, when t ∈ [i, i + 1) and x ∈ {x ∈ Rd : |x − x(t)| ≤ R}, we have |x| ≤ |x − x(t)| + |x(t) − x(0)| + |x(0)| ≤ R + t + |x(0)| ≤ C ′ + i. Then we have,  i+1  ⌊T ⌋−1  2d 1 (6.14) ≥ |u(t, x)| d−2 dxdt ′+i C i |x−x(t)|≤R i=0 ⌊T ⌋−1

≥c

 i=0

1 ≥c C′ + i

 0

⌊T ⌋

1 dt. C′ + t

(6.15)

By using of (6.14), (6.15) and Theorem 2.9, we obtain 2d  C ′ + ⌊T ⌋   T  |u(t, x)| d−2 c log ≤ dxdt ≤ CE(u0 , u1 ) < ∞. C′ |x| 0 Rd Since u is a soliton-like solution, by Theorem 4.1 we have E(u0 , u1 ) < ∞. Noting that T > 0 is arbitrary and the constants C, R and c are independent of T , letting T tend to infinity, we derive a contradiction. This completes the proof of the theorem.  7. Low-to-high frequency cascade solution In this section, we will use Theorem 4.1 to preclude the low-to-high frequency cascade in the sense of Theorem 1.3. More precisely, we will prove Theorem 7.1 (No Low-to-High Cascade). There is no u : R × Rd → R such that u solves (NLWH), and satisfies the properties of a low-to-high frequency cascade solution in the sense of Theorem 1.3. Proof. Suppose u is a global solution that is low-to-high frequency cascade, we aim to derive a contradiction. By the definition of the low-to-high frequency cascade, we may choose a sequence {tn } ⊂ R with tn → ∞ ˙ 1−ϵ × H˙ x−ϵ ) for some ϵ > 0. such that N (tn ) → ∞ as n → ∞. From Theorem 4.1, we know that u ∈ L∞ t (R; Hx Using (1.2) followed by H¨ older’s inequality, we have, for all n ∈ N and η > 0,  |ξ|2 |ˆ u(tn , ξ)|2 + |ˆ ut (tn , ξ)|2 dξ |ξ|≤c(η)N (tn )

 |ξ|

.

2sc

2

c

2(1−ϵ)

|ˆ u(tn , ξ)| dξ

|ξ|

|ξ|≤c(η)N (tn )

2

|ˆ u(tn , ξ)| dξ

|ξ|≤c(η)N (tn )



|ξ|2(sc −1) |ˆ ut (tn , ξ)|2 dξ

+

 ϵ+sϵ −1  c

|ξ|≤c(η)N (tn ) ϵ

sc −1  ϵ+s −1

 ϵ+sϵ −1  c

sc −1  ϵ+s −1 c

|ξ|−2ϵ |ˆ ut (tn , ξ)|2 dξ

|ξ|≤c(η)N (tn )

2(sc −1)

ϵ

c −1 ϵ+sc −1 . . η ϵ+sc −1 ∥(u, ut )∥Lϵ+s ∞ (R;H ˙ 1−ϵ ×H ˙ −ϵ ) . η x

x

(7.1)

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On the other hand, by Chebyshev’s inequality, we have, for all η > 0 and n ∈ N,  |ξ|2 |ˆ u(tn , ξ)|2 + |ˆ ut (tn , ξ)|2 dξ |ξ|≥c(η)N (tn )  u(tn , ξ)|2 + |ξ|2(sc −1) |ˆ ut (tn , ξ)|2 dξ ≤ [c(η)N (tn )]−2(sc −1) |ξ|2sc |ˆ Rd −2(sc −1)

. [c(η)N (tn )]

∥(u, ut )∥2L∞ (R;H˙ sc ×H˙ sc −1 ) . [c(η)N (tn )]−2(sc −1) . x

(7.2)

x

We now estimate the nonlinear term in the energy. Note that using Sobolev’s inequality and the interpolation ˙ sc with u ∈ L∞ t Hx ,  |u(tn , x)|2 |u(tn , y)|2 dxdy . ∥∇u(tn )∥2L2x ∥u∥2L∞ H˙ sc . ∥∇u(tn )∥2L2x . (7.3) γ x t |x − y| d d R ×R Noticing (7.1), (7.2) and Plancherel’s theorem in (7.3), we estimate the energy as    E(u(tn ), ut (tn )) . |ξ|2 |ˆ u(tn )|2 dξ + |ˆ ut (tn )|2 dξ + |ξ|2 |ˆ u(tn )|2 dξ d d d R R R  2 2 . |ξ| |ˆ u(tn )| dξ + |ξ|2 |ˆ u(tn )|2 dξ |ξ|≤c(η)N (tn ) |ξ|≥c(η)N (tn )   + |ˆ ut (tn )|2 dξ + |ˆ ut (tn )|2 dξ |ξ|≤c(η)N (tn )

.η

ϵ ϵ+sc −1

|ξ|≥c(η)N (tn ) −2(sc −1)

+ [c(η)N (tn )]

,

(7.4)

for all η > 0 and n ∈ N. Letting n → ∞ in (7.4) and using the conservation of energy, by N (tn ) → ∞, we have, for all η > 0, ϵ E(u(0), ut (0)) . η ϵ+sc −1 . Taking η → 0, we obtain E(u(0), ut (0)) = 0. Thus u ≡ 0 contradicting our assumption that u is a blow-up solution.  Acknowledgments The author would like to express his sincere gratitude to the anonymous referees for their invaluable comments and suggestions which helped improve the paper greatly. This research was supported by the National Natural Science Foundation of China (No. 11301489), the Distinguished Youth Science Foundation of Shanxi Province (2015021001), the Outstanding Youth Foundation of North University of China (No. JQ201604), the Youth Academic Leaders Support Program of North University of China, and the Research Fund for the Doctoral Program of Higher Education (20121420120004). References [1] A. Bulut, Maximizers for the Strichartz inequalities for the wave equation, Differential Integral Equations 23 (2010) 1035–1072. [2] A. Bulut, Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation, J. Funct. Anal. 263 (2012) 1609–1660. [3] M. Christ, M. Weinstein, I. Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation, J. Funct. Anal. 100 (1991) 87–109. [4] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schr¨ odinger equation in R3 , Ann. of Math. 167 (2008) 767–865. [5] C. Kenig, Global Well-Posedness, Scattering and Blow Up for the Energy-Critical, Focusing, Non-Linear Schrodinger and Wave Equations, in: Lecture Notes, 2010. [6] C. Kenig, The Concentration-Compactness/Rigidity Method for Critical Dipsersive and Wave Equations, in: Lecture Notes, 2010. [7] C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear Schr¨ odinger equation in the radial case, Invent. Math. 166 (2006) 645–675.

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