This study explored the time asymptotic behavior of the Schrödinger equation with an inhomogeneous energy-critical nonlinearity. The approach follows the concentration-compactness method due to Kenig and Merle. To address the primary challenge posed by the singular inhomogeneous term, we utilized Caffarelli-Kohn-Nirenberg weighted inequalities. This work notably expanded the existing literature by applying these techniques to higher spatial dimensions without requiring any spherically symmetric assumption.
Citation: Saleh Almuthaybiri, Radhia Ghanmi, Tarek Saanouni. On the nonlinear Schrödinger equation with critical source term: global well-posedness, scattering and finite time blowup[J]. AIMS Mathematics, 2024, 9(11): 30230-30262. doi: 10.3934/math.20241460
[1] | Saleh Almuthaybiri, Radhia Ghanmi, Tarek Saanouni . Correction: On the nonlinear Schrödinger equation with critical source term: global well-posedness, scattering and finite time blowup. AIMS Mathematics, 2025, 10(2): 2413-2414. doi: 10.3934/math.2025112 |
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[5] | Saleh Almuthaybiri, Tarek Saanouni . On coupled non-linear Schrödinger systems with singular source term. AIMS Mathematics, 2024, 9(10): 27871-27895. doi: 10.3934/math.20241353 |
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[7] | Lingfei Li, Yingying Xie, Yongsheng Yan, Xiaoqiang Ma . Going beyond the threshold: Blowup criteria with arbitrary large energy in trapped quantum gases. AIMS Mathematics, 2022, 7(6): 9957-9975. doi: 10.3934/math.2022555 |
[8] | Xinyi Zhang, Jian Zhang . On Schrödinger-Poisson equations with a critical nonlocal term. AIMS Mathematics, 2024, 9(5): 11122-11138. doi: 10.3934/math.2024545 |
[9] | Khaled Kefi, Nasser S. Albalawi . Three weak solutions for degenerate weighted quasilinear elliptic equations with indefinite weights and variable exponents. AIMS Mathematics, 2025, 10(2): 4492-4503. doi: 10.3934/math.2025207 |
[10] | Salim A. Messaoudi, Mohammad M. Al-Gharabli, Adel M. Al-Mahdi, Mohammed A. Al-Osta . A coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities: Existence, uniqueness, blow-up and a large-time asymptotic behavior. AIMS Mathematics, 2023, 8(4): 7933-7966. doi: 10.3934/math.2023400 |
This study explored the time asymptotic behavior of the Schrödinger equation with an inhomogeneous energy-critical nonlinearity. The approach follows the concentration-compactness method due to Kenig and Merle. To address the primary challenge posed by the singular inhomogeneous term, we utilized Caffarelli-Kohn-Nirenberg weighted inequalities. This work notably expanded the existing literature by applying these techniques to higher spatial dimensions without requiring any spherically symmetric assumption.
We consider the Cauchy problem for the Schrödinger equation with an inhomogeneous nonlinearity
i∂∂tu+Δu+|x|−τ|u|p−1u=0; | (INLS) |
u|t=0=u0. | (1.1) |
In this context, the wave function u is a complex-valued function defined on the variable (t,x)∈R×RN. Additionally, the singular inhomogeneous term is given by |⋅|−τ, where τ>0.
The inhomogeneous nonlinear equation of Schrödinger type describes beam propagation in nonlinear optics and plasma physics. In fact, stable high-power propagation can be realized in a plasma by introducing a preliminary laser beam that forms a channel with reduced electron density, thereby decreasing the nonlinearity within that channel [1,2,3]. In the context of the optical nonlinear Schrödinger equation, light energy can be confined, enabling the transmission of complex structured beams and solitons [4,5]. Additional references include [6,7,8].
The well-posedness of the inhomogeneous nonlinear Schrödinger equation (INLS) has been explored by numerous authors. The existence of energy subcritical solutions was first established in [9]. This result was later revisited in [10], where solutions in Strichartz spaces were examined under additional restrictions for N=2,3. The dichotomy between global existence and scattering versus finite time blowup below the ground state threshold was addressed in [11,12,13] using the concentration-compactness argument by Kenig and Merle [14]. This work was further developed in [15] employing the Dodson-Murphy method [16], and the spherically symmetric assumption was relaxed in [17]. Additional discussions on more general inhomogeneous terms can be found in [18,19]. The finite time blowup of solutions without radial or finite time variance assumptions was investigated in [20,21]. Recently, the Sobolev critical regime has also been considered, with local well-posedness studied in [22,23,24,25]. Scattering for spherically symmetric initial data was demonstrated in [26,27] in the three-dimensional case, while the radial assumption for this case was removed in [28]. A result indicating non-scattering was presented in [29]. For a numerical perspective, a quantitative analysis of solutions to the three-dimensional cubic nonlinear Schrödinger equation above the mass-energy threshold is provided in [33], which introduces a new blowup criterion and predicts the asymptotic behavior of solutions across various initial data classes, including modulated ground states, Gaussian, super-Gaussian, off-centered Gaussian, and oscillatory Gaussian, along with several conjectures regarding the scattering threshold.
The motivation of this note is to extend the findings of [26,27,28] to higher spatial dimensions and to eliminate the radial assumption. Specifically, the scattering threshold was demonstrated in [26,27] for three spatial dimensions. The novel contribution here is to establish the scattering threshold for N≥4 without assuming spherical symmetry. This indicates that every energy-critical solution to (INLS) asymptotically approaches a solution of the linear Schrödinger equation as t→∞. The methodology follows the roadmap laid out by Kenig and Merle in [14].
The remainder of the paper is organized as follows: Section 2 presents the main result along with some useful estimates. Section 3 provides auxiliary results. Section 4 is dedicated to proving global existence and scattering. Finally, Section 5 addresses the finite time blowup.
Here and henceforth, the Lebesgue and Sobolev spaces equipped with the standard norms are denoted by
Lr:=Lr(RN),˙H1:={f∈S′(RN),∇f∈L2},˙H1rd:={f∈˙H1,f(⋅)=f(|⋅|)};‖⋅‖r:=‖⋅‖Lr,‖⋅‖:=‖⋅‖2,‖⋅‖˙H1:=‖∇⋅‖. |
Finally, one denotes by (T−,T+) the maximal existence interval of an eventual energy solution to (INLS).
This section contains the main contribution of this note and some useful standard estimates.
Let us denote the free Schrödinger kernel:
eitΔu:=F−1(e−it|⋅|2Fu), | (2.1) |
where F is the Fourrier transform. Thanks to the Duhamel formula, solutions to (INLS) are fixed points of the integral function
f(u(t)):=eitΔu0−i∫t0ei(t−s)Δ(|x|−τ|u|p−1u)ds. | (2.2) |
Solutions of the problem (INLS), formally satisfy the conservation of the energy
E(u(t)):=∫RN|∇u(t,x)|2dx−21+p∫RN|x|−τ|u(t,x)|1+pdx=E(u0). | (2.3) |
If u resolves the equation (INLS), then so does the family uκ:=κ2−τp−1u(κ2⋅,κ⋅),κ>0. Moreover, there is only one invariant Sobolev norm under the above dilatation, precisely
‖uκ(t)‖˙Hsc=‖u(κ2t)‖˙Hsc,sc:=N2−2−τp−1. |
In the sequel, we will focus on the energy-critical regime
sc=1⇔p=pc:=1+2(2−τ)N−2,N≥3. | (2.4) |
We will consider the next assumption on the inhomogeneous term
0<τ<min{6−N2,4N}or2+NN<τ<2. | (2.5) |
Let us define the potential energy
P[u]:=∫RN|x|−τ|u|1+pdx. | (2.6) |
Take the associated ground state
φ(x):=(1+|x|2−τ(N−τ)(N−2))−N−22−τ. | (2.7) |
Thanks to [35, Theorem 4.3] and [34, Remark 2.1], one has
Δφ+|x|−τφp=0; | (2.8) |
1C∗:=inf0≠u∈˙H1‖∇u‖(P[u])11+p=‖∇φ‖(P[φ])11+p. | (2.9) |
Finally, we denote for short the Sobolev embedding exponent
2∗:=2NN−2,N≥3. | (2.10) |
From now on, we hide the time variable for simplicity, spreading it out only when necessary.
The main contribution of this note is the next dichotomy of global existence and scattering versus finite time blowup of energy critical solutions under the ground state threshold.
Theorem 2.1. Let N>3, τ satisfy (2.5), and p=pc. Let u0∈˙H1, satisfying:
E(u0)<E(φ). | (2.11) |
Then,
1) The solution of (INLS) is global and scatters if
‖∇u0‖<‖∇φ‖. | (2.12) |
2) The solution of (INLS) blows up in finite or infinite time if u0∈L2 and
‖∇u0‖>‖∇φ‖. | (2.13) |
In view of the results stated in the above theorem, some comments are in order.
● φ denotes a ground state solution to (2.7);
● This work complements [26,27,28] to higher space dimensions N>3;
● The local existence obtained here complements [22,23,24,25], where the data was supposed to be in H1;
● Due to the use of fixed point argument in the small data theory, for τ<1, one needs the condition p≥2. This gives the restriction 0<2τ≤6−N. So one assumes that N≤5. Moreover, the use of the Sobolev embedding ˙H1↪L2NN−2 restricts the space dimension to N∈{3,4,5};
● For τ<1, the condition τ<4N is because one needs the inequality p−1−τ>0 in the local theory;
●
min{6−N2,4N}={43,ifN=3,1,ifN=4,12,ifN=5; |
● The condition τ<6−N2 doesn't appear in [26] because only N=3 is treated and so 0<τ<min{6−N2,4N} reads 0<τ<43;
● In the local theory, for τ>1, we use some weighted Strichartz spaces in the spirit of [27]. The choice of γ=(−2+N2)− done in [27] is not possible for N≥4 because of the necessary condition γ<1. So, the proof is different and we get the extra restriction τ>2+NN;
● The blowup in finite or infinite time means that sup[0,T+)‖∇u(t)‖=∞;
● The radial assumption is not needed for the blowup;
● If one assumes that xu0∈L2 or u0 radial, the finite time blowup holds;
● This work complements [14] to the inhomogeneous case.
By contradiction, if the first part of Theorem 2.1 fails, then there is a minimal non-scattering solution under the ground state threshold which possesses certain compactness properties as follows.
Proposition 2.1. If the first part of Theorem 2.1 fails, there exists a maximal solution to (INLS), denoted by u∈C([0,T+),˙H1) and a frequency scale function λ:[0,T+)↦R+, such that inft∈[0,T+)λ(t)≥1, and
supt∈[0,T+)‖∇u(t)‖<‖∇φ‖; | (2.14) |
‖u‖S(0,T+)=∞; | (2.15) |
{1λ(t)N−22u(t,⋅λ(t)),t∈[0,T+)}is pre-compact in˙H1. | (2.16) |
To complete the proof of the first part of Theorem 2.1, one proves that the type of solution appearing in the statement of Proposition 2.1 cannot exist. This is achieved in Subsection 4.2.
Remark 2.1. In Proposition 2.1, there is no moving spatial center x(t) in the parametrization of the minimal non-scattering solution. Indeed, thanks to Proposition 4.1, the profiles with |xn|λn→∞ correspond to scattering solutions. By arguments in [36], we can arrange the frequency scale function to be bounded below.
Remark 2.2. Proposition 2.1 is an adaptation of [28, Theorem 1.2], and the idea of the proof is somehow similar. Indeed, we aim to generalize [28, Theorem 1.2] for higher space dimensions and for more general inhomogeneous term, namely N≥3 and b satisfying (2.5) rather than N=3 and τ=1.
For the reader's convenience, we recall some known and useful tools which play an important role in the proof of the main result. To start, we recall the homogeneous Sobolev embedding [37, Theorem 1.38], for N≥3,
‖u‖2∗≤CN‖∇u‖,for allu∈˙H1. | (2.17) |
The following Caffarelli-Kohn-Nirenberg weighted interpolation inequalities [38,39], will be useful.
Lemma 2.1. Let N≥1, 1<p≤q<∞, and −Nq<b≤a<Nq′. Assume that a−b−1=N(1q−1p). Then,
‖|⋅|bf‖q≤C‖|⋅|a∇f‖p. |
Recall the associated Bernstein estimates to the standard Littlewood–Paley projections PM, see [40, Subsection 11.2],
‖|∇|sPMf‖r≃Ms‖PMf‖r,for all1≤r≤∞; | (2.18) |
‖PMf‖r1≃MN(1r2−1r1)‖PMf‖r2,for all1≤r2≤r1≤∞. | (2.19) |
The next refined Fatou argument [41, Lemma 11.3] will be useful.
Lemma 2.2. Let a functional sequence satisfy:
lim supn→∞‖fn‖r<∞andfn→f almost everywhere onRN. |
Then,
limn∫RN(|fn|r−|fn−f|r−|f|r)dx=0. | (2.20) |
The next linear profile decomposition for bounded radial sequences in ˙H1 is a key tool for the scattering proof [41,42].
Proposition 2.2. Take (un) as a bounded sequence in ˙H1. Then, for any M∈N, there exist a subsequence denoted also by (un) and
1) For any 1≤j≤M, a profile ψj∈˙H1;
2) For any 1≤j≤M, a sequence (tjn,λjn,xjn)∈R×R+×RN satisfying
tjn≡0ortjn→∞andxjn≡0or|xjn|→∞, | (2.21) |
and for 1≤i≠j≤M and n→∞,
log(λjnλkn+tjn(λjn)2−tkn(λkn)2λjnλkn+|xjn−xkn|λjnλkn)→∞; | (2.22) |
3) A sequence of remainders WMn∈˙H1, such that
un=M∑j=1fjn(eitjnΔψj)+WMn:=1(λjn)N−22M∑j=1[eitjnΔψj](⋅−xjnλjn)+WMn. | (2.23) |
Moreover,
limM→∞[lim supn→∞‖∇ei⋅ΔWMn‖S(R)]=0. | (2.24) |
For fixed M, one has the next Pythagorean expansions
‖∇un‖2=M∑j=1‖∇ψj‖2+‖∇WMn‖2+on(1);E(un)=M∑j=1E(eitjnΔψj)+E(WMn)+on(1). |
Now, let us collect some standard estimates related to the Schrödinger problem.
Definition 2.1
A pair (q,r) is said admissible if q,r≥2, (q,r,N)≠(2,∞,2) and N(12−1r)=2q. One says for short (q,r)∈Λ. Let I⊂R be an interval, and one denotes the Strichartz space by
Ω(I):=⋂(q,r)∈ΛLq(I,Lr). |
Let us now state some Strichartz estimates [43].
Proposition 2.3. Let N≥2 and u0∈L2. Then,
1) ‖ei⋅Δu0‖Ω(I)≲‖u0‖;
2) ‖u−eitΔu0‖Ω(I)≲inf(˜q,˜r)∈Λ‖i∂∂tu+Δu‖L˜q′(I,L˜r′).
Let us give some Strichartz estimates adapted to the weighted Lebesgue spaces [24].
Definition 2.2. Take N≥3 and 0≤γ<1. A pair of real numbers (q,r) is γ-admissible if
{N(12−1r)+γ=2q;γ2<1q≤12;γ2≤1r<12. |
Take the set Λγ:={(q,r), γ-admissible } and the weighted Strichartz norm
‖⋅‖Sγ(I):=sup(q,r)∈Λγ‖⋅‖Lq(I,Lr(|x|−rγ)). |
The next Strichartz estimate was proved in [24, Proposition 1.5].
Proposition 2.4. Let N≥3 and 0≤γ,˜γ<1, and a time slab I⊂R. Take (q,r)∈Λγ and (˜q,˜r)∈Λ˜γ. Then,
‖ei⋅Δf‖Lq(I,Lr(|x|−rγ))≲‖f‖; | (2.25) |
‖∫⋅0ei(⋅−τ)Δh(τ,⋅)dτ‖Lq(I,Lr(|x|−rγ))≲‖h‖L˜q′(I,L˜r′(|x|˜γ˜r′)),forq>˜q′. | (2.26) |
Define the variance potential
Vψ:=∫RNψ(x)|u(⋅,x)|2dx, | (2.27) |
where ψ:RN→R is a smooth function. Let also the Morawetz action be
Mψ=2ℑ∫RNˉu(∇ψ⋅∇u)dx:=2ℑ∫RNˉu(ψjuj)dx, | (2.28) |
where here and in the sequel, the repeated index are summed. Let us give a Morawetz type estimate [45, Lemma 4.5].
Proposition 2.5. Take u∈C([0,T],˙H1) as the local solution to (INLS). Let ψ:RN→R be a smooth function. Then, the following equality holds on [0,T],
V″ψ[u]=M′ψ[u]=4∫RN∂l∂kψℜ(∂ku∂lˉu)dx−∫RNΔ2ψ|u|2dx−2pc−2pc∫RNΔψ|x|−τ|u|pcdx+4pc∫RN∇ψ⋅∇(|x|−τ)|u|pcdx. |
This section proves the profile decomposition, a local theory and a variational analysis.
In this subsection, one proves Proposition 2.2. Taking account of [41,42], it is sufficient to prove that
P[un]=M∑j=1P[eitjnΔψj]+P[WMn]+on(1). |
For this, denoting the sequence (˜ψ1l)n:=eit1nΔψ1, one needs to establish that
limn∫RN|x|−τ(|un|1+p−|un−(˜ψ1l)n|1+p−|(˜ψ1l)n|1+p)dx:=limnIn=0. |
One of the two next scenarios happens. The first one is t1n→∞. The second one is t1n≡0. Take the first case. Recall two useful inequalities. The first one [35] reads, for any m≥2,
||x|m−|x−y|m−|y|m|≤m2m−1(|x−y|m−1|y|+|x−y||y|m−1). | (3.1) |
The second one follows from Lemma 2.1,
‖|x|−1f‖r≲‖∇f‖r,for all1<r<N. | (3.2) |
Now, by (3.2) and (3.1), via Sobolev embeddings and Hölder estimate, one writes
In≤(1+p)2p∫RN|x|−τ(|un−(˜ψ1l)n|p|(˜ψ1l)n|+|un−(˜ψ1l)n||(˜ψ1l)n|p)dx≲∫RN((|x|−1|un−(˜ψ1l)n|)τ|un−(˜ψ1l)n|p−τ|(˜ψ1l)n|+(|x|−1|(˜ψ1l)n|)τ|un−(˜ψ1l)n||(˜ψ1l)n|p−τ)dx≲‖|x|−1(un−(˜ψ1l)n)‖τ‖un−(˜ψ1l)n‖p−τ2∗‖(˜ψ1l)n‖2∗+‖|x|−1(˜ψ1l)n‖τ‖un−(˜ψ1l)n‖2∗‖(˜ψ1l)n‖p−τ2∗≲‖un−(˜ψ1l)n‖τ˙H1‖un−(˜ψ1l)n‖p−τ2∗‖(˜ψ1l)n‖2∗+‖(˜ψ1l)n‖τ˙H1‖un−(˜ψ1l)n‖2∗‖(˜ψ1l)n‖p−τ2∗. | (3.3) |
Using the free Schrödinger operator dispersive estimate [44], ‖eitΔ⋅‖r≤CtN(12−1r)‖⋅‖r′, for all r≥2, one gets limnIn=0. In the second case, the claim follows by (3.3) via (2.24) and (2.20). This finishes the proof.
One discusses two cases depending on the inhomogeneous index.
0<τ<min{6−N2,4N}. Here and hereafter, one takes, for an interval I⊂R, the spaces
S(I):=L2(2+N)N−2(I×RN); | (3.4) |
W(I):=L2(2+N)N−2(I,L2N(2+N)4+N2); | (3.5) |
∇W(I):={u:∇xu∈W(I)}. | (3.6) |
This choice implies in particular that
∇W(I)↪S(I); | (3.7) |
(2(2+N)N−2,2N(2+N)4+N2)∈Λ. | (3.8) |
Take also the spaces
Wτ(I):=L2(2+N)(1+τ)N(1+τ)−2(I,L2N(2+N)(1+τ)4+N2(1+τ)); | (3.9) |
∇Wτ(I):={u:∇xu∈Wτ(I)}. | (3.10) |
This choice implies in particular that
W0(I)=W(I); | (3.11) |
(2(2+N)(1+τ)N(1+τ)−2,2N(2+N)(1+τ)4+N2(1+τ))∈Λ. | (3.12) |
This subsection contains two parts.
● Global solution for small data. The problem (INLS) has a local solution in the energy space which is global for small data.
Proposition 3.1. Let 0∈I a real interval and u0∈˙H1. Then, there exists δ>0 such that if ‖ei⋅Δu0‖S(I)≤δ, then, there is a unique solution to (INLS) in C(I,˙H1). Moreover, ‖u‖S(I)≤2δ and ‖∇u‖Wτ(I)∩W(I)<∞.
Proof. We proceed with a fixed point argument. Take the Duhamel integral function (2.2). Let also, for a,b>0, the space
Xa,b:={u∈C(I,˙H1),‖u‖L∞(I,˙H1)≤2A,‖u‖S(I)≤a,‖∇u‖Wτ(I)∩W(I)≤b}, |
endowed with the complete distance d(u,v):=‖u−v‖S(I). By the Strichartz estimate, one writes for u,v∈Xa,b and w:=u−v,
d(f(u),f(v))≲‖∇[|x|−τ(|u|p−1u−|v|p−1v)]‖L2(I,L2N2+N)≲‖|x|−τ−1(|u|p−1+|v|p−1)w‖L2(I,L2N2+N)+‖|x|−τ|u|p−1∇w‖L2(I,L2N2+N)+‖|x|−τ(|u|p−2+|v|p−2)∇vw‖L2(I,L2N2+N):=(I)+(II)+(III). |
Thus, by (3.2), one gets
(II)≲‖(|x|−1u)τ|u|p−1−τ∇w‖L2(I,L2N2+N)≲‖|x|−1u‖τWτ(I)‖u‖p−1−τS(I)‖∇w‖Wτ(I)≲‖∇u‖τWτ(I)‖u‖p−1−τS(I)‖∇w‖Wτ(I)≲bτap−1−τd(u,v). | (3.13) |
Here, one used the Hölder estimate via the identities
12=p−1−τ2(2+N)N−2+1+τ2(2+N)(1+τ)N(1+τ)−2; | (3.14) |
2+N2N=p−1−τ2(2+N)N−2+1+τ2N(2+N)(1+τ)4+N2(1+τ). | (3.15) |
Moreover, in order to estimate (I), it is sufficient to consider the following term, by use of (3.2) and the Hölder estimate via (3.14)–(3.15),
(I)1:=‖|x|−τ−1|u|p−1w‖L2(I,L2N2+N)=‖(|x|−1u)τ|u|p−1−τ(|x|−1w)‖L2(I,L2N2+N)≲‖|x|−1u‖τWτ(I)‖u‖p−1−τS(I)‖|x|−1w‖Wτ(I)≲‖∇u‖τWτ(I)‖u‖p−1−τS(I)‖∇w‖Wτ(I)≲bτap−1−τd(u,v). | (3.16) |
Furthermore, in order to estimate (III), it is sufficient to consider the following term, by use of (3.2), Sobolev embeddings, and the Hölder estimate via (3.14)–(3.15),
(III)1:=‖|x|−τ|u|p−2∇vw‖L2(I,L2N2+N)=‖(|x|−1w)τ|u|p−2∇v|w|1−τ‖L2(I,L2N2+N)≲‖|x|−1w‖τWτ(I)‖u‖p−2S(I)‖∇v‖Wτ(I)‖w‖1−τS(I)≲‖u‖p−2S(I)‖∇v‖Wτ(I)‖w‖Wτ(I)≲bap−2d(u,v). | (3.17) |
Here, one takes the case τ≤1 and used the assumption pc≥2, which reads
τ≤6−N2. | (3.18) |
Now, if τ>1, one has p≥2 and p>1+τ, so, one writes
(III)1:=‖|x|−τ|u|p−2∇vw‖L2(I,L2N2+N)=‖|u|p−1−τ(|x|−1u)τ−1∇v(|x|−1w)‖L2(I,L2N2+N)≲‖|x|−1u‖τ−1Wτ(I)‖u‖p−1−τS(I)‖∇v‖Wτ(I)‖|x|−1w‖Wτ(I)≲‖∇u‖τ−1Wτ(I)‖u‖p−1−τS(I)‖∇v‖Wτ(I)‖∇w‖Wτ(I)≲bτap−1−τd(u,v). | (3.19) |
Now, regrouping the identities (3.13) to (3.19), it follows that
d(f(u),f(v))≤c(bτap−1−τ+bap−2)d(u,v). | (3.20) |
So, f is a contraction for small 0<a,b≪1. Next, one proves the stability f(Xa,b)⊂Xa,b. Taking v=0 in (3.20) via Strichartz estimates, one gets for the choice b:=2cA and 0<a≪1,
‖∇f(u)‖Wτ(I)∩W(I)≤c‖∇u0‖+d(f(u),f(0))≤cA+c(bτap−1−τ+bap−2)b | (3.21) |
≤[12+c((2cA)τap−1−τ+2cAap−2)]b | (3.22) |
<b. | (3.23) |
Now, by (3.2) via the Hölder estimate and Sobolev embedding via an absorption argument, one gets for δ:=a2≪1,
‖f(u)‖S(I)≤‖ei⋅Δu0‖S(I)+‖(|x|−1u)τ|u|p−1−τu‖L2(I,L2N2+N)≤δ+c‖|x|−1u‖τWτ(I)‖u‖p−1−τS(I)‖u‖Wτ(I)≤δ+c‖∇u‖τWτ(I)‖u‖p−1−τS(I)‖u‖Wτ(I)≤δ+cb1+τap−1−τ≤2δ=a. | (3.24) |
Finally, with Sobolev embeddings and arguing as in (3.23), for 0<a≪1, one gets
‖f(u)‖L∞(I,˙H1)≤A+c(bτap−1−τ+bap−2)b≤2A. | (3.25) |
The stability f(Xa,b)⊂Xa,b follows by (3.23)–(3.25). The proof is closed via a classical Picard argument.
● Long-time perturbation. The second part of this section deals with the next result.
Proposition 3.2. Let T>0 and I:=[0,T]. Take u∈C(I,˙H1) as a solution to (INLS) and ˜u∈L∞(I,˙H1), satisfying for some ϵ,A>0,
‖˜u‖L∞T(˙H1)∩S(I)≤A;i˜ut+Δ˜u+|x|−τ|˜u|p−1˜u=e; | (3.26) |
\max\Big\{\|\nabla e\|_{\Omega'(I)}, \|e^{{ \text i}\cdot{\Delta}} [u_0-\tilde u_0]\|_{W_{\tau}(I)}\Big\}\leq\epsilon. | (3.27) |
Then, there exists \epsilon_0: = \epsilon_0(A) , satisfying for any 0 < \epsilon < \epsilon_0 ,
\|u\|_{S(I)}\leq C(A). |
Proof. Taking w: = u-\tilde u and \mathcal N[u]: = |x|^{-{\tau}}|u|^{p-1}u , one gets
\begin{eqnarray*} i w_t+\Delta w & = &i {\frac\partial{\partial t}}u+\Delta u-(i{\tilde u}_t+\Delta\tilde u)\\ & = &\mathcal N[{\tilde u}]-\mathcal N[w+\tilde u]-e. \end{eqnarray*} |
Taking account of the Duhamel integral formula (2.2), one writes
\begin{eqnarray} w(t) & = &e^{{ \text i}(t-t_k)\Delta}w(t_k)+i\int_{t_k}^te^{{ \text i}(t-\tau)\Delta}\Big(\mathcal N[{\tilde u}]-\mathcal N[w+\tilde u]\Big)\, d\tau\\ &+&i\int_{t_k}^te^{{ \text i}(t-\tau)\Delta}e(\tau)\, d\tau. \end{eqnarray} | (3.28) |
Here, one picks a partition
\begin{align} t_0 = 0, \quad I: = \bigcup\limits_{0\leq k\leq K}[t_k, t_{1+k}): = \bigcup\limits_k I_k; \end{align} | (3.29) |
\begin{align} \|\nabla\tilde u\|_{W_{\tau}(t_k, t_{1+k})} < \eta\ll 1, \quad \mbox{for all}\quad 0\leq k\leq K. \end{align} | (3.30) |
Indeed, arguing as in the local theory with a Bootstrap argument and Sobolev embeddings via (3.13), (3.26) and (3.27), one has
\begin{eqnarray*} \|\nabla\tilde u\|_{W_{\tau}(t_k, t_{1+k})} &\lesssim&\|\tilde u(t_k)\|_{\dot H^1}+\|\nabla\mathcal N[\tilde u]\|_{L^2(I_k, L^\frac{2N}{2+N})}+\|\nabla e\|_{\Omega'(I_k)}\\ &\lesssim&A+\|\nabla \tilde u\|_{W_{\tau}(I_k)}^{1+{{\tau}}}\|\tilde u\|_{S(I_k)}^{p-1-{{\tau}}}+\epsilon\lesssim A. \end{eqnarray*} |
With a Picard fixed point argument and arguing as in the the local theory, one solves the previous integral equation in I_0 . So, with Proposition 3.1,
\|w\|_{S(I_0)}\leq2\epsilon\quad\mbox{and}\quad \|\nabla w\|_{W_{\tau}(I_0)}\leq C(\epsilon, A). |
Letting t = t_1 in the previous integral equality (3.28) and applying e^{{ \text i}(t-t_1)\Delta} , one gets
\begin{eqnarray*} e^{{ \text i}(t-t_1)\Delta}w(t_1) & = &e^{{ \text i}(t-t_0)\Delta}w(t_0)+i\int_{t_0}^{t_1}e^{{ \text i}(t-\tau)\Delta}\Big(\mathcal N[{\tilde u}]-\mathcal N[w+\tilde u]\Big)\, d\tau\\ &+&i\int_{t_0}^{t_1}e^{{ \text i}(t-\tau)\Delta}e(\tau)\, d\tau. \end{eqnarray*} |
Taking account of the proof of Proposition 3.1 via (3.27), one gets
\|e^{{ \text i}(t-t_1)\Delta}w(t_1)\|_{S(I_1)}\leq \|e^{{ \text i}(t-t_0)\Delta}w(t_0)\|_{S(I)}+2c\epsilon\leq c\epsilon+2c\epsilon. |
Iterating this process, it follows that
\|e^{{ \text i}(t-t_k)\Delta}w(t_k)\|_{S(I_k)}\leq\|e^{{ \text i}(t-t_0)\Delta}w(t_0)\|_{S(I)}+c2^k\epsilon\leq c2^{1+k}\epsilon. |
Now, applying Proposition 3.1 to the above Duhamel integral formula (3.28), we get
\begin{align} \|w\|_{S(I_k)}&\leq c\epsilon2^{2+k};\\ \|w\|_{S(I)}&\leq c\epsilon\sum\limits_{k = 0}^K2^{2+k}\leq 4c(-1+2^{1+K})\epsilon. \end{align} | (3.31) |
Finally, one ends the proof by the triangle inequality in (3.31) via (3.26),
\|u\|_{S(I)}\leq \|w\|_{S(I)}+\|\tilde u\|_{S(I)}\lesssim C(A). |
\frac{2+N}{N} < {\tau} < 2 .
● Global solution for small data. The problem (INLS) has a local solution in the energy space which is global for small data. We start with some notations.
Let 0 < \varepsilon\ll 1 , \gamma: = 1-\varepsilon , and the real numbers
\begin{align} r_0&: = \frac{2N}{N-2-\varepsilon} = \Big(\frac{2N}{N-2}\Big)^+ ; \end{align} | (3.32) |
\begin{align} \frac{1}{r_1}& = \frac{1}{r_0}+\frac{1}{N}\Longleftrightarrow r_1 = \frac{2N}{N-\varepsilon}; \end{align} | (3.33) |
\begin{align} (q_0&, r_1)\in\Lambda_\gamma \Longleftrightarrow q_0 = \frac{4}{2-\varepsilon}; \end{align} | (3.34) |
\begin{align} (q_2&, r_2)\in\Lambda_\gamma\quad\mbox{to be picked later}. \end{align} | (3.35) |
Here and hereafter, if I is a real interval, one takes the weighted Lebesgue spaces
\begin{align} S(I): = L^{q_0}\Big(I, L^{r_0}(|x|^{-r_0\gamma})\Big); \end{align} | (3.36) |
\begin{align} W(I): = L^{q_0}\Big(I, L^{r_1}(|x|^{-r_0\gamma})\Big); \end{align} | (3.37) |
\begin{align} M(I): = L^{q_2}\Big(I, L^{r_2}(|x|^{-r_2\gamma})\Big). \end{align} | (3.38) |
Since N\geq4 gives \frac{N}{r_0} > \gamma , by Lemma 2.1, one gets
\begin{align} \|\cdot\|_{S(I)}\lesssim\|\nabla\cdot\|_{W(I)}. \end{align} | (3.39) |
Let us state the small data global existence result.
Proposition 3.3. Let N\geq4 , {\frac{2+N}{N} < {\tau} < 2} , 0\in I be a real interval and u_0\in\dot H^1 . Then, there exists \delta > 0 such that if \|e^{{ \text i}\cdot{\Delta}}u_0\|_{S(I)}\leq\delta , there is a unique solution to (INLS) in C(I, \dot H^1) . Moreover, \|u\|_{S(I)}\leq2 \delta and \|\nabla u\|_{M(I)\cap W(I)} < \infty .
Proof. Let f be the Duhamel integral function given in (2.2). Let also, for a, b > 0 , the space
X_{a, b}: = \Big\{u\in C(I, \dot H^1), \quad \|u\|_{L^\infty(I, \dot H^1)}\leq 2A, \quad \|u\|_{S(I)}\leq a, \quad \|\nabla u\|_{M(I)\cap W(I)}\leq b\Big\}, |
endowed with the complete distance
\begin{align} d(u, v): = \|u-v\|_{S(I)}. \end{align} | (3.40) |
Take 0\leq\tilde \gamma < 1 and (\tilde q, \tilde r)\in \Lambda_{\tilde\gamma} to be picked later. Let u, v\in X_{a, b} and w: = u-v . By the Strichartz estimate in Proposition 2.4 via (3.39), one writes
\begin{align} d(f(u), f(v)) &\lesssim\Big\|\nabla\Big[|x|^{-{\tau}}(|u|^{p-1}u-|v|^{p-1}v)\Big]\Big\|_{L^{\tilde{q}'}\big(I, L^{\tilde{r}'}(|x|^{\tilde\gamma\tilde{r}'})\big)}\\ &\lesssim\||x|^{-{\tau}-1}(|u|^{p-1}+|v|^{p-1})w\|_{L^{\tilde{q}'}\big(I, L^{\tilde{r}'}(|x|^{\tilde\gamma\tilde{r}'})\big)}+\||x|^{-{\tau}}|u|^{p-1}\nabla w\|_{L^{\tilde{q}'}\big(I, L^{\tilde{r}'}(|x|^{\tilde\gamma\tilde{r}'})\big)}\\ &+\||x|^{-{\tau}}(|u|^{p-2}+|v|^{p-2})\nabla vw\|_{L^{\tilde{q}'}\big(I, L^{\tilde{r}'}(|x|^{\tilde\gamma\tilde{r}'})\big)}\\ &: = (I)+(II)+(III). \end{align} | (3.41) |
In order to estimate (I) , it is sufficient to consider the following term
\begin{eqnarray} (I)_1 &: = &\||x|^{-{\tau}-1}|u|^{p-1}w\|_{L^{\tilde{q}'}\big(I, L^{\tilde{r}'}(|x|^{\tilde\gamma\tilde{r}'})\big)}\\ & = &\||x|^{-{\tau}-1 +\tilde\gamma}|u|^{p-1}w\|_{L^{\tilde{q}'}\big(I, L^{\tilde{r}'}\big)} . \end{eqnarray} | (3.42) |
Using the Hölder estimate via Lemma 2.1 and (3.42), we write
\begin{align} (I)_1 & = \|(|x|^{-\gamma-1}u)^{p-1-\theta}(|x|^{-\gamma}u)^{\theta}|x|^{-\gamma-1}w\|_{L^{\tilde{q}'}\big(I, L^{\tilde{r}'}\big)}\\ &\lesssim\||x|^{-\gamma-1} u\|_{L^{q_2}(I, L^{r_2})}^{p-1-\theta}\||x|^{-\gamma}u\|_{L^{q_0}(I, L^{r_0})}^{\theta}\||x|^{-\gamma-1} w\|_{L^{q_2}(I, L^{r_2})}\\ &\lesssim\||x|^{-\gamma-1} w\|_{L^{q_2}L^{r_2}}^{p-\theta}\||x|^{-\gamma}u\|_{L^{q_0}L^{r_0}}^{\theta}\\ &\lesssim\|{\nabla} u\|_{M(I)}^{p-1-\theta}\|u\|_{S(I)}^{\theta}\|{\nabla} w\|_{M(I)}\\ &\lesssim b^{p-1-\theta} a^{\theta}d(u, v). \end{align} | (3.43) |
Here, one needs the identities
\begin{align} \frac1{\tilde{r}'}& = \frac{\theta}{r_0}+ \frac{p-\theta}{r_2}; \end{align} | (3.44) |
\begin{align} \frac1{\tilde{q}'}& = \frac{\theta}{q_0}+ \frac{p-\theta}{q_2}; \end{align} | (3.45) |
\begin{align} 0 < \theta& = \tilde\gamma +p(1+\gamma)-1-{\tau} < p{-1}, \end{align} | (3.46) |
with the inequalities
\begin{align} 0 < \gamma, \tilde\gamma < 1; \end{align} | (3.47) |
\frac\gamma2 < \frac1{q_0}\leq\frac12, \quad \frac\gamma2\leq\frac1{r_1} < \frac12; | (3.48) |
\begin{align} 0 < \frac{\tilde\gamma}2 < \frac1{\tilde q}\leq\frac12, \quad \frac{\tilde\gamma}2\leq\frac1{\tilde r} < \frac12. \end{align} | (3.49) |
Note that (3.44) with (3.46) gives (3.45). Now, one picks
\begin{align} {\frac{2+N}{N} < {\tau} < 2}\quad\mbox{and}\quad { 0 < \varepsilon < \frac{{\tau} N-N-2}{N-2}} . \end{align} | (3.50) |
The choice (3.50) implies that
\begin{align} {\tilde\gamma: = {\tau}-p-\varepsilon}\in(0, 1) . \end{align} | (3.51) |
Moreover, one picks
\begin{align} \theta: = \tilde\gamma +p(1+\gamma)-1-{\tau} = p(1-\varepsilon)-1-\varepsilon . \end{align} | (3.52) |
With (3.51) and (3.52), the inequalities in (3.46) are satisfied for 0 < \varepsilon\ll 1 . Now, let us choose
\begin{align} \frac1{\tilde{q}} & = \frac12\Big(2- \frac{2\theta}{q_0}- {\gamma(p-\theta)}\Big)^-\\ & = \frac12\Big(2- {\theta}[N(\frac12-\frac1{r_0}-\frac1N)+\gamma]- {\gamma(p-\theta)}\Big)^-. \end{align} | (3.53) |
By (3.32), (3.53) implies that
\begin{align} \frac1{\tilde{q}} & = \frac12\Big(2- {\theta}(\frac\varepsilon2+\gamma)- {\gamma(p-\theta)}\Big)^-\\ & = \frac12\Big(2- {\theta}\frac\varepsilon2- {\gamma p}\Big)^-\\ & = \frac12\Big(2- (1-\varepsilon)p-\frac\varepsilon2(p(1-\varepsilon)-1-\varepsilon)\Big)^-\\ & = \frac12\Big(2-p+\frac\varepsilon2(1+p)(1+\varepsilon)\Big)^-\\ &: = {\frac{2-p+\varepsilon'}2} . \end{align} | (3.54) |
Now, one checks the requested assumptions on the above choice. Compute
\begin{align} \frac1{r_1} & = \frac1N+\frac1{r_0}\\ & = \frac12-\frac\varepsilon{2N}\in[\frac\gamma2, \frac12) = [\frac12-\frac\varepsilon2, \frac12). \end{align} | (3.55) |
Moreover,
\begin{align} \frac2{q_0} & = N(\frac12-\frac1N-\frac1{r_0})+\gamma\\ & = 1-\frac{\varepsilon}{2}\in(\gamma, 1]. \end{align} | (3.56) |
Furthermore, by (3.51) and (3.54), because {{\tau} < 2} implies that 2-{p}\in({\tau}-p, 1] , it follows that for \varepsilon_0\to0 ,
\begin{align} (\tilde\gamma, 1]\ni\frac2{\tilde{q}} & = {2-p+\varepsilon'} . \end{align} | (3.57) |
Also, by taking \varepsilon, \varepsilon'\to0 , one gets \frac2{\tilde{r}} = {1-\frac2N(2-{\tau}+\varepsilon'+\varepsilon)} < 1 because {\tau} < 2 . So, we need to check that
\begin{align} \tilde\gamma\leq\frac2{\tilde{r}} &\Longleftrightarrow -\tilde\gamma+1-\frac2N\Big(2-p+\varepsilon'-\tilde\gamma\Big)\geq0\\ &\Longleftrightarrow (N-2){\tau} < N-4+Np. \end{align} | (3.58) |
The last line is clearly satisfied because
\begin{align} {\tau} < 2. \end{align} | (3.59) |
Let us see the couple (q_2, r_2) . By (3.54) and (3.34), for 0 < \varepsilon\ll 1 , one writes
\begin{align} (\gamma, 1]\ni\frac2{q_2} & = \frac1{p-\theta}\Big(\frac2{\tilde q'}-\frac{2\theta}{q_0}\Big) \end{align} | (3.60) |
\begin{align} & = \frac1{p-\theta}\Big(2-(2-p+\varepsilon')-\theta\frac{2-\varepsilon}2\Big)\\ & = 1-\frac1{p-\theta}\Big(\varepsilon'-\theta\frac{\varepsilon}2\Big)\\ &\Longleftrightarrow {\varepsilon\theta < 2\varepsilon' < \varepsilon(2p-\theta).} \end{align} | (3.61) |
The identity (3.61) is possible because of (3.52) and taking \varepsilon\ll 1 . Moreover, the equality \gamma+N(\frac12-\frac1{r_2}) = \frac2{q_2} via (3.61) implies that
\begin{align} [\gamma, 1)\ni\frac2{r_2} & = 1-\frac2N\Big(1-\frac1{p-\theta}\Big(\varepsilon'-\theta\frac{\varepsilon}2\Big)-\gamma\Big)\\ & = 1-\frac2N\Big(\varepsilon-\frac1{p-\theta}\Big(\varepsilon'-\theta\frac{\varepsilon}2\Big)\Big). \end{align} | (3.62) |
Now, the identity (3.62) is equivalent to
\begin{align} \varepsilon(p-\theta) > \varepsilon'-\frac12\varepsilon\theta; \end{align} | (3.63) |
\begin{align} \frac2N\Big(\varepsilon-\frac1{p-\theta}\Big(\varepsilon'-\theta\frac{\varepsilon}2\Big)\Big) < \varepsilon. \end{align} | (3.64) |
The condition (3.63) is satisfied by (3.61) and (3.64) is equivalent to
\begin{align} \varepsilon(2p-\theta-N(p-\theta)) < 2\varepsilon'. \end{align} | (3.65) |
This is clearly possible via (3.61) because \theta < p by (3.52),
Now, (3.55) and (3.56) imply that (q_0, r_1)\in\Lambda^\gamma . Also, (3.57) and (3.58) imply that (\tilde q, \tilde r)\in\Lambda^{\tilde\gamma} . Finally, (3.60) and (3.61) imply that (q_2, r_2)\in\Lambda^\gamma . Thus, (3.43) follows under the assumption
\begin{align} {\frac{2+N}N < {\tau} < 2}. \end{align} | (3.66) |
Now, using again the Hölder estimate and Lemma 2.1 via (3.41), one has
\begin{align} (II) & = \|(|x|^{-\gamma-1}u)^{p-1-\theta}(|x|^{-\gamma}u)^{\theta}(|x|^{-\gamma}\nabla w)\|_{L^{\tilde{q}'}\big(I, L^{\tilde{r}'}\big)}\\ &\lesssim\||x|^{-\gamma-1} u\|_{L^{q_2}(I, L^{r_2})}^{p-1-\theta}\||x|^{-\gamma}u\|_{L^{q_0}(I, L^{r_0})}^{\theta}\||x|^{-\gamma} \nabla w\|_{L^{q_2}(I, L^{r_2})}\\ &\lesssim\|\nabla u\|_{M(I)}^{p-1-\theta}\|u\|_{S(I)}^{\theta}\| \nabla w\|_{M(I)}\\ &\lesssim b^{p-1-\theta} a^{\theta}d(u, v). \end{align} | (3.67) |
In order to estimate (III) , it is sufficient to consider the following term, where using again the Hölder estimate and Lemma 2.1 via (3.41), one obtains
\begin{align} (III)_1 & = \| |x|^{-{\tau}+\tilde{\gamma}}|u|^{p-2}\nabla v w\|_{L^{\tilde{q}'}\big(I, L^{\tilde{r}'}\big)}\\ & = \|(|x|^{-\gamma}\nabla v)( |x|^{-\gamma} u)^{p-2}(|x|^{-\gamma-1}w)^{p-1-\theta}(|x|^{-\gamma}w)^{\theta-p+2}\|_{L^{\tilde{q}'}\big(I, L^{\tilde{r}'}\big)}\\ &\lesssim\||x|^{-\gamma}\nabla v\|_{L^{q_2}(I, L^{r_2})}\||x|^{-\gamma} u\|_{L^{q_0}(I, L^{r_0})}^{p-2}\||x|^{-\gamma-1}w\|_{L^{q_2}(I, L^{r_2})}^{p-1-\theta}\||x|^{-\gamma}w\|_{L^{q_0}(I, L^{r_0})}^{\theta-p+2}\\ &\lesssim\|\nabla v\|_{M(I)}\|\nabla w\|_{M(I)}^{p-1-\theta}\|u\|_{S(I)}^{p-2}\| w\|_{S(I)}^{\theta-p+2}\\ &\lesssim b^{p-1-\theta} a^{\theta}d(u, v). \end{align} | (3.68) |
Indeed, (3.46) implies that p-2 < \theta < p-1 . Now, regrouping the identities (3.41), (3.43), (3.67) and (3.68), it follows that
\begin{align} d(f(u), f(v)) &\lesssim b^{p-1-\theta}a^\theta d(u, v). \end{align} | (3.69) |
So, f is a contraction for small 0 < a\ll 1 . Next, one proves the stability f(X_{a, b})\subset X_{a, b} . Taking v = 0 in (3.69) via Strichartz estimates, one gets for the choice b: = 2cA and 0 < a\ll 1 ,
\begin{align} \|\nabla f(u)\|_{M(I)\cap W(I)} &\leq c\|\nabla u_0\|+d(f(u), f(0))\\ &\leq cA+cb^{p-\theta}a^{\theta}\\ &\leq \Big(\frac12+(2cA)^{p-1-\theta} a^{\theta}\Big)b\\ & < b. \end{align} | (3.70) |
Arguing as previously, for 0 < a\ll 1 , one gets
\begin{align} \|f(u)\|_{L^\infty(I, \dot H^1)}\leq A+cb^{p-\theta}a^{\theta}\leq 2A. \end{align} | (3.71) |
Now, for \delta: = \frac a2\ll 1 , one uses (2.2) via (3.39) and (3.70) to write
\begin{eqnarray} \|f(u)\|_{S(I)} &\leq&\|e^{{ \text i}\cdot\Delta}u_0\|_{S(I)}+\|\nabla f(u)\|_{M(I)}\\ &\leq&\delta+ca^{\theta} b^{p-\theta}\\ &\leq&2\delta = a. \end{eqnarray} | (3.72) |
The stability f(X_{a, b})\subset X_{a, b} follows by (3.70), (3.71) and (3.72). The proof is closed via a classical Picard argument.
● Long-time perturbation. The second part of this section deals with the next result.
Proposition 3.4. Let T > 0 and I: = [0, T] . Take u\in {C(I, \dot H^1)} as a solution to (INLS) and \tilde u\in {L^\infty(I, \dot H^1)} , satisfying for some \epsilon, A > 0 ,
\|\tilde u\|_{L^\infty_T(\dot H^1)\cap S(I)}\leq A; | (3.73) |
\begin{array}{c} i{\tilde u}_t+\Delta\tilde u+|x|^{-{\tau}}|\tilde u|^{p-1}\tilde u = e ;\\ \max\Big\{\|\nabla e\|_{L^{\tilde{q}'}\big(I, L^{\tilde{r}'}(|x|^{\tilde\gamma\tilde{r}'})\big)}, \|\nabla e^{{ \text i}\cdot{\Delta}} [u_0-\tilde u_0]\|_{M(I)\cap W(I)}\Big\}\leq\epsilon. \end{array} | (3.74) |
Then, there exists \epsilon_0: = \epsilon_0(A) , satisfying for any 0 < \epsilon < \epsilon_0 ,
\|u\|_{S(I)}\leq C(A). |
The proof is omitted because it follows like Proposition 3.2.
In this section, one prepares some estimates related to the stability of the assumptions (2.11)–(2.13) by the flow of(INLS). Take \varphi\in\dot H^1 to be the ground state of (2.8), which is a minimizer of (2.9). The Eq (2.7) gives
\begin{align} \|\nabla\varphi\|^2& = P[\varphi]; \end{align} | (3.75) |
\begin{align} \|\nabla\varphi \|& = C_*^{-\frac{1+p}{p-1}} = C_*^{-\frac{N-{\tau}}{2-{\tau}}}; \end{align} | (3.76) |
\begin{align} E(\varphi)& = \frac{2-{\tau}}{N-{\tau}}C_*^{-2\frac{N-{\tau}}{2-{\tau}}}. \end{align} | (3.77) |
Let us give the first result of this section.
Lemma 3.1. For \delta\in(0, 1) , there exists \tilde\delta: = \tilde\delta(\delta, N)\in(0, 1) such that if u\in \dot H^1 satisfies
\begin{align} \|\nabla u\| < \|\nabla\varphi \|; \end{align} | (3.78) |
\begin{align} E(u) < (1-\delta)E(\varphi), \end{align} | (3.79) |
then,
\begin{gather} \|\nabla u\|^2 < (1-\tilde\delta)\|\nabla\varphi \|^2; \end{gather} | (3.80) |
\begin{gather} \|\nabla u\|^2-P[u] \geq \tilde\delta\|\nabla u\|^2; \end{gather} | (3.81) |
\begin{gather} E(u)\geq0. \end{gather} | (3.82) |
Proof. Take the real function f(x): = x-\frac2{1+p}C_*^{1+p}x^{\frac{1+p}2} . Then,
\begin{align} f(\|\nabla u\|^2) & = \|\nabla u\|^2-\frac2{1+p}C_*^{1+p}\|\nabla u\|^{1+p}\\ &\leq\|\nabla u\|^2-\frac{2}{1+p}P[u] \\ &\leq E(u) \end{align} | (3.83) |
\begin{align} &\leq (1-\delta)E(\varphi). \end{align} | (3.84) |
The equation f'(x) = 0 is equivalent to x = x^* = C_*^{-\frac{2(1+p)}{p-1}} = \|\nabla\varphi \|^2. Moreover, by (3.77), one has f(x^*) = E(\varphi) . Now, since f is positive and strictly increasing on [0, x^*] , one has (3.80) and (3.82) by (3.83) and (3.84). Now, let the real function g(x): = x-C_*^{1+p}x^{\frac{1+p}2} . Then,
\begin{align} \|\nabla u\|^2-P[u] &\geq \|\nabla u\|^2-(C_*\|\nabla u\|)^{1+p}\\ & = g(\|\nabla u\|^2). \end{align} | (3.85) |
Moreover, g(x) = 0 if, and only if, x = 0 or x = x^* . Thus, g(x)\gtrsim x on [0, (1-\tilde\delta)x^*] . So, (3.80) gives (3.81).
Corollary 3.1. If u\in \dot H^1 satisfies \|\nabla u\| < \|\nabla\varphi \| . Then, E(u)\geq 0 .
Proof. The case E(u)\geq E(\varphi) = \frac{2-{\tau}}{N-{\tau}}C_*^{-2\frac{N-{\tau}}{2-{\tau}}} > 0 is clear. Otherwise, Lemma 3.1 gives the result.
With Lemma 3.1 via a continuity argument and the conservation of the energy, one has the following energy trapping.
Proposition 3.5. For \delta\in(0, 1) , there exists \tilde\delta\in(0, 1) such that if u_0\in \dot H^1 satisfies:
\begin{align} \|\nabla u_0\| < \|\nabla\varphi \|; \end{align} | (3.86) |
\begin{align} E(u_0) < (1-\delta)E(\varphi), \end{align} | (3.87) |
then the maximal solution to (INLS) satisfies, for any t\in [0, T^+) ,
\begin{gather} \|\nabla u(t)\|^2 < (1-\tilde\delta)\|\nabla\varphi \|^2; \end{gather} | (3.88) |
\begin{gather} \|\nabla u(t)\|^2-P[u(t)] \geq \tilde\delta\|\nabla u(t)\|^2; \end{gather} | (3.89) |
\begin{gather} E(u)\geq0; \end{gather} | (3.90) |
\begin{gather} E(u(t))\simeq \|\nabla u(t)\|^2\simeq\|\nabla u_0\|^2. \end{gather} | (3.91) |
Proof. For the last point, since E(u(t))\leq \|\nabla u(t)\|^2 , by (3.89), one has
\begin{eqnarray*} E(u(t)) &\geq&(1-\frac2{1+p})\|\nabla u(t)\|^2+\frac2{1+p}(\|\nabla u(t)\|^2-P[u(t)] )\\ &\geq&(1-\frac2{1+p})\|\nabla u(t)\|^2. \end{eqnarray*} |
The rest of the proof follows by Lemma 3.1 via the conservation of the energy and a continuity argument.
Now, one gives a result similar to Lemma 3.1, in the complementary of the assumption (3.78).
Lemma 3.2. For \delta\in(0, 1) , there exists \tilde\delta: = \tilde\delta(\delta, N)\in(0, 1) such that if u\in \dot H^1 satisfies (3.79) and
\begin{align} \|\nabla u\| > \|\nabla\varphi \|, \end{align} | (3.92) |
then
\begin{align} \|\nabla u\|^2& > (1+\tilde\delta)\|\nabla\varphi \|^2; \end{align} | (3.93) |
\begin{align} \|\nabla u\|^2&-P[u]\leq -\tilde\delta\|\nabla\varphi \|^2. \end{align} | (3.94) |
Proof. The proof of (3.93) is omitted because it is similar to Lemma 3.1. For (3.94), one writes
\begin{eqnarray*} 2\Big(\|\nabla u\|^2-P[u]\Big) & = &(1+p)E(u)-(p-1)\|\nabla u\|^2\\ & < &(1+p)(1-\delta)E(\varphi)-(p-1)\|\nabla\varphi \|^2\\ & < &(1-\delta)(p-1)\|\nabla\varphi \|^2-(p-1)\|\nabla\varphi \|^2\\ & < &-\delta(p-1)\|\nabla\varphi \|^2. \end{eqnarray*} |
This closes the proof.
In this section, one proves the global existence and energy scattering, namely, the first part of Theorem 2.1. The proof follows with contradiction. To begin, one proves that if the first part of Theorem 2.1 fails, then Proposition 2.1 holds. Then, one shows that the scenarios in Proposition 2.1 don't happen.
We treat the case 0 < {\tau} < \min\{\frac{6-N}{2}, \frac4N\} because the case \frac{2+N}{N} < {\tau} < 2 follows similarly. For 0 < \lambda_n < \infty and x_n\in \mathbb{R}^N , one defines the operator
\begin{align} f_n[\psi] &: = \frac1{\lambda_n^\frac{N-2}2}\psi(\frac{\cdot-x_n}{\lambda_n}). \end{align} | (4.1) |
The next result is essential in proving Proposition 2.1 was established in [28, Proposition 3.3] in three space dimensions.
Proposition 4.1. Let the sequences 0 < \lambda_n < \infty , x_n\in \mathbb{R}^N , and t_n\in \mathbb{R} , such that
\begin{align} |\frac{x_n}{\lambda_n}|\to\infty, \quad\mathit{\mbox{and}}\quad t_n\equiv0\quad\mathit{\mbox{or}}\quad t_n\to\pm\infty. \end{align} | (4.2) |
Take \psi\in\dot H^1 and the sequence
\begin{align} \psi_n&: = f_n[e^{{ \text i}t_n\Delta}\psi] = e^{{ \text i}\lambda_n^2t_n\Delta}f_n[\psi]. \end{align} | (4.3) |
Then, for any n > > 1 , there is a global solution to (INLS) denoted by v_n\in C(\mathbb{R}, \dot H^1) satisfying
\begin{align} v_n(0) = \psi_n, \quad \|v_n\|_{\nabla W_{\tau}( \mathbb{R})}\lesssim 1. \end{align} | (4.4) |
Moreover, for all \varepsilon > 0 , there is n_\varepsilon\in \mathbb{N} and \chi\in C_c^\infty(\mathbb{R}\times \mathbb{R}^N) such that
\begin{align} \|\lambda_n^{\frac{N-2}2}v_n(\lambda_n^2(t-t_n), \lambda_nx+x_n)-\chi\|_{\nabla W_{\tau}( \mathbb{R})} < \varepsilon, \quad \forall n > n_0. \end{align} | (4.5) |
Proof. Let a smooth function be
\begin{equation} \chi_n(x): = \left\{ \begin{array}{ll} 1, \quad |x+\frac{x_n}{\lambda_n}|\geq\frac12|\frac{x_n}{\lambda_n}|;\\ 0, \quad |x+\frac{x_n}{\lambda_n}| < \frac14|\frac{x_n}{\lambda_n}|. \end{array}, \quad |\partial^\alpha \chi_n|\lesssim |\frac{x_n}{\lambda_n}|^{-|\alpha|}. \right. \end{equation} | (4.6) |
Take the sequence of slabs
\begin{align} I_{n, T}: = [a_{n, T}^-, a_{n, T}^+]: = [-\lambda_n^2(t_n+T), \lambda_n^2(-t_n+T)]; \end{align} | (4.7) |
\begin{align} I_{n, T}^+: = (a_{n, T}^+, \infty), \quad I_{n, T}^-: = (-\infty, a_{n, T}^-). \end{align} | (4.8) |
Taking account of [40, Appendix A.2], let a Littlewood-Paley frequency cutoff be
\begin{align} P_{n}: = P_{|\frac{x_n}{\lambda_n}|^{-\theta}\leq\cdot\leq|\frac{x_n}{\lambda_n}|^{\theta}}, \quad \theta\in(0, 1). \end{align} | (4.9) |
Let us denote the sequence of approximate solutions to (INLS),
\begin{equation} \tilde v_{n, T}(t): = \left\{ \begin{array}{lll} f_n[\chi_nP_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi], \quad t\in I_{n, T};\\ e^{{ \text i}(t-a_{n, T}^+)\Delta}[\tilde v_{n, T}(a_{n, T}^+)], \quad t\in I_{n, T}^+;\\ e^{{ \text i}(t-a_{n, T}^-)\Delta}[\tilde v_{n, T}(a_{n, T}^-)], \quad t\in I_{n, T}^-. \end{array} \right. \end{equation} | (4.10) |
Using the long-time perturbation result in Proposition 3.2, one proves the existence of the solutions v_n .
● Proof of the condition
\begin{align} \limsup\limits_{T\to\infty}\limsup\limits_{n\to\infty}\|\tilde v_{n, T}\|_{L^\infty( \mathbb{R}, \dot H^1)\cap\nabla W_{\tau}( \mathbb{R})}\lesssim1. \end{align} | (4.11) |
One has directly from (4.6) via the Hölder estimate,
\begin{align} \|\chi_n\|_\infty+\|\nabla\chi_n\|_N &\lesssim1+|\frac{x_n}{\lambda_n}|^{-1}|B(|\frac{x_n}{\lambda_n}|)|^\frac1N \\ &\lesssim1. \end{align} | (4.12) |
Now, by the Hölder estimate, it follows that on I_{n, T} ,
\begin{align} \|\tilde v_{n, T}\|_{\dot H^1} & = \|f_n[\chi_nP_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi]\|_{\dot H^1} \\ &\lesssim\|\chi_n(\frac{\cdot-x_n}{\lambda_n})e^{{ \text i}(t+\lambda_n^{2}t_n)\Delta}[f_n(P_n\psi)]\|_{\dot H^1} \\ &\lesssim\|\chi_n\|_\infty\|e^{{ \text i}(t+\lambda_n^{2}t_n)\Delta}[f_n(P_n\psi)]\|_{\dot H^1}+\|\nabla\chi_n\|_N\|e^{{ \text i}(t+\lambda_n^{2}t_n)\Delta}[f_n(P_n\psi)]\|_{2^*} . \end{align} | (4.13) |
Take \phi_{P_n} , a bump function associated to the projector P_n . By Strichartz and Bernstein estimates and Sobolev embedding via (4.13), it follows that on I_{n, T} ,
\begin{align} \|\tilde v_{n, T}\|_{\dot H^1} &\lesssim\|f_n(P_n\psi)\|_{\dot H^1}\\ &\lesssim\|P_n\psi\|_{\dot H^1}\\ &\lesssim|\frac{x_n}{\lambda_n}|^{-\theta} \|\phi_{P_n}\|_{N}\|\psi\|_{2^*}\\ &\lesssim1. \end{align} | (4.14) |
Also, by the Hölder and Strichartz estimates via Sobolev embedding and (4.13), it follows that
\begin{align} \|\tilde v_{n, T}\|_{\nabla W_{\tau}(I_{n, T})} & = \|f_n[\chi_nP_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi]\|_{\nabla W_{\tau}(I_{n, T})} \\ &\lesssim\|\chi_n(\frac{\cdot-x_n}{\lambda_n})e^{{ \text i}(t+\lambda_n^{2}t_n)\Delta}[f_n(P_n\psi)]\|_{\nabla W_{\tau}(I_{n, T})} \\ &\lesssim\|\chi_n\|_\infty\|e^{{ \text i}(t+\lambda_n^{2}t_n)\Delta}[f_n(P_n\psi)]\|_{\nabla W_{\tau}(I_{n, T})}+\|\nabla\chi_n\|_N\|e^{{ \text i}(t+\lambda_n^{2}t_n)\Delta}[f_n(P_n\psi)]\|_{S(I_{n, T})}\\ &\lesssim\|f_n(P_n\psi)\|_{\dot H^1} . \end{align} | (4.15) |
Thus, by (4.14) and (4.15), one gets
\begin{align} \|\tilde v_{n, T}\|_{L^\infty(I_{n, T}, \dot H^1)\cap\nabla W_{\tau}(I_{n, T})} &\lesssim1. \end{align} | (4.16) |
Thus, (4.11) follows by (4.10) and (4.16) via Strichartz estimates.
● Proof of the condition
\begin{align} \limsup\limits_{T\to\infty}\limsup\limits_{n\to\infty}\|\tilde v_{n, T}(0)-\psi_n\|_{\dot H^1} = 0. \end{align} | (4.17) |
Let us take two cases: the first one is t_n = 0\in I_{n, T} . So,
\begin{align} \|\tilde v_{n, T}(0)-\psi_n\|_{\dot H^1} = \|(1-\chi_nP_n)\psi\|_{\dot H^1}\to0. \end{align} | (4.18) |
In a second case, one assumes that t_n\to+\infty and 0\in I_{n, T}^+ . So, by (4.10), one writes
\begin{align} \|\tilde v_{n, T}(0)-\psi_n\|_{\dot H^1} & = \|e^{-ia_{n, T}^+\Delta}[\tilde v_{n, T}(a_{n, T}^+)] -\psi_n\|_{\dot H^1}\\ & = \|e^{-ia_{n, T}^+\Delta}[f_n(\chi_nP_ne^{{ \text i}(\lambda_n^{-2}a_{n, T}^++t_n)\Delta}\psi] -\psi_n\|_{\dot H^1}\\ & = \|f_n[e^{{ \text i}t_{n}\Delta}e^{-iT\Delta}\chi_nP_ne^{{ \text i}T\Delta}\psi] -f_n[e^{{ \text i}t_{n}\Delta}\psi]\|_{\dot H^1}\\ & = \|(1-\chi_nP_n)e^{{ \text i}T\Delta}\psi\|_{\dot H^1}\to0. \end{align} | (4.19) |
● Proof of the condition
\begin{align} \limsup\limits_{T\to\infty}\limsup\limits_{n\to\infty}\|\tilde e_{n, T}\|_{L^2( \mathbb{R}, \dot W^{1, \frac{2N}{2+N}})} &: = \limsup\limits_{T\to\infty}\limsup\limits_{n\to\infty}\|(i\partial_t+\Delta)\tilde v_{n, T}+|x|^{-\tau}|\tilde v_{n, T}|^{p^c-1}\tilde v_{n, T}\|_{L^2( \mathbb{R}, \dot W^{1, \frac{2N}{2+N}})}\\ & = 0. \end{align} | (4.20) |
Let us split the error into two parts as follows \tilde e_{n, T}: = \tilde e_{n, T}^l+\tilde e_{n, T}^{nl} . First one writes on I_{n, T} ,
\begin{align} \tilde e_{n, T}^l & = (i\partial_t+\Delta)\tilde v_{n, T}\\ & = (i\partial_t+\Delta)\Big(f_n[\chi_nP_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi]\Big)\\ & = (i\partial_t+\Delta)\Big([\chi_n(\frac{x-x_n}{\lambda_n})e^{{ \text i}(\lambda_n^{2}t_n+t)\Delta}f_nP_n\psi]\Big)\\ & = \Delta(\chi_n(\frac{x-x_n}{\lambda_n}))e^{{ \text i}(\lambda_n^{2}t_n+t)\Delta}[f_nP_n\psi]+2\nabla(\chi_n(\frac{x-x_n}{\lambda_n}))\cdot e^{{ \text i}(\lambda_n^{2}t_n+t)\Delta}\nabla[f_nP_n\psi]. \end{align} | (4.21) |
Moreover,
\begin{align} \tilde e_{n, T}^{nl} & = |x|^{-\tau}|\tilde v_{n, T}|^{p^c-1}\tilde v_{n, T}\\ & = |x|^{-\tau}|f_n[\chi_nP_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi]|^{p^c-1}f_n[\chi_nP_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi]\\ & = \lambda_n^{-(2{-\tau})}f_n\Big(|\lambda_nx+x_n|^{-\tau}\chi_n^{p^c}|P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi|^{p^c-1}P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi\Big). \end{align} | (4.22) |
Now, by Hölder and Bernstein estimates via (4.6) and (4.21), one writes for 0 < \theta\ll 1 ,
\begin{align} \|\nabla\tilde e_{n, T}^l\|_{\Omega'(I_{n, T})} &\lesssim\sum\limits_{k = 1}^3\|\partial^k[\chi_n(\frac{x-x_n}{\lambda_n})]e^{{ \text i}(\lambda_n^{2}t_n+t)\Delta}\partial^{3-k}[f_nP_n\psi]\|_{L^1(I_{n, T}, L^2)}\\ &\lesssim|I_{n, T}|\sum\limits_{k = 1}^3\lambda_n^{-k}\|\partial^k\chi_n(\frac{x-x_n}{\lambda_n})\|_\infty\|\partial^{3-k}[f_nP_n\psi]\|_{L^\infty(I_{n, T}, L^2)}\\ &\lesssim\lambda_n^2T\sum\limits_{k = 1}^3\lambda_n^{-k}|\frac{x_n}{\lambda_n}|^{-k}\lambda_n^{k-2}\|\partial^{3-k}[P_n\psi]\|_{L^\infty(I_{n, T}, L^2)}\\ &\lesssim T\sum\limits_{k = 1}^3|\frac{x_n}{\lambda_n}|^{-k+(3-k)\theta}\to0\quad\mbox{as}\quad n\to\infty. \end{align} | (4.23) |
Moreover, by (4.22),
\begin{align} \|\nabla\tilde e_{n, T}^{nl}\|_{\Omega'(I_{n, T})} &\lesssim\lambda_n^{-(2{-\tau})}\Big\|\nabla\Big[f_n\Big(|\lambda_nx+x_n|^{-\tau}\chi_n^{p^c}|P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi|^{p^c-1}P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi\Big)\Big]\Big\|_{L^2(I_{n, T}, L^{\frac{2N}{2+N}})}\\ &\lesssim\lambda_n^{\tau}\sqrt T\Big\|\nabla\Big[\chi_n^{p^c}|\lambda_nx+x_n|^{-\tau}|P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi|^{p^c}\Big]\Big\|_{L^\infty(I_{n, T}, \dot W^{1, \frac{2N}{2+N}})}. \end{align} | (4.24) |
Now, taking account of (4.6), one has
\begin{align} \|\chi_n^{p^c}|\lambda_nx+x_n|^{-\tau}\|_{\infty} &\lesssim |x_n|^{-\tau}; \end{align} | (4.25) |
\begin{align} \|\nabla[\chi_n^{p^c}|\lambda_nx+x_n|^{-\tau}]\|_{\infty} &\lesssim |x_n|^{-\tau}|\frac{x_n}{\lambda_n}|^{-1} . \end{align} | (4.26) |
So, by Hölder and Bernstein estimates via (4.24) and (4.26), one writes
\begin{align} \|\nabla\tilde e_{n, T}^{nl}\|_{\Omega'(I_{n, T})} &\lesssim\lambda_n^{\tau}|x_n|^{-\tau}|\frac{x_n}{\lambda_n}|^{-1}\|[P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi]^{p^c}\|_{L^\infty(I_{n, T}, L^{\frac{2N}{2+N}})}\\ &+\lambda_n^{\tau}\|\chi_n^{p^c}|\lambda_nx+x_n|^{-\tau}|P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi|^{p^c-1}\nabla(P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi)\|_{L^\infty(I_{n, T}, L^{\frac{2N}{2+N}})}\\ &: = (A)+(B) . \end{align} | (4.27) |
Using Hölder and Bernstein estimates and Sobolev embedding, one gets
\begin{align} (A) & = \lambda_n^{\tau}|x_n|^{-\tau}|\frac{x_n}{\lambda_n}|^{-1}\||P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi|^{p^c}\|_{L^\infty(I_{n, T}, L^{\frac{2N}{2+N}})}\\ &\lesssim\lambda_n^{\tau}|x_n|^{-\tau}|\frac{x_n}{\lambda_n}|^{-1}\|P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi\|_{L^\infty(I_{n, T}, \dot H^{1})}^{p^c}\\ &\lesssim|\frac{x_n}{\lambda_n}|^{{-\tau}-1+\theta}\|\psi\|_{\dot H^{1}}^{p^c}\to0\quad\mbox{as}\quad n\to\infty. \end{align} | (4.28) |
Also, by Hölder and Bernstein estimates and Sobolev embedding, one gets, via the Strichartz estimates,
\begin{align} (B) & = \lambda_n^{\tau}\|\chi_n^{p^c}|\lambda_nx+x_n|^{-\tau}|P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi|^{p^c-1}\nabla(P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi)\|_{L^\infty(I_{n, T}, L^{\frac{2N}{2+N}})}\\ &\lesssim|\frac{x_n}{\lambda_n}|^{-\tau}\|P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi\|_{L^\infty(I_{n, T}, L^{N(p^c-1)})}^{p^c-1}\|\nabla(P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi)\|_{L^\infty(I_{n, T}, L^{2})}\\ &\lesssim|\frac{x_n}{\lambda_n}|^{{-\tau}+\theta\frac{(N-2)(1{-\tau})}{2(2{-\tau})}}\|e^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi\|_{L^\infty(I_{n, T}, L^{2^*})}^{p^c-1}\|\nabla(P_ne^{{ \text i}(\lambda_n^{-2}t+t_n)\Delta}\psi)\|_{L^\infty(I_{n, T}, L^{2})}\\ &\lesssim|\frac{x_n}{\lambda_n}|^{{-\tau}+\theta(p^c+\frac{(N-2)(1{-\tau})}{2(2{-\tau})})}\|\psi\|_{\dot H^{1}}^{p^c}\to0\quad\mbox{as}\quad n\to\infty. \end{align} | (4.29) |
Taking \theta < \min\{1+b, \frac b{p^c+\frac{(N-2)(1{-\tau})}{2(2{-\tau})}}\} , the proof is finished by (4.27)–(4.29). Now, one turns on I_{n, T}^+ . In such a case, one has via (4.10),
\begin{align} \tilde e_{n, T} & = (i\partial_t+\Delta)\tilde v_{n, T}+|x|^{-\tau}|\tilde v_{n, T}|^{p^c-1}\tilde v_{n, T}\\ & = |x|^{-\tau}|\tilde v_{n, T}|^{p^c-1}\tilde v_{n, T}\\ & = |x|^{-\tau}|e^{{ \text i}(t-a_{n, T}^+)\Delta}[\tilde v_{n, T}(a_{n, T}^+)]|^{p^c-1}e^{{ \text i}(t-a_{n, T}^+)\Delta}[\tilde v_{n, T}(a_{n, T}^+)]. \end{align} | (4.30) |
Arguing as in (3.13), one writes, via (4.11),
\begin{align} \|\tilde e_{n, T}\|_{\Omega'(I_{n, T})} &\lesssim\|e^{{ \text i}(t-a_{n, T}^+)\Delta}[\tilde v_{n, T}(a_{n, T}^+)]\|_{S(I_{n, T})}^{p^c-1{-\tau}}\|e^{{ \text i}(t-a_{n, T}^+)\Delta}[\tilde v_{n, T}(a_{n, T}^+)]\|_{W_{\tau}(I_{n, T})}^{1+b}\\ &\lesssim\|e^{{ \text i}(t-a_{n, T}^+)\Delta}[\tilde v_{n, T}(a_{n, T}^+)]\|_{S(I_{n, T})}^{p^c-1{-\tau}}\\ &\lesssim\|e^{{ \text i}t\Delta}[f_n(\chi_nP_ne^{{ \text i}T\Delta}\psi)]\|_{S(0, \infty)}^{p^c-1{-\tau}}\\ &\lesssim\|f_ne^{{ \text i}\lambda_n^{-2}t\Delta}[\chi_nP_ne^{{ \text i}T\Delta}\psi]\|_{S(0, \infty)}^{p^c-1{-\tau}}. \end{align} | (4.31) |
Moreover, with a change of variable via (4.31) and Sobolev embedding with Strichartz estimates, one gets
\begin{align} \|\tilde e_{n, T}\|_{\Omega'(I_{n, T})} &\lesssim\|e^{{ \text i}t\Delta}[\chi_nP_ne^{{ \text i}T\Delta}\psi]\|_{S(0, \infty)}^{p^c-1{-\tau}}\\ &\lesssim\|e^{{ \text i}t\Delta}[\chi_nP_ne^{{ \text i}T\Delta}\psi]\|_{\nabla W_{\tau}(0, \infty)}^{p^c-1{-\tau}}\\ &\lesssim\|\nabla[\chi_nP_n-1]e^{{ \text i}T\Delta}\psi\|^{p^c-1{-\tau}}+\|e^{{ \text i}t\Delta}\psi\|_{W_{\tau}(T, \infty)}^{p^c-1{-\tau}}. \end{align} | (4.32) |
The proof is achieved by the dominated convergence theorem and (4.32).
Now, (4.4) follows with a direct application of Proposition 3.4 which gives also
\begin{align} \limsup\limits_{T\to\infty}\limsup\limits_n\|\tilde v_{n, T}-v_n\|_{\nabla W_{\tau}( \mathbb{R})} = 0. \end{align} | (4.33) |
In the rest, one proves (4.5), which is reduced via (4.33) to
\begin{align} \limsup\limits_{T\to\infty}\limsup\limits_n\|\chi_ne^{{ \text i}t\Delta}P_n\psi-\chi\|_{\nabla W_{\tau}( \mathbb{R})} = 0. \end{align} | (4.34) |
For large T > > 1 and -T < t < T , by (4.10) and the dominated convergence theorem, one has for n\to\infty ,
\begin{align} \|\lambda_n^{\frac{N-2}2}\tilde v_{n, T}(\lambda_n^2(t-t_n), \lambda_nx+x_n)-\chi\|_{\nabla W_{\tau}( \mathbb{R})} & = \|\chi_ne^{{ \text i}t\Delta}P_n\psi-\chi\|_{\nabla W_{\tau}( \mathbb{R})}\\ &\to\|e^{{ \text i}t\Delta}\psi-\chi\|_{\nabla W_{\tau}( \mathbb{R})}. \end{align} | (4.35) |
So, (4.5) follows by (4.35) via a density argument. Moreover, for t > T , one has via (4.10),
\begin{align} \lambda_n^{\frac{N-2}2}\tilde v_{n, T}(\lambda_n^2(t-t_n), \lambda_nx+x_n) & = f_n^{-1}e^{{ \text i}\lambda_n^2(t-T)\Delta}f_n\chi_ne^{{ \text i}T\Delta}P_n\psi\\ & = e^{{ \text i}t\Delta}(e^{-iT\Delta}\chi_ne^{{ \text i}T\Delta})P_n\psi. \end{align} | (4.36) |
So, (4.5) follows by (4.36) via a density argument.
Now, one returns to the proof of Proposition 2.1. One says that the statement (SC)(u_0) holds if: For u_0\in\dot H^{1} satisfying (2.11) and (2.12), the corresponding solution to (INLS) is global and satisfies:
\begin{align} u\in S( \mathbb{R}), \end{align} | (4.37) |
where S(\mathbb{R}) is the space defined in (3.4) and (3.36), respectively. Using Sobolev embeddings and the Strichartz estimate, one writes for 0 < T < T^+ ,
\begin{align} \|e^{{ \text i}\cdot{\Delta}}u_0\|_{S(0, T)}\lesssim \|\nabla e^{{ \text i}\cdot{\Delta}}u_0\|_{W_{\tau}(0, T)}\lesssim\|\nabla u_0\|. \end{align} | (4.38) |
Thus, if \|\nabla u_0\|\ll 1 , by the small data theory in Proposition 3.3, (SC)(u_0) holds. Now, for each \delta > 0 , one defines the quantities
\begin{align} S_\delta: = \Big\{u_0\in\dot H^1, \quad E(u_0) < \delta\quad\mbox{and}\quad \|\nabla u_0\| < \|\nabla\varphi \|\Big\}; \end{align} | (4.39) |
\begin{align} E_c: = \sup\Big\{\delta > 0\quad\mbox{s. t}\quad u_0\in S_\delta\Rightarrow \, (SC)(u_0)\quad\mbox{ holds}\Big\}. \end{align} | (4.40) |
If the first part of Theorem 2.1 fails, it follows that
\begin{equation} E_c < E(\varphi). \end{equation} | (4.41) |
Then, there is a sequence u_n of solutions to (INLS) such that the data u_{n, 0}\in \dot H^{1} satisfies
\begin{align} \|\nabla u_{n, 0}\| < \|\nabla\varphi \|; \end{align} | (4.42) |
\begin{align} E(u_{n, 0}) \to E_c\quad\mbox{as}\quad n\to\infty; \end{align} | (4.43) |
\begin{align} \|u_{n}\|_{S( \mathbb{R})} = \infty\quad\mbox{ for any } n. \end{align} | (4.44) |
Now, using the profile decomposition in Proposition 2.2, one writes
\begin{align} u_{n, 0} & = \sum\limits_{j = 1}^Mf_n^j(e^{{ \text i}{t_n^j\Delta}}\psi^j)+ W^M_n\\ &: = \frac1{(\lambda^j_n)^{\frac{N-2}2}}\sum\limits_{j = 1}^M[e^{{ \text i}{t_n^j\Delta}}\psi^j](\frac{\cdot-x_n^j}{\lambda^j_n})+ W^M_n. \end{align} | (4.45) |
Using Proposition 3.4 and Proposition 4.1 and following lines in [28, Theorem 1.2], one has only one profile, (t_n, x_n)\equiv(0, 0) and \|W_n^1\|_{\dot H^1}\to0 . Then, there exists 0 < \lambda_n < \infty such that \|\lambda_n^\frac{N-2}2u_{0, n}(\lambda_n\cdot)-\psi\|_{\dot H^1}\to0 . Taking account of Propositions 3.4 and 3.5, the solution to (INLS) with datum \psi is the solution needed. See [28, Theorem 1.2] for more details.
Let u\in C([0, T^+), \dot H^1) and a frequency scale function \lambda:[0, T^+)\mapsto \mathbb{R}_+ , such that \inf_{t\in [0, T^+)}\lambda(t)\geq1 , given in Proposition 2.1. One discusses two cases.
To preclude the finite-time blowup scenario, one needs the following reduced Duhamel formula [40, Proposition 8.7], which is a consequence of the compactness properties.
Lemma 4.1. The following weak limit holds in \dot H^1 for T\to T^+ ,
\begin{align} i\int_t^Te^{{ \text i}(t-s)\Delta}\Big[|x|^{-\tau}|u|^{p^c-1}u\Big]\, ds\rightharpoonup u(t). \end{align} | (4.46) |
Now, assume that T^+ < \infty . By (4.46) via Hölder, Hardy, and Bernstein estimates, one writes for M > 0 ,
\begin{align} \|P_M u(t)\| &\lesssim \|P_M [|x|^{-\tau}|u|^{p^c-1}u]\|_{L^1((0, T^+), L^2)}\\ &\lesssim M(T^+-t)\|(|x|^{-1}u)^{\tau}|u|^{p^c{-\tau}}\|_{L^\infty((0, T^+), L^\frac{2N}{2+N})}\\ &\lesssim M(T^+-t)\||x|^{-1}u\|_{L^\infty((0, T^+), L^2)}^{\tau}\|u\|_{L^\infty((0, T^+), L^{2^*})}^{p^c{-\tau}}\\ &\lesssim M(T^+-t)\|u\|_{L^\infty((0, T^+), \dot H^1)}^{\tau}\|u\|_{L^\infty((0, T^+), L^{2^*})}^{p^c{-\tau}} . \end{align} | (4.47) |
So, by the Bernstein inequality for the high frequencies via (4.47), one gets
\begin{align} \|u(t)\| &\lesssim \|P_M u(t)\|+\|(1-P_M) u(t)\|\\ &\lesssim M(T^+-t)+M^{-1} . \end{align} | (4.48) |
Now, taking account of the mass conservation and letting t be close to T^+ , it follows that u = 0 which contradicts T^+ < \infty and closes the proof.
In this subsection, one assumes that T^+ = \infty . Let us give some notations in the spirit of [16]. Take, for R > > 1 , the radial function defined on \mathbb{R}^N by
\zeta:x\mapsto\left\{ \begin{array}{ll} \frac12|x|^2, \quad\mbox{if}\quad |x|\leq R/2 ;\\ R|x|, \quad\mbox{if}\quad |x| > R. \end{array} \right. |
Moreover, one assumes that in the centered annulus C(R/2, R): = \{x\in \mathbb{R}^N, \; R/2 < |x| < R\} ,
\partial_r\zeta > 0, \quad\partial^2_r\zeta \geq 0\quad\mbox{and}\quad |\partial^\alpha \zeta|\leq C_\alpha R|\cdot|^{1-\alpha}, \; \forall\, |\alpha| \geq 1. |
Here, \partial_r \zeta: = \frac\cdot{|\cdot|}\cdot\nabla \zeta denotes the radial derivative. Note that on the centered ball of radius R/2 , one has
\zeta_{jk} = \delta_{jk}, \quad \Delta \zeta = N\quad\mbox{and}\quad \Delta^2 \zeta = 0. |
Moreover, by the radial identity
\begin{equation} \partial_j\partial_k = \Big(\frac{\delta_{jk}}r-\frac{x_jx_k}{r^3}\Big)\partial_r+\frac{x_jx_k}{r^2}\partial_r^2, \end{equation} | (4.49) |
one gets for |x| > R ,
\begin{align} \zeta_{jk} = \frac R{|x|}\Big(\delta_{jk}-\frac{x_jx_k}{|x|^2}\Big); \end{align} | (4.50) |
\begin{align} \Delta \zeta = \frac{(N -1)R}{|x|}; \end{align} | (4.51) |
\begin{align} |\Delta^2 \zeta|\lesssim\frac R{|x|^3}. \end{align} | (4.52) |
Using Cauchy Schwarz and Hardy estimates via (3.89) and (3.91), one has
\begin{align} |M_\zeta| = 2\Big|\Im\int_{ \mathbb{R}^N} \bar u(\nabla\zeta\cdot\nabla u)\, dx\Big|\lesssim R^2 E(u). \end{align} | (4.53) |
Taking account of the identity (4.49), one has
\begin{align} \Re\Big(\int_{B^c(R/2)}\partial_l\partial_k\zeta \partial_{k}u\partial_{l}\bar u\, dx\Big) & = \Re\int_{B^c(R/2)}\Big[\Big(\frac{\delta_{lk}}r-\frac{x_lx_k}{r^3}\Big)\partial_r\zeta +\frac{x_lx_k}{r^2}\partial_r^2\zeta \Big]\partial_{k}u\partial_{l}\bar u\, dx\\ & = \int_{B^c(R/2)}\Big(|\nabla u|^2-\frac{|x\cdot\nabla u|^2}{|x|^2}\Big)\frac{\partial_r\zeta }{|x|}\, dx +\int_{B^c(R/2)}\frac{|x\cdot\nabla u|^2}{|x|^2}\partial_r^2\zeta \, dx\\ & = \int_{B^c(R/2)}|\not\nabla u|^2\frac{\partial_r\zeta }{|x|}\, dx+\int_{B^c(R/2)}\frac{|x\cdot\nabla u|^2}{|x|^2}\partial_r^2\zeta\, dx , \end{align} | (4.54) |
where the angular gradient is
\not\nabla : = \nabla -\frac{x\cdot\nabla}{|x|^2}x . |
Now, by (4.54) via Proposition 2.5, one writes
\begin{align} M_\zeta'[u] & = 4\Big(\|\nabla u\|_{L^2(B(R/2))}^2-\int_{B(R/2)}|x|^{-\tau}|u|^{p^c}\, dx\Big)\\ &+\int_{B^c(R/2)}|\not\nabla u|^2\frac{\partial_r\zeta }{|x|}\, dx+\int_{B^c(R/2)}\frac{|x\cdot\nabla u|^2}{|x|^2}\partial_r^2\zeta\, dx-\int_{B^c(R/2)}\Delta^2\zeta|u|^2\, dx\\ &-\int_{B^c(R/2)}\Big(\frac{4b}{p^c}\frac{\nabla\zeta\cdot x}{|x|^2}+2\frac{p^c-2}{p^c}\Delta \zeta\Big) |x|^{-\tau}|u|^{p^c}\, dx. \end{align} | (4.55) |
So, (4.55) via (3.89), (3.91) and Sobolev embeddings implies that
\begin{align} M_\zeta'[u] &\geq4\Big(\|\nabla u\|_{L^2(B(R/2))}^2-\int_{B(R/2)}|x|^{-\tau}|u|^{p^c}\, dx\Big)-c\int_{B^c(R/2)}\Big(|x|^{-2}|u|^2+|x|^{-\tau}|u|^{p^c}\Big)\, dx\\ &\geq4\Big(\|\nabla u\|^2-\int_{ \mathbb{R}^N}|x|^{-\tau}|u|^{p^c}\, dx\Big)-c\int_{B^c(R/2)}\Big(|\nabla u|^2+|x|^{-2}|u|^2+|x|^{-\tau}|u|^{p^c}\Big)\, dx\\ &\gtrsim E(u)-c\int_{B^c(R/2)}\Big(|\nabla u|^2+|x|^{-2}|u|^2+|x|^{-\tau}|u|^{p^c}\Big)\, dx. \end{align} | (4.56) |
So, (4.53) via (4.56) gives
\begin{align} E(u) &\lesssim \frac{R^2}T+\int_{B^c(R/2)}\Big(|\nabla u|^2+|x|^{-2}|u|^2+|x|^{-\tau}|u|^{p^c}\Big)\, dx. \end{align} | (4.57) |
Finally, one picks T: = R^3\to\infty , so (2.16) via (4.57) gives E(u) = 0 . This contradiction finishes the proof.
In this section, one proves the second part of Theorem 2.1. Let us denote \phi_A: = A^2\phi(\frac\cdot A) , for A > 0 , where \phi\in C_0^\infty(\mathbb{R}^N) is radial and satisfies
\phi(x) = \left\{ \begin{array}{ll} \frac12|x|^2, \quad |x|\leq1 ;\\ 0, \quad |x|\geq2, \end{array} \right.\quad\mbox{and}\quad \phi''\leq1. |
A calculus gives
\phi_A''\leq1, \quad \phi_A'(r)\leq r\quad\mbox{and}\quad \Delta\phi_A\leq N. |
By the localized variance identity [46, Corollary 3.2], one has
\begin{eqnarray} M_A'' & = &-\int_{ \mathbb{R}^N}\Delta^2\phi_A|u|^2\, dx+4\int_{ \mathbb{R}^N}\partial_l\partial_k\phi_A\Re(\partial_ku\partial_l\bar u)\, dx\\ &+&\frac{4}{1+p}\int_{ \mathbb{R}^N}\nabla \phi_A\cdot\nabla(|x|^{-{\tau}})|u|^{1+p}\, dx-2\frac{p-1}{1+p}\int_{ \mathbb{R}^N}\Delta\phi_A |x|^{-{\tau}}|u|^{1+p}\, dx\\ & = &4\Big(\|\nabla u\|_{L^2(|x| < A)}^2-\int_{|x| < A} |x|^{-{\tau}}|u|^{1+p}\, dx\Big)-\int_{ \mathbb{R}^N}\Delta^2\phi_A|u|^2\, dx\\ &+&\frac{4}{1+p}\int_{|x| > A}\nabla \phi_A\cdot\nabla(|x|^{-{\tau}})|u|^{1+p}\, dx-2\frac{p-1}{1+p}\int_{|x| > A}\Delta\phi_A |x|^{-{\tau}}|u|^{1+p}\, dx\\ &+&4\int_{|x| > A}\partial_l\partial_k\phi_A\Re(\partial_ku\partial_l\bar u)\, dx. \end{eqnarray} | (5.1) |
Recall the next radial identities
\begin{align} \frac{\partial^2}{\partial x_j\partial x_k}&: = \partial_{jk}^2 = \Big(\frac{\delta_{jk}}r-\frac{x_jx_k}{r^3}\Big)\partial_r+\frac{x_jx_k}{r^2}\partial_r^2; \end{align} | (5.2) |
\begin{align} \Delta& = \partial_r^2+\frac{N-1}r\partial_r; \end{align} | (5.3) |
\begin{align} \nabla& = \frac x{r}\partial_r . \end{align} | (5.4) |
Since \phi is radial, we have from the above identities
\begin{align} \partial_{jk}^2\phi_A\partial_ku(t)\partial_j\bar u(t)& = |\nabla u|^2\frac{\phi_A'}r+\Big(\phi_A''-\frac{\phi_A'}r\Big)\frac{|x\cdot\nabla u|^2}{r^2}; \end{align} | (5.5) |
\begin{align} \nabla\phi_A \cdot \nabla(|x|^{-{\tau}})& = -{\tau}|x|^{-{\tau}}\frac{\phi_A'}{r} . \end{align} | (5.6) |
Thus, (5.1), (5.5), and (5.6) give
\begin{eqnarray} M_A'' & = &4\Big(\|\nabla u\|^2-P[u]\Big)-\int_{ \mathbb{R}^N}\Delta^2\phi_A|u|^2\, dx-4\Big(\int_{|x| > A}|\nabla u|^2\, dx-\int_{|x| > A} |x|^{-{\tau}}|u|^{1+p}\, dx\Big)\\ &+&4\int_{|x| > A}\Big(|\nabla u|^2\frac{\phi_A'}r+\Big(\phi_A''-\frac{\phi_A'}r\Big)\frac{|x\cdot\nabla u|^2}{r^2}\Big)\, dx\\ &-&\frac{4{\tau}}{1+p}\int_{|x| > A}\frac{\phi_A'}r|x|^{-{\tau}}|u|^{1+p}\, dx-2\frac{p-1}{1+p}\int_{|x| > A}\Delta\phi_A |x|^{-{\tau}}|u|^{1+p}\, dx\\ &\leq&4\Big(\|\nabla u\|^2-P[u]\Big)-\int_{ \mathbb{R}^N}\Delta^2\phi_A|u|^2\, dx+4\int_{|x| > A}\Big(\frac{\phi_A'}r-1\Big)\Big(|\nabla u|^2-\frac{|x\cdot\nabla u|^2}{r^2}\Big)\, dx\\ &-&\frac{4{\tau}}{1+p}\int_{|x| > A}\frac{\phi_A'}r|x|^{-{\tau}}|u|^{1+p}\, dx-2\frac{p-1}{1+p}\int_{|x| > A}\Delta\phi_A |x|^{-{\tau}}|u|^{1+p}\, dx+4\int_{|x| > A}|x|^{-{\tau}}|u|^{1+p}\, dx\\ &\lesssim&\|\nabla u\|^2-P[u]+A^{-2}+A^{-{\tau}}\|u\|_{1+p}^{1+p}. \end{eqnarray} | (5.7) |
Now, if we assume that \sup_{[0, T^+)}\|\nabla u(t)\| < \infty , (5.7) implies that
\begin{align} M_A'' &\lesssim\|\nabla u\|^2-P[u]+A^{-2}+A^{-{\tau}}. \end{align} | (5.8) |
Thus, (3.94) and (5.8) give M_A''\leq -c < 0 , for large A > > 1 . Integrating this inequality twice in time, it follows that u is non-global. This ends the proof of the second part of Theorem 2.1.
The primary contribution of this note is Theorem 2.1, which complements the findings of [26,27,28] to higher spatial dimensions and removes the radial assumption. While the scattering threshold was established in [26,27,28] for three spatial dimensions, the novelty of this work lies in demonstrating the scattering threshold for space dimensions larger than four without the requirement of spherical symmetry. The approach follows the road-map outlined by Kenig and Merle in [14].
Saleh Almuthaybiri: formal analysis, funding acquisition; Radhia Ghanmi: investigation, methodology, mriting; Tarek Saanouni: project administration, resources, supervision, validation, review. All authors have read and approved the final version of the manuscript for publication.
The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).
The second author Radhia Ghanmi is grateful for Laboratory of partial differential equations and applications LR03ES04, University of Tunis El Manar, 2092 Tunis, Tunisia. The second and third authors don’t receive any fund for this publication.
The authors declare no conflict of interest.
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