On blowup solutions to the focusing L 2-supercritical nonlinear fractional Schrödinger equation

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nonlinear fractional Schrödinger equation

van Duong Dinh

To cite this version:

van Duong Dinh. On blowup solutions to the focusing L 2-supercritical nonlinear fractional

Schrödinger equation. Journal of Mathematical Physics, American Institute of Physics (AIP), 2018,

59 (071506). �hal-01936024�

nonlinear fractional Schr¨ odinger equation

Van Duong Dinh

^a)

We study dynamical properties of blowup solutions to the focusing L

²

-supercritical nonlinear fractional Schr¨ odinger equation i∂

t

u−(−∆)

^s

u = −|u|

^α

u on [0, +∞)× R

^d

, where d ≥ 2,

_2d−1^d

≤ s < 1,

^4s_d

< α <

_d−2s^4s

and the initial data u(0) = u

0

∈ H ˙

^sc

∩ H ˙

^s

is radial with the critical Sobolev exponent s

c

. To this end, we establish a compactness lemma related to the equation by means of the profile decomposition for bounded sequences in ˙ H

^sc

∩ H ˙

^s

. As a result, we obtain the ˙ H

^sc

-concentration of blowup solutions with bounded ˙ H

^sc

-norm and the limiting profile of blowup solutions with critical ˙ H

^sc

-norm.

Keywords: Nonlinear fractional Schr¨ odinger equation; Blowup; Concentration;

Limiting profile

I. INTRODUCTION

In this paper, we consider the Cauchy problem for the focusing L

²

-supercritical nonlinear fractional Schr¨ odinger equation

i∂

t

u − (−∆)

^s

u = −|u|

^α

u, on [0, +∞) × R

^d

,

u(0) = u

₀

, (1.1)

where u : [0, +∞) × R

^d

→ C , s ∈ (0, 1)\{1/2} and α > 0. The operator (−∆)

^s

is the fractional Laplacian which is the Fourier multiplier by |ξ|

^2s

. The fractional Schr¨ odinger equation was discovered by N. Laskin

²⁴

as a result of extending the Feynmann path integral, from the Brownian-like to L´ evy-like quantum mechanical paths. The fractional Schr¨ odinger equation also appears in the study of water waves equations (see e.g. Refs. 21 and 26).

The study of the nonlinear fractional Schr¨ odinger equation has attracted a lot of interest in the last decade (see e.g. Refs. 2, 6–8, 13–15, 19, 21, 23, 27, and 29 and references cited therein).

The equation (1.1) enjoys the scaling invariance

u

_λ

(t, x) := λ

^2sα

u(λ

^2s

t, λx), λ > 0.

A calculation shows

ku

λ

(0)k

H˙^γ

= λ

^γ+2sα−d2

ku

0

k

H˙^γ

. From this, we define the critical Sobolev exponent

s

c

:= d 2 − 2s

α , (1.2)

as well as the critical Lebesgue exponent α

_c

:= 2d

d − 2s

_c

= dα

2s . (1.3)

a)Institut de Math´ematiques de Toulouse UMR5219, Universit´e Toulouse CNRS, 31062 Toulouse Cedex 9, France.; Electronic mail:dinhvan.duong@math.univ-toulouse.fr

By definition, we have the Sobolev embedding ˙ H

^sc

, → L

^αc

. The equation (1.1) is called L

²

-subcritical (L

²

-critical or L

²

-supercritical) if s

c

< 0 (s

c

= 0 or s

c

> 0) respectively.

The local well-posedness for (1.1) in Sobolev spaces with non-radial initial data was studied in Ref. 19 (see also Ref. 10). In the non-radial setting, the unitary group e

^{−it(−∆)s}

enjoys Strichartz estimates (see Ref. 5 or Ref. 10):

ke

^{−it(−∆)s}

ψk

_Lp(R,L^q)

. k|∇|

^γp,q

ψk

_L2

, where (p, q) satisfies the Schr¨ odinger admissible condition

p ∈ [2, ∞], q ∈ [2, ∞), (p, q, d) 6= (2, ∞, 2), 2 p + d

q ≤ d 2 , and

γ

p,q

= d 2 − d

q − 2s p .

It is easy to see that the condition

²_p

+

^d_q

≤

^d₂

implies γ

_p,q

> 0 for all Schr¨ odinger admissible pairs (p, q) except (p, q) = (∞, 2). This means that for non-radial data, Strichartz estimates for e

^{−it(−∆)s}

have a loss of derivatives except for (p, q) = (∞, 2). This makes the study of local well-posedness in the non-radial case more difficult. The local theory for (1.1) showed in Refs. 10 and 19 is much weaker than the one for classical nonlinear Schr¨ odinger equation, i.e. s = 1. In particular, in the ˙ H

^s

-subcritical case (i.e. s

c

< s) the equation (1.1) is locally well-posed in H

^s

only for dimensions d = 1, 2, 3. The loss of derivatives in Strichartz estimates can be removed if one considers radial initial data. More precisely, we have for d ≥ 2,

_2d−1^d

≤ s < 1 and ψ radial,

ke

^{−it(−∆)s}

ψk

_Lp(R,L^q)

. kψk

_L2

, provided that (p, q) satisfies the fractional admissible condition

p ∈ [2, ∞], q ∈ [2, ∞), (p, q) 6=

2, 4d − 2 2d − 3

, 2s

p + d q = d

2 .

These Strichartz estimates with no loss of derivatives allow us to show a better local theory for (1.1) with radial initial data. We refer the reader to Section II for more details.

The existence of blowup solutions to (1.1) was studied numerically in Ref. 23. Later, Boulenger-Himmelsbach-Lenzmann

²

established blowup criteria for radial H

^s

solutions to (1.1). Note that in Ref. 2, they considered H

^2s

solutions due to the lack of a full local theory at the time of consideration. Thanks to the local theory given in Section II, we can recover H

^s

solutions by approximation arguments. More precisely, they proved the following:

Theorem 1.1 (Ref. 2). Let d ≥ 2, s ∈ (1/2, 1) and α > 0. Let u

0

∈ H

^s

be radial and assume that the corresponding solution to (1.1) exists on the maximal forward time interval [0, T ).

• Mass-critical case: If s

_c

= 0 or α =

^4s_d

and E(u

₀

) < 0, then the solution u either blows up in finite time, i.e. T < +∞ or blows up infinite time, i.e. T = +∞ and

ku(t)k

H˙^s

≥ ct

^s

, ∀t ≥ t

_∗

, for some C > 0 and t

_∗

> 0 depending only on u

₀

, s and d.

• Mass-supercritical and energy-subcritical case: If 0 < s

c

< s or

^4s_d

< α <

_d−2s^4s

and α < 4s and either E(u

0

) < 0, or if E(u

0

) ≥ 0, we assume that

E

^sc

(u

0

)M

^s−sc

(u

0

) < E

^sc

(Q)M

^s−sc

(Q), ku

0

k

^s_˙^c

H^s

ku

0

k

^s−s_L2 ^c

> kQk

^s_˙^c

H^s

kQk

^s−s_L2 ^c

,

where Q is the unique (up to symmetries) positive radial solution to the elliptic equa- tion

(−∆)

^s

Q + Q − |Q|

^α

Q = 0, then the solution blows up in finite time, i.e. T < +∞.

• Energy-critical case: If s

c

= s or α =

_d−2s^4s

and α < 4s and either E(u

0

) < 0, or if E(u

0

) ≥ 0, we assume that

E(u

0

) < E(W ), ku

0

k

H˙^s

> kW k

H˙^s

,

where W is the unique (up to symmetries) positive radial solution to the elliptic equa- tion

(−∆)

^s

W − |W |

^d−2s4s

W = 0, then the solution blows up in finite time, i.e. T < +∞.

Here M (u) and E(u) are the conserved mass and energy respectively.

The blowup criteria of Boulenger-Himmelsbach-Lenzmann

²

naturally lead to the study of dynamical properties such as blowup rate, concentration and limiting profile,.. of blowup solutions to (1.1).

In the mass-critical case s

c

= 0 or α =

^4s_d

, the dynamics of blowup H

^s

solutions was recently considered in Ref. 11 (see also Ref. 13). The study of blowup H

^s

solutions to the focusing mass-critical nonlinear fractional Schr¨ odinger equation is connected to the notion of ground state which is the unique (up to symmetries) positive radial solution of the elliptic equation

(−∆)

^s

Q + Q − |Q|

^4sd

Q = 0. (1.4) Note that the existence and uniqueness (modulo symmetries) of ground state to (1.4) were shown in Refs. 14 and 15. Using the sharp Gagliardo-Nirenberg inequality

kf k

^4sd+2

L^4sd+2

≤ C

GN

kf k

_L^4sd2

kf k

²_H˙s

, with

C

_GN

= 2s + d

d kQk

⁻_L2^4sd

,

the conservation of mass and energy show that if u

0

∈ H

^s

satisfies ku

0

k

_L2

< kQk

_L2

, then the corresponding solution exists globally in time. This suggests that kQk

_L2

is the critical mass for formation of singularities. To study dynamical properties of blowup H

^s

solutions to the mass-critical (1.1), the author in Ref. 11 proved a compactness lemma related to the equation by means of the profile decomposition for bounded sequences in H

^s

.

Proposition 1.2 (Compactness lemma

¹¹

). Let d ≥ 1 and 0 < s < 1. Let (v

n

)

_n≥1

be a bounded sequence in H

^s

such that

lim sup

n→∞

kv

n

k

H˙^s

≤ M, lim sup

n→∞

kv

n

k

L^4sd+2

≥ m.

Then there exists a sequence (x

_n

)

_n≥1

in R

^d

such that up to a subsequence, v

n

(· + x

n

) * V weakly in H

^s

,

for some V ∈ H

^s

satisfying

kV k

_L^4sd2

≥ d d + 2s

m

^4sd+2

M

²

kQk

_L^4sd2

.

Thanks to this compactness lemma, the author in Ref. 11 showed that the L

²

-norm of blowup solutions must concentrate by an amount which is bounded from below by kQk

L²

at the blowup time. He also showed the limiting profile of blowup solutions with minimal mass ku

0

k

_L2

= kQk

_L2

, that is, up to symmetries of the equation, the ground state Q is the profile for blowup solutions with minimal mass.

The main goal of this paper is to study dynamical properties of blowup solutions to (1.1) in the mass-supercritical and energy-subcritical case with initial data in ˙ H

^sc

∩ H ˙

^s

. To this end, we first show the local well-posedness for (1.1) with initial data in ˙ H

^sc

∩ H ˙

^s

. For data in H

^s

, the local well-posedness in non-radial and radial cases was showed in Refs. 11 and 19. In the non-radial setting, the inhomogeneous Sobolev embedding W

^s,q

, → L

^r

plays a crucial role (see e.g. Ref. 19). Since we are considering data in ˙ H

^sc

∩ H ˙

^s

, the inhomogeneous Sobolev embedding does not help. We thus have to rely on Strichartz estimates without loss of derivatives and the homogeneous Sobolev embedding ˙ W

^s,q

, → L

^r

. We hence restrict ourself to radially symmetric initial data, d ≥ 2 and

_2d−1^d

≤ s < 1 for which Strichartz estimates without loss of derivatives are available. After the local theory is established, we show the existence of blowup ˙ H

^sc

∩ H ˙

^s

solutions. The existence of blowup H

^s

solutions for (1.1) was shown in Ref. 2 (see Theorem 1.1). Note that the conservation of mass plays a crucial role in the argument of Ref. 2. In our consideration, the lack of mass conservation laws makes the problem more difficult. We are only able to show blowup criteria for negative energy intial data in ˙ H

^sc

∩ H ˙

^s

with an additional assumption

sup

t∈[0,T)

ku(t)k

H˙^sc

< ∞, (1.5) where [0, T ) is the maximal forward time of existence. In the mass-critical case s

_c

= 0, this assumption holds trivially by the conservation of mass. We refer to Section II for more details. To study blowup dynamics for data in ˙ H

^sc

∩ H ˙

^s

, we prove the profile decomposition for bounded sequences in ˙ H

^sc

∩ H ˙

^s

which is proved by following the argument of Ref. 20 (see also Refs. 12 and 18). This profile decomposition allows us to study the variational structure of the sharp constant to the Gagliardo-Nirenberg inequality

kf k

^α+2_Lα+2

≤ A

GN

kf k

^α_H˙sc

kf k

²_H˙s

. (1.6) We will see in Proposition 3.2 that the sharp constant A

GN

is attained at a function U ∈ H ˙

^sc

∩ H ˙

^s

of the form

U(x) = aQ(λx + x

0

),

for some a ∈ C

^∗

, λ > 0 and x

0

∈ R

^d

, where Q is a solution to the elliptic equation (−∆)

^s

Q + (−∆)

^sc

Q − |Q|

^α

Q = 0.

Moreover,

A

GN

= α + 2 2 kQk

^−α_˙

H^sc

.

The sharp Gagliardo-Nirenberg inequality (1.6) together with the conservation of energy yield the global existence for solutions satisfying

sup

t∈[0,T)

ku(t)k

H˙^sc

< kQk

H˙^sc

.

Another application of the profile decomposition is the compactness lemma, that is, for any bounded sequence (v

_n

)

_n≥1

in ˙ H

^sc

∩ H ˙

^s

satisfying

lim sup

n→∞

kv

n

k

H˙^s

≤ M, lim sup

n→∞

kv

n

k

L^α+2

≥ m,

there exists a sequence (x

_n

)

_n≥1

in R

^d

such that up to a subsequence, v

n

(· + x

n

) * V weakly in ˙ H

^sc

∩ H ˙

^s

, for some V ∈ H ˙

^sc

∩ H ˙

^s

satisfying

kV k

^α_H˙sc

≥ 2 α + 2

m

^α+2

M

²

kQk

^α_H˙sc

.

As a consequence, we show that the ˙ H

^sc

-norm of blowup solutions satisfying (1.5) must concentrate by an amount which is bounded from below by kQk

H˙^sc

at the blowup time (see Theorem 4.1). We finally show in Theorem 5.2 the limiting profile of blowup solutions with critical norm

sup

t∈[0,T)

ku(t)k

H˙^sc

= kQk

H˙^sc

. (1.7) The paper is organized as follows. In Section II, we recall Strichartz estimates and show the local well-posednesss for data in ˙ H

^sc

∩ H ˙

^s

. We also prove blowup criteria for negative energy data in ˙ H

^sc

∩ H ˙

^s

as well as the profile decomposition of bounded sequences in ˙ H

^sc

∩ H ˙

^s

. In Section III, we give some applications of the profile decomposition including the sharp Gagliardo-Nirenberg inequality (1.6) and the compactness lemma. In Section IV, we show the ˙ H

^sc

-concentration of blowup solutions. Finally, the limiting profile of blowup solutions with critical norm (1.7) will be given in Section V.

II. PRELIMINARIES

A. Homogeneous Sobolev spaces

We recall the definition of homogeneous Sobolev spaces needed in the sequel (see e.g.

Refs. 1, 16 or 28). Denote S

0

the subspace of the Schwartz space S consisting of functions φ satisfying D

^β

φ(0) = 0 for all ˆ β ∈ N

^d

, where ˆ · is the Fourier transform on S. Given γ ∈ R and 1 ≤ q ≤ ∞, the generalized homogeneous Sobolev space ˙ W

^γ,q

is defined as a closure of S

₀

under the norm

kuk

W˙^γ,q

:= k|∇|

^γ

uk

L^q

< ∞.

Under this setting, the spaces ˙ W

^γ,q

are Banach spaces. We shall use ˙ H

^γ

:= ˙ W

^γ,2

. Note that the spaces ˙ H

^γ1

and ˙ H

^γ2

cannot be compared for the inclusion. Nevertheless, for γ

1

< γ < γ

2

, the space ˙ H

^γ

is an interpolation space between ˙ H

^γ1

and ˙ H

^γ2

.

B. Strichartz estimates

We next recall Strichartz estimates for the fractional Schr¨ odinger equation. To do so, we define for I ⊂ R and p, q ∈ [1, ∞] the mixed norm

kuk

_Lp(I,L^q)

:= Z

I

Z

R^d

|u(t, x)|

^q

dx

^p_q

¹_p

,

with a usual modification when either p or q are infinity. The unitary group e

^{−it(−∆)s}

enjoys several types of Strichartz estimates, for instance non-radial Strichartz estimates,

radial Strichartz estimates and weighted Strichartz estimates (see e.g. Ref. 6). We only

recall here two types: non-radial and radial Strichartz estimates.

• Non-radial Strichartz estimates (see e.g. Refs. 5 and 10): for d ≥ 1 and s ∈ (0, 1)\{1/2}, the following estimates hold:

ke

^{−it(−∆)s}

ψk

L^p(R,L^q)

. k|∇|

^γp,q

ψk

L²

,

Z

t 0

e

−i(t−τ)(−∆)^s

f (τ )dτ

_Lp(

R,L^q)

. k|∇|

^{γp,q−γa0,b0−2s}

f k

_La0(R,L^b0)

, where (p, q) and (a, b) are Schr¨ odinger admissible pairs, i.e.

p ∈ [2, ∞], q ∈ [2, ∞), (p, q, d) 6= (2, ∞, 2), 2 p + d

q ≤ d 2 , and

γ

p,q

= d 2 − d

q − 2s p ,

and similarly for γ

_a⁰_,b⁰

. As mentioned in the introduction, these Strichartz estimates have a loss of derivatives except for (p, q) = (a, b) = (∞, 2).

• Radial Strichartz estimates (see e.g. Refs. 3, 17 or 22): for d ≥ 2 and

d

2d−1

≤ s < 1, the following estimates hold:

ke

^{−it(−∆)s}

ψk

_Lp(R,L^q)

. kψk

_L2

, (2.1)

Z

t 0

e

−i(t−τ)(−∆)^s

f (τ )dτ

_Lp(

R,L^q)

. kf k

_La0

(R,L^b0)

, (2.2) where ψ and f are radially symmetric and (p, q), (a, b) sastisfy the fractional admissible condition:

p ∈ [2, ∞], q ∈ [2, ∞), (p, q) 6=

2, 4d − 2 2d − 3

, 2s

p + d q = d

2 . (2.3)

C. Local well-posedness

In this subsection, we show the local well-posedness for (1.1) with initial data in ˙ H

^sc

∩ H ˙

^s

. Before entering some details, let us recall the local well-posedness for (1.1) with initial data in H

^s

.

Proposition 2.1 (Local well-posedness in H

^s11

). Let



 

 

 

 



 

 

 

 



d = 1,

¹₃

< s <

¹₂

, 0 < α <

_1−2s^4s

, u

0

∈ H

^s

non-radial, d = 1,

¹₂

< s < 1, 0 < α < ∞, u

0

∈ H

^s

non-radial, d = 2,

¹₂

< s < 1, 0 < α <

_2−2s^4s

, u

₀

∈ H

^s

non-radial, d = 3,

³₅

≤ s ≤

³₄

, 0 < α <

_3−2s^4s

, u

₀

∈ H

^s

radial, d = 3,

³₄

< s < 1, 0 < α <

_3−2s^4s

, u

0

∈ H

^s

non-radial, d ≥ 4,

_2d−1^d

≤ s < 1, 0 < α <

_d−2s^4s

, u

0

∈ H

^s

radial.

(2.4)

Then the equation (1.1) is locally well-posed in H

^s

. In addition, the maximal forward time of existence satisfies either T = +∞ or T < +∞ and lim

_t↑T

kuk

H˙^s

= ∞. Moreover, the solution enjoys the conservation of mass and energy, i.e. M (u(t)) = M (u

₀

) and E(u(t)) = E(u

0

) for all t ∈ [0, T ), where

M (u(t)) = Z

|u(t, x)|

²

dx, E(u(t)) = 1

2 Z

|(−∆)

^s/2

u(t, x)|

²

dx − 1 α + 2

Z

|u(t, x)|

^α+2

dx.

We now give the local well-posedness for (1.1) with initial data in ˙ H

^sc

∩ H ˙

^s

.

Proposition 2.2 (Local well-posedness in ˙ H

^sc

∩ H ˙

^s

). Let d ≥ 2,

_2d−1^d

≤ s < 1 and

4s

d

≤ α <

_d−2s^4s

. Let

p = 4s(α + 2)

α(d − 2s) , q = d(α + 2)

d + αs . (2.5)

Then for any u

0

∈ H ˙

^sc

∩ H ˙

^s

radial, there exist T > 0 and a unique solution u to (1.1) satisfying

u ∈ C([0, T ), H ˙

^sc

∩ H ˙

^s

) ∩ L

^p_loc

([0, T ), W ˙

^sc,q

∩ W ˙

^s,q

).

The maximal forward time of existence satisfies either T = +∞ or T < +∞ and lim

_t↑T

ku(t)k

H˙^sc

+ ku(t)k

H˙^s

= ∞. Moreover, the solution enjoys the conservation of energy, i.e. E(u(t)) = E(u

₀

) for all t ∈ [0, T ).

Remark 2.3. When s

c

= 0 or α =

^4s_d

, Proposition 2.2 is a consequence of Proposition 2.1 since ˙ H

⁰

= L

²

and L

²

∩ H ˙

^s

= H

^s

.

Proof of Proposition 2.2. It is easy to check that (p, q) satisfies the fractional admissible condition (2.3). We next choose (m, n) so that

1 p

⁰

= 1

p + α m , 1

q

⁰

= 1 q + α

n . We see that

θ := α m − α

p = 1 − (d − 2s)α

4s > 0, q ≤ n = dq d − sq . The later fact ensures the Sobolev embedding ˙ W

^s,q

, → L

ⁿ

. Consider

X := n

u ∈ C(I, H ˙

^sc

∩ H ˙

^s

) ∩ L

^p

(I, W ˙

^sc,q

∩ W ˙

^s,q

) : kuk

_L∞(I,H˙^sc∩H˙^s)

+kuk

_Lp(I,W˙^sc,q∩W˙^s,q)

≤ M o , equipped with the distance

d(u, v) := ku − vk

_L∞(I,L²)

+ ku − vk

_Lp(I,L^q)

,

where I = [0, ζ] and M, ζ > 0 to be determined later. Thanks to Duhamel’s formula, it suffices to show that the functional

Φ(u)(t) := e

^{−it(−∆)s}

u

0

+ i Z

t

0

e

−i(t−τ)(−∆)^s

|u(τ)|

^α

u(τ)dτ is a contraction on (X, d). Thanks to Strichartz estimates (2.1) and (2.2),

kΦ(u)k

_L∞(I,H˙^sc∩H˙^s)

+ kΦ(u)k

_Lp(I,W˙^sc,q∩W˙^s,q)

. ku

0

k

H˙^sc∩H˙^s

+ k|u|

^α

uk

_Lp0

(I,W˙^sc,q0

∩W˙^s,q0)

, kΦ(u) − Φ(v)k

L^∞(I,L²)

+ kΦ(u) − Φ(v)k

L^p(I,L^q)

. k|u|

^α

u − |v|

^α

vk

_Lp0

(I,L^q0)

.

By the fractional derivatives (see e.g. Proposition 3.1 of Ref. 9) and the choice of (m, n), the H¨ older inequality implies

k|u|

^α

uk

_Lp0

(I,W˙^sc,q0

∩W˙^s,q0)

. kuk

^α_Lm(I,Lⁿ)

kuk

_Lp(I,W˙ ^sc,q∩W˙^s,q)

. |I|

^θ

kuk

^α_Lp(I,Lⁿ)

kuk

_Lp(I,W˙^sc,q∩W˙^s,q)

. |I|

^θ

kuk

^α_Lp(I,W˙s,q)

kuk

_Lp(I,W˙^sc,q∩W˙^s,q)

.

Similarly,

k|u|

^α

u − |v|

^α

vk

_Lp0

(I,L^q0)

.

kuk

^α_Lm(I,Lⁿ)

+ kvk

^α_Lm(I,Lⁿ)

ku − vk

_Lp(I,L^q)

. |I|

^θ

kuk

^α_{Lp(I,W˙ s,q)}

+ kvk

^α_Lp(I,W˙s,q)

ku − vk

_Lp(I,L^q)

. This shows that for all u, v ∈ X , there exists C > 0 independent of ζ and u

₀

∈ H ˙

^sc

∩ H ˙

^s

such that

kΦ(u)k

_L∞(I,H˙^sc∩H˙^s)

+ kΦ(u)k

_Lp(I,W˙^sc,q∩W˙ ^s,q)

≤ Cku

0

k

H˙^sc∩H˙^s

+ Cζ

^θ

M

^α+1

, (2.6) d(Φ(u), Φ(v)) ≤ Cζ

^θ

M

^α

d(u, v).

If we set M = 2Cku

0

k

H˙^sc∩H˙^s

and choose ζ > 0 so that Cζ

^θ

M

^α

≤ 1

2 ,

then Φ is a strict contraction on (X, d). This proves the existence of solution u ∈ C(I, H ˙

^sc

∩ H ˙

^s

) ∩ L

^p

(I, W ˙

^sc,q

∩ W ˙

^s,q

).

Note that by radial Strichartz estimates, the solution belongs to L

^a

(I, W ˙

^sc,b

∩ W ˙

^s,b

) for any fractional admissible pairs (a, b). The blowup alternative is easy since the time of existence depends only on the ˙ H

^sc

∩ H ˙

^s

-norm of initial data. The conservation of energy follows from

the standard approximation. The proof is complete.

Corollary 2.4 (Blowup rate). Let d ≥ 2,

_2d−1^d

≤ s < 1,

^4s_d

≤ α <

_d−2s^4s

and u

0

∈ H ˙

^sc

∩ H ˙

^s

be radial. Assume that the corresponding solution u to (1.1) given in Proposition 2.2 blows up at finite time 0 < T < +∞. Then there exists C > 0 such that

ku(t)k

H˙^sc∩H˙^s

> C (T − t)

^s−sc2s

, (2.7)

for all 0 < t < T .

Proof. Let 0 < t < T . If we consider (1.1) with initial data u(t), then it follows from (2.6) and the fixed point argument that if for some M > 0,

Cku(t)k

H˙^sc∩H˙^s

+ C(ζ − t)

^θ

M

^α+1

≤ M, then ζ < T . Thus,

Cku(t)k

H˙^sc∩H˙^s

+ C(T − t)

^θ

M

^α+1

> M, for all M > 0. Choosing M = 2Cku(t)k

H˙^sc∩H˙^s

, we see that

(T − t)

^θ

ku(t)k

^α_H˙sc∩H˙s

> C.

This implies

ku(t)k

H˙^sc∩H˙^s

> C (T − t)

^θα

,

which is exactly (2.7) since

_α^θ

=

^{4s−α(d−2s)}_4αs

=

^s−s_2s^c

. The proof is complete.

D. Blowup criteria

In this subsection, we prove blowup criteria for H ˙

^sc

∩ H ˙

^s

solutions to the mass- supercritical and energy-subcritical (1.1). For initial data in H

^s

, Boulenger-Himmelsbach- Lenzmann proved blowup criteria for the equation (see Theorem 1.1 for more details). The main difficulty in our consideration is that the conservation of mass is no longer available.

We overcome this difficulty by assuming that the solution satisfies the uniform bound (1.5).

More precisely, we have the following:

Proposition 2.5 (Blowup criteria). Let d ≥ 2,

_2d−1^d

≤ s < 1,

^4s_d

< α <

_d−2s^4s

and α < 4s.

Let u

₀

∈ H ˙

^sc

∩ H ˙

^s

be radial satisfying E(u

₀

) < 0. Assume that the corresponding solution to (1.1) defined on a maximal forward time interval [0, T ) satisfies (1.5). Then the solution u blows up in finite time, i.e. T < +∞.

Remark 2.6. The condition α < 4s comes from the radial Sobolev embedding (a analogous condition appears in Ref. 2 (see again Theorem 1.1)).

Proof of Proposition 2.5. Let χ : [0, ∞) → [0, ∞) be a smooth function such that χ(r) =

r

²

if r ≤ 1,

0 if r ≥ 2, and χ

⁰⁰

(r) ≤ 2 for r ≥ 0.

For a given R > 0, we define the radial function χ

R

: R

^d

→ R by ϕ

R

(x) = ϕ

R

(r) := R

²

χ(r/R), |x| = r.

It is easy to see that

2 − ϕ

⁰⁰_R

(r) ≥ 0, 2 − ϕ

⁰_R

(r)

r ≥ 0, 2d − ∆ϕ

_R

(x) ≥ 0, ∀r ≥ 0, ∀x ∈ R

^d

. Moreover,

k∇

^j

ϕ

R

k

L^∞

. R

^2−j

, j = 0, · · · , 4, and

supp(∇

^j

ϕ

R

) ⊂

{|x| ≤ 2R} for j = 1, 2, {R ≤ |x| ≤ 2R} for j = 3, 4.

Now let u ∈ H ˙

^sc

∩ H ˙

^s

be a solution to (1.1). We define the local virial action by M

ϕ_R

(t) := 2

Z

∇ϕ

R

(x) · Im(u(t, x)∇u(t, x))dx.

The virial action M

ϕ_R

(t) is well-defined. Indeed, we first learn from the H¨ older inequality and the Sobolev embedding ˙ H

^sc

, → L

^αc

that

kuk

_L2(|x|.R)

. R

^sc

kuk

_Lαc(|x|.R)

. R

^sc

kuk

H˙^sc(|x|.R)

. (2.8) Using the fact supp(∇ϕ

R

) ⊂ {|x| . R}, (2.8) and the estimate given in Lemma A.1 of Ref.

2, we have

|M

ϕ_R

(t)| ≤ C(χ, R)

k|∇|

¹²

u(t)k

²_L2(|x|.R)

+ ku(t)k

_L2(|x|.R)

k|∇|

¹²

u(t)k

_L2(|x|.R)

≤ C(χ, R)

ku(t)k

²⁻_L2(|x|^1s

.R)

ku(t)k

^1s_˙

H^s(|x|.R)

+ ku(t)k

²⁻_L2(|x|^2s1

.R)

ku(t)k

^2s1_˙

H^s(|x|.R)

(2.9)

≤ C(χ, R)

ku(t)k

²⁻_˙ ^1s

H^sc(|x|.R)

ku(t)k

^1s_˙

H^s(|x|.R)

+ ku(t)k

²⁻_˙ ^2s1

H^sc(|x|.R)

ku(t)k

^2s1_˙

H^s(|x|.R)

.

This shows that M

_ϕR

(t) is well-defined for all t ∈ [0, T ). Note that in the case χ(r) = r

²

or ϕ

R

(x) = |x|

²

, we have formally the virial identity (see Lemma 2.1 of Ref. 2):

M

_|x|⁰ 2

(t) = 8sku(t)k

²_H˙s

− 4dα

α + 2 ku(t)k

^α+2_Lα+2

= 4dαE(u(t)) − 2(dα − 4s)ku(t)k

²_H˙s

. (2.10) We also have from Lemma 2.1 of 2 that for any t ∈ [0, T ),

M

_ϕ⁰

R

(t) = − Z

∞

0

m

^s

Z

∆

²

ϕ

_R

|u

_m

(t)|

²

dxdm + 4

d

X

j,k=1

Z

∞ 0

m

^s

Z

∂

_jk²

ϕ

_R

∂

_j

u

_m

(t)∂

_k

u

_m

(t)dxdm

− 2α α + 2

Z

∆ϕ

R

|u(t)|

^α+2

dx, where

u

m

(t) := c

s

1

−∆ + m u(t) = c

s

F

⁻¹

u(t) ˆ

|ξ|

²

+ m

, m > 0, with

c

s

:=

r sin πs π . Since ϕ

R

(x) = |x|

²

for |x| ≤ R, we use (2.10) to write

M

_ϕ⁰_R

(t) = 8sku(t)k

²_H˙s

− 4dα

α + 2 ku(t)k

^α+2_Lα+2

− 8sku(t)k

²_H˙s(|x|>R)

+ 4dα

α + 2 ku(t)k

^α+2_Lα+2(|x|>R)

− Z

∞

0

m

^s

Z

|x|>R

∆

²

ϕ

R

|u

m

(t)|

²

dxdm +4

∞

X

j,k=1

Z

∞ 0

m

^s

Z

|x|>R

∂

_jk²

ϕ

R

∂

j

u

m

(t)∂

k

u

m

(t)dxdm

− 2α α + 2

Z

|x|>R

∆ϕ

_R

|u(t)|

^α+2

dx

= 4dαE(u(t)) − 2(dα − 4s)ku(t)k

²_H˙s

+4

∞

X

j,k=1

Z

∞ 0

m

^s

Z

|x|>R

∂

_jk²

ϕ

_R

∂

_j

u

_m

(t)∂

_k

u

_m

(t)dxdm − 8sku(t)k

²_H˙s

(|x|>R)

− Z

∞

0

m

^s

Z

|x|>R

∆

²

ϕ

R

|u

m

(t)|

²

dxdm + 2α α + 2

Z

|x|>R

(2d − ∆ϕ

R

)|u(t)|

^α+2

dx.

Using

∂

_jk²

=

δ

jk

− x

j

x

k

r

²

∂

r

r + x

j

x

k

r

²

∂

_r²

, we write

4

∞

X

j,k=1

Z

∞ 0

m

^s

Z

|x|>R

∂

_jk²

ϕ

_R

∂

_j

u

_m

(t)∂

_k

u

_m

(t)dxdm = 4 Z

∞

0

m

^s

Z

|x|>R

ϕ

⁰⁰_R

|∇u

_m

(t)|

²

dxdm.

Note that (see (2.12) in Ref. 2) Z

∞

0

m

^s

Z

|∇f

m

|

²

dxdm = Z

sin πs π

Z

∞ 0

m

^s

(|ξ|

²

+ m)

²

dm

|ξ|

²

| f ˆ (ξ)|

²

dξ = skf k

²_H˙s

.

We thus get 4

∞

X

j,k=1

Z

∞ 0

m

^s

Z

|x|>R

∂

_jk²

ϕ

R

∂

j

u

m

(t)∂

k

u

m

(t)dxdm

= 8sku(t)k

²_H˙s

(|x|>R)

− 4 Z

∞

0

m

^s

Z

|x|>R

(2 − ϕ

⁰⁰_R

)|∇u

_m

(t)|

²

dxdm

≤ 8sku(t)k

²_H˙s

(|x|>R)

.

Thanks to Lemma A.2 of Ref. 2, the definition of ϕ

R

and the uniform bound (1.5), we estimate

Z

∞ 0

m

^s

Z

|x|>R

∆

²

ϕ

_R

|u

_m

(t)|

²

dxdm

. k∆

²

ϕ

_R

k

^s_L∞

k∆ϕ

_R

k

^1−s_L∞

kuk

²_L2(|x|.R)

. R

^−2s

R

^2sc

ku(t)k

²_H˙sc(|x|

.R)

. R

^−2(s−sc)

. We thus obtain

M

_ϕ⁰

R

(t) ≤ 4dαE(u(t)) − 2(dα − 4s)ku(t)k

²_H˙s

+ CR

^−2(s−sc)

+ 2α

α + 2 Z

|x|>R

(2d − ∆ϕ

R

)|u(t)|

^α+2

dx.

Since k2d − ∆ϕ

R

k

L^∞

. 1, it remains to bound ku(t)k

^α+2_Lα+2(|x|>R)

. To do this, we make use of the argument of Ref. 25 (see also Ref. 12). Consider for A > 0 the annulus C = {A < |x| ≤ 2A}, we claim that for any > 0,

ku(t)k

^α+2_Lα+2(|x|>R)

≤ ku(t)k

²_H˙s

+ C()A

^−2(s−sc)

. (2.11) To show (2.11), we recall the radial Sobolev embedding (see e.g. Ref. 4):

sup

x6=0

|x|

^d2−β

|f (x)| ≤ C(d, β)kf k

H˙^β

,

for all radial functions f ∈ H ˙

^β

( R

^d

) with

¹₂

< β <

^d₂

. Thanks to radial Sobolev embedding and (2.8), we have

ku(t)k

^α+2_Lα+2(C)

.

sup

C

|u(t, x)|

^α

ku(t)k

²_L2(C)

. A

⁻

(

^d₂−β

)

^α

ku(t)k

^α_H˙β(C)

ku(t)k

²_L2(C)

1

2 < β < d 2

. A

⁻

(

^d₂^−β

)

^α

ku(t)k

β s

H˙^s(C)

ku(t)k

¹⁻

β s

L²(C)

α

ku(t)k

²_L2(C)

1

2 < β < s < d 2

. A

⁻

(

^d₂−β

)

^α

ku(t)k

αβ s

H˙^s(C)

ku(t)k (

^1−βs

)

^α+2

L²(C)

. A

^−ϑ

ku(t)k

^αβ_˙^s

H^s(C)

, (2.12)

where

ϑ :=

d 2 − β

α −

1 − β

s

α + 2

s

c

. It is easy to check that

ϑ = 2(s − s

c

)

1 − αβ 2s

.

By our assumption α < 4s, we can choose

¹₂

< β < s so that ϑ > 0. We next apply the Young inequality to have for any > 0,

A

^−ϑ

ku(t)k

αβ s

H˙^s(C)

. ku(t)k

²_H˙s(C)

+ C()A

^{−2s−αβ2sϑ}

= ku(t)k

²_H˙s(C)

+ C()A

^−2(s−sc)

. This combined with (2.12) prove (2.11). We now write

Z

|x|>R

|u(t)|

^α+2

dx =

∞

X

j=0

Z

2^jR<|x|≤2^j+1R

|u(t)|

^α+2

dx, and apply (2.11) with A = 2

^j

R to get

Z

|x|>R

|u(t)|

^α+2

dx ≤

∞

X

j=0

ku(t)k

²_{H˙s(2jR<|x|≤2j+1R)}

+ C()

∞

X

j=0

(2

^j

R)

^−2(s−sc)

≤ ku(t)k

²_H˙s

(|x|>R)

+ C()R

^−2(s−sc)

. This shows that for any > 0,

ku(t)k

^α+2_Lα+2(|x|>R)

≤ ku(t)k

²_H˙s(|x|>R)

+ C()R

^−2(s−sc)

, and hence

M

_ϕ⁰_R

(t) ≤ 4dαE(u(t))− 2(dα −4s)ku(t)k

²_H˙s

+ O

R

^−2(s−sc)

+ ku(t)k

²_H˙s

+ C()R

^−2(s−sc)

. By the conservation of energy with E(u

₀

) < 0 and the fact dα > 4s, we take > 0 small enough and R > 0 large enough to obtain

M

_ϕ⁰

R

(t) ≤ 2dαE(u

₀

) − δku(t)k

²_H˙s

, (2.13) where δ := dα − 4s > 0. We now follow the argument of Ref. 2. Since E(u

0

) < 0, we learn from (2.13) that M

_ϕ⁰_R

(t) ≤ −c for c > 0. From this, we conclude that M

ϕ_R

(t) < 0 for all t > t

1

for some sufficiently large time t

1

1. Taking integration over [t

1

, t], we have

M

ϕ_R

(t) ≤ −δ Z

t

t1

ku(τ)k

²_H˙s

dτ ≤ 0, ∀t ≥ t

1

. (2.14) We have from (2.9) and the assumption (1.5) that

|M

_ϕR

(t)| ≤ C(χ, R) ku(t)k

^1s_˙

H^s

+ ku(t)k

^2s1_˙

H^s

. (2.15)

We also have

ku(t)k

H˙^s

& 1, ∀t ≥ 0. (2.16) Indeed, suppose it is not true. Then there exists a sequence (t

n

)

n

⊂ [0, +∞) such that ku(t

n

)k

H˙^s

→ 0 as n → ∞. Thanks to the Gagliardo-Nirenberg inequality (1.6) and the assumption (1.5), we see that ku(t

n

)k

_Lα+2

→ 0. We thus get E(u(t

n

)) → 0, which is a contradiction to E(u(t)) = E(u

₀

) < 0. This shows (2.16). Combining (2.15) and (2.16), we obtain

|M

_ϕR

(t)| ≤ C(χ, R)ku(t)k

^1s_˙

H^s

. (2.17)

Therefore, (2.14) and (2.17) yield M

ϕ_R

(t) ≤ C(χ, R)

Z

t t1

|M

ϕ_R

(τ)|

^2s

dτ, ∀t ≥ t

1

. By nonlinear integral inequality, we get

M

ϕ_R

(t) . C(χ, R)|t − t

∗

|

^1−2s

,

for s > 1/2 with some t

_∗

< +∞. Therefore, M

ϕ_R

(t) → −∞ as t ↑ t

_∗

. Hence the solution

cannot exist for all times t ≥ 0. The proof is complete.

E. Profile decomposition

In this subsection, we recall the profile decomposition for bounded sequences in ˙ H

^sc

∩ H ˙

^s

. Theorem 2.7 (Profile decomposition). Let d ≥ 1, 0 < s < 1 and

^4s_d

< α < 2

^?

, where

2

^?

:=

4s

d−2s

if d > 2s,

∞ if d ≤ 2s. (2.18)

Let (v

n

)

n≥1

be a bounded sequence in H ˙

^sc

∩ H ˙

^s

. Then there exist a subsequence still denoted (v

n

)

_n≥1

, a family (x

^j_n

)

_j≥1

of sequences in R

^d

and a sequence (V

^j

)

_j≥1

of functions in H ˙

^sc

∩ H ˙

^s

such that

• for every k 6= j,

|x

^k_n

− x

^j_n

| → ∞, as n → ∞, (2.19)

• for every l ≥ 1 and every x ∈ R

^d

, v

_n

(x) =

l

X

j=1

V

^j

(x − x

^j_n

) + v

_n^l

(x),

with

lim sup

n→∞

kv

_n^l

k

L^q

→ 0, as l → ∞, (2.20) for every q ∈ (α

_c

, 2 + 2

^?

), where α

_c

is given in (1.3). Moreover,

kv

n

k

²_H˙sc

=

l

X

j=1

kV

^j

k

²_H˙sc

+ kv

^l_n

k

²_H˙sc

+ o

n

(1), (2.21)

kv

_n

k

²_H˙s

=

l

X

j=1

kV

^j

k

²_H˙s

+ kv

_n^l

k

²_H˙s

+ o

_n

(1), (2.22) as n → ∞.

Remark 2.8. In the case s

c

= 0 or α =

^4s_d

, Theorem 2.7 is exactly Theorem 3.1 in Ref.

11 due to the fact ˙ H

⁰

= L

²

and L

²

∩ H ˙

^s

= H

^s

.

Proof of Theorem 2.7. The proof is based on the argument of Ref. 20 (see also Refs. 12 and 18). For reader’s convenience, we give some details. Since ˙ H

^sc

∩ H ˙

^s

is a Hilbert space, we denote Ω(v

n

) the set of functions obtained as weak limits of sequences of the translated v

n

(· + x

n

) with (x

n

)

_n≥1

a sequence in R

^d

. Set

η(v

n

) := sup{kvk

H˙^sc

+ kvk

H˙^s

: v ∈ Ω(v

n

)}.

Clearly,

η(v

_n

) ≤ lim sup

n→∞

kv

_n

k

H˙^sc

+ kv

_n

k

H˙^s

.

We will show that there exist a sequence (V

^j

)

_j≥1

of Ω(v

n

) and a family (x

^j_n

)

_j≥1

of sequences in R

^d

such that for every k 6= j,

|x

^k_n

− x

^j_n

| → ∞,

as n → ∞ and up to a subsequence, we can write for every l ≥ 1 and every x ∈ R

^d

, v

n

(x) =

l

X

j=1

V

^j

(x − x

^j_n

) + v

^l_n

(x),

with η(v

^l_n

) → 0 as l → ∞. Moreover, (2.21) and (2.22) hold as n → ∞.

Indeed, if η(v

n

) = 0, then we take V

^j

= 0 for all j ≥ 1 and the proof is done. Otherwise we choose V

¹

∈ Ω(v

_n

) such that

kV

¹

k

H˙^sc

+ kV

¹

k

H˙^s

≥ 1

2 η(v

n

) > 0.

By definition, there exists a sequence (x

¹_n

)

_n≥1

in R

^d

such that up to a subsequence, v

n

(· + x

¹_n

) * V

¹

weakly in ˙ H

^sc

∩ H ˙

^s

.

Set v

_n¹

(x) := v

n

(x) − V

¹

(x − x

¹_n

). It follows that v

¹_n

(· + x

¹_n

) * 0 weakly in ˙ H

^sc

∩ H ˙

^s

and thus

kv

n

k

²_H˙sc

= kV

¹

k

²_H˙sc

+ kv

_n¹

k

²_H˙sc

+ o

n

(1), kv

n

k

²_H˙s

= kV

¹

k

²_H˙s

+ kv

¹_n

k

²_H˙s

+ o

_n

(1),

as n → ∞. We next replace (v

n

)

n≥1

by (v

_n¹

)

n≥1

and repeat the same argument. If η(v

_n¹

) = 0, then we take V

^j

= 0 for all j ≥ 2 and the proof is done. Otherwise there exist V

²

∈ Ω(v

¹_n

) and a sequence (x

²_n

)

_n≥1

in R

^d

such that

kV

²

k

H˙^sc

+ kV

²

k

H˙^s

≥ 1

2 η(v

¹_n

) > 0, and

v

_n¹

(· + x

²_n

) * V

²

weakly in ˙ H

^sc

∩ H ˙

^s

.

Set v

²_n

(x) := v

_n¹

(x) − V

²

(x − x

²_n

). It follows that v

²_n

(· + x

²_n

) * 0 weakly in ˙ H

^sc

∩ H ˙

^s

and kv

_n¹

k

²_H˙sc

= kV

²

k

²_H˙sc

+ kv

_n²

k

²_H˙sc

+ o

_n

(1),

kv

_n¹

k

²_H˙s

= kV

²

k

²_H˙s

+ kv

²_n

k

²_H˙s

+ o

n

(1), as n → ∞. We now show that

|x

¹_n

− x

²_n

| → ∞,

as n → ∞. Indeed, if it is not true, then up to a subsequence, x

¹_n

− x

²_n

→ x

₀

as n → ∞ for some x

0

∈ R

^d

. Rewriting

v

¹_n

(x + x

²_n

) = v

_n¹

(x + (x

²_n

− x

¹_n

) + x

¹_n

),

and using the fact v

_n¹

(· + x

¹_n

) converges weakly to 0, we see that V

²

= 0. This implies that η(v

¹_n

) = 0, which is a contradiction. An argument of iteration and orthogonal extraction allows us to construct the family (x

^j_n

)

j≥1

of sequences in R

^d

and the sequence (V

^j

)

j≥1

of functions in ˙ H

^sc

∩ H ˙

^s

satisfying the claim above. Moreover, the convergence of the series P

∞

j≥1

kV

^j

k

²_H˙sc

+ kV

^j

k

²_H˙s

implies that

kV

^j

k

²_H˙sc

+ kV

^j

k

²_H˙s

→ 0, as j → ∞.

By construction,

η(v

^j_n

) ≤ 2 kV

^j+1

k

H˙^sc

+ kV

^j+1

k

H˙^s

,

which shows that η(v

^j_n

) → 0 as j → ∞. It remains to show (2.20). To this end, we introduce for R > 1 a function φ

R

∈ S satisfying ˆ φ

R

: R

^d

→ [0, 1] and

φ ˆ

R

(ξ) =

1 if 1/R ≤ |ξ| ≤ R, 0 if |ξ| ≤ 1/2R ∨ |ξ| ≥ 2R.

We write

v

_n^l

= φ

R

∗ v

_n^l

+ (δ − φ

R

) ∗ v

_n^l

,

where δ is the Dirac function and ∗ is the convolution operator. Let q ∈ (α

_c

, 2 + 2

^?

) be fixed. By Sobolev embedding and the Plancherel formula,

k(δ − φ

_R

) ∗ v

^l_n

k

_Lq

. k(δ − φ

_R

) ∗ v

_n^l

k

H˙^β

. Z

|ξ|

^2β

|(1 − φ ˆ

_R

(ξ))ˆ v

^l_n

(ξ)|

²

dξ

1/2

. Z

|ξ|≤1/R

|ξ|

^2β

|ˆ v

^l_n

(ξ)|

²

dξ

^1/2

+ Z

|ξ|≥R

|ξ|

^2β

|ˆ v

^l_n

(ξ)|

²

dξ

^1/2

. R

^sc−β

kv

^l_n

k

H˙^sc

+ R

^β−s

kv

_n^l

k

H˙^s

,

where β =

^d₂

−

^d_q

∈ (s

c

, s). Besides, the H¨ older interpolation inequality yields kφ

R

∗ v

_n^l

k

L^q

. kφ

R

∗ v

_n^l

k

αc q

L^αc

kφ

R

∗ v

_n^l

k

¹⁻

αc q

L^∞

. kv

_n^l

k

αc q

H˙^sc

kφ

_R

∗ v

_n^l

k

¹⁻

αc q

L^∞

. Observe that

lim sup

n→∞

kφ

R

∗ v

^l_n

k

L^∞

= sup

x_n

lim sup

n→∞

|φ

R

∗ v

^l_n

(x

_n

)|.

By the definition of Ω(v

_n^l

), we see that lim sup

n→∞

kφ

R

∗ v

^l_n

k

L^∞

≤ sup n Z

φ

R

(−x)v(x)dx

: v ∈ Ω(v

_n^l

) o . The Plancherel formula then implies

Z

φ

R

(−x)v(x)dx =

Z

φ ˆ

R

(ξ)ˆ v(ξ)dξ

. kkξ|

^−sc

φ ˆ

R

k

_L2

k|ξ|

^sc

vk ˆ

_L2

. R

^d2−sc

k φ ˆ

_R

k

H˙^−sc

kvk

H˙^sc

. R

^2sα

η(v

^l_n

).

Thus, for every l ≥ 1, lim sup

n→∞

kv

_n^l

k

L^q

. lim sup

n→∞

k(δ − φ

R

) ∗ v

^l_n

k

L^q

+ lim sup

n→∞

kφ

R

∗ v

_n^l

k

L^q

. R

^sc−β

kv

_n^l

k

H˙^sc

+ R

^β−s

kv

^l_n

k

H˙^s

+ kv

^l_n

k

αc q

H˙^sc

h

R

^2sα

η(v

_n^l

) i(

1−^α_q^c

) . Choosing R =

η(v

^l_n

)

⁻¹

_2s^α−

for some > 0 small enough, we learn that lim sup

n→∞

kv

_n^l

k

L^q

. η(v

^l_n

)

^(β−sc)

(

_2s^α−

) kv

^l_n

k

H˙^sc

+ η(v

^l_n

)

^(s−β)

(

_2s^α−

) kv

^l_n

k

H˙^s

+η(v

_n^l

)

^2sα

(

^1−αcq

) kv

_n^l

k

αc q

H˙^sc

.

Letting l → ∞ and using the uniform boundedness of (v

_n^l

)

_l≥1

in ˙ H

^sc

∩ H ˙

^s

together with the fact that η(v

_n^l

) → 0 as l → ∞, we obtain

lim sup

n→∞

kv

_n^l

k

L^q

→ 0, as l → ∞.

This completes the proof of Theorem 2.7.