Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces

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wave equations in Sobolev spaces

van Duong Dinh

To cite this version:

van Duong Dinh. Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev

spaces. International Journal of Apllied Mathematics, Academic Publications, 2018, 31 (4), pp.483-

525. �hal-01426761v2�

Schr¨ odinger and wave equations in Sobolev spaces

Van Duong Dinh

Abstract

In this paper, we establish the local well-posedness results in sub-critical and critical cases for the pure power-type nonlinear fractional Schr¨odinger and wave equations onR^d, namely

i∂tu+ Λ^σu+µ|u|^ν−1u= 0, u|t=0=ϕ, and

∂²tv+ Λ^2σv+µ|v|^ν−1v= 0, v|t=0=ϕ, ∂tv|t=0=φ, where σ ∈ (0,∞)\{1}, ν > 1, µ ∈ {±1} and Λ = √

−∆ is the Fourier multiplier by|ξ|.

For the nonlinear fractional Schr¨odinger equation, we extend the previous results in [22] for σ ≥ 2. These results cover the well-known results for Schr¨odinger equation σ = 2 given in [4]. In the case σ∈(0,2)\{1}, we show the local well-posedness in the sub-critical case forν > 1 in contrast toν≥2 whend= 1, and ν≥3 whend≥2 of [22]. These results also generalize the ones of [11] when d = 1 and of [18] when d ≥ 2, where the authors considered the cubic fractional Schr¨odinger equation withσ∈(1,2). To our knowledge, the nonlinear fractional wave equation does not seem to have been much considered, up to [37]

on the scattering operator withσan even integer and [6], [7] in the context of the damped fractional wave equation.

1 Introduction and main results

Letσ∈(0,∞)\{1}. We consider the Cauchy fractional Schr¨odinger and wave equations posed onR^d, d≥1, namely

i∂tu+ Λ^σu+µ|u|^ν−1u, u_|t=0 =ϕ, (NLFS) and

∂_t²v+ Λ^2σv+µ|v|^ν−1v, v_|t=0=ϕ, ∂tv_|t=0=φ, (NLFW) whereν >1 and µ∈ {±1}. The operator Λ^σ= (√

−∆)^σ is the Fourier multiplier by|ξ|^σ where

∆ =Pd

j=1∂_j²is the free Laplace operator onR^d. The numberµ= 1 (resp. µ=−1) corresponds to the defocusing case (resp. focusing case). When σ ∈(0,2)\{1}, the fractional Schr¨odinger equation was discovered by N. Laskin (see [26], [27]) owing to the extension of the Feynman path integral, from the Brownian-like to L´evy-like quantum mechanical paths. This equation also appears in the water wave models (see [23]). When σ ∈ [2,∞), the (NLFS) generalizes the well-known nonlinear Schr¨odinger equationσ= 2 or the fourth-order nonlinear Schr¨odinger equationσ= 4 (see e.g. [30] and references therein). In the caseσ∈(0,2)\{1}, the fractional wave equation, introduced in [8], reflects the L´evy stable process and the fractional Brownian motion. In the other side, whenσ∈[2,∞), the (NLFW) can be seen as a generalization of the fourth-order nonlinear wave equation (see e.g. [31]).

The study of nonlinear fractional Schr¨odinger and wave equations has attracted a lot of interest in the past several years (see [9], [10], [11], [17], [18], [20], [22], [23], [29], [30], [37] and references therein). It is well known that (see [15], [24], [5] or [34]) the (NLFS) and (NLFW) are locally well-posed inH^γ withγ≥d/2 provided the nonlinearity is sufficiently regular. The main purpose of this note is to prove the local well-posedness for (NLFS) and (NLFW) forγ∈[0, d/2).

1

For the (NLFS), we extend the previous results in [22] forσ≥2. These results cover the well- known results for Schr¨odinger equationσ= 2 given in [4]. In the caseσ∈(0,2)\{1}, we show the local well-posedness in the sub-critical case forν >1 in contrast toν≥2 whend= 1, andν ≥3 whend≥2 of [22]. These results generalize the ones of [11] whend= 1, and of [18] whend≥2, where the authors considered the cubic fractional Schr¨odinger equation with σ ∈ (1,2). We also give the global existence in energy space, namelyH^σ/2under some assumptions. Moreover, in the critical case, we prove the global existence and scattering with small homogeneous data instead of inhomogeneous one in [22]. To our knowledge, the (NLFW) does not seem to have been much considered, up to [37] on the scattering operator withσ an even integer and [6], [7]

in the context of the damped fractional wave equation.

Let us introduce some standard notations (see the appendix of [16], Chapter 5 of [36] or Chapter 6 of [2]). Let χ0 ∈ C₀^∞(R^d) be such that χ0(ξ) = 1 for |ξ| ≤ 1 and supp(χ0) ⊂ {ξ ∈ R^d,|ξ| ≤ 2}. We set χ(ξ) := χ0(ξ)−χ0(2ξ). It is easy to see that χ ∈ C₀^∞(R^d) and supp(χ) ⊂ {ξ ∈ R^d,1/2 ≤ |ξ| ≤ 2}. We denote the Littlewood-Paley projections by P0 :=

χ0(D), PN :=χ(N⁻¹D) withN = 2^k, k∈Zwhereχ0(D), χ(N⁻¹D) are the Fourier multipliers byχ0(ξ) and χ(N⁻¹ξ) respectively. Givenγ∈Rand 1≤q≤ ∞, one defines the Sobolev and Besov spaces as

H_q^γ :=n

u∈S⁰ | kuk_Hq^γ :=k hΛi^γukL^q <∞o

, hΛi:=p 1 + Λ², B_q^γ :=n

u∈S⁰ | kuk_Bq^γ :=kP0ukL^q+ X

N∈2^N

N^2γkPNuk²_Lq

^1/2

<∞o ,

where S⁰ is the space of tempered distributions. Now, let S0 be a subspace of the Schwartz space S consisting of functions φsatisfying D^αφ(0) = 0 for allˆ α∈ N^d where ˆ· is the Fourier transform onS and S0⁰ its topological dual space. One can seeS0⁰ asS⁰/P whereP is the set of all polynomials onR^d. The homogeneous Sobolev and Besov spaces are defined by

H˙_q^γ :=n

u∈S0⁰ | kukH˙_q^γ:=kΛ^γukL^q <∞o , B˙_q^γ :=n

u∈S0⁰ | kukB˙_q^γ:= X

N∈2^Z

N^2γkPNuk²_Lq

^1/2

<∞o .

We again refer the reader to the appendix of [16], Chapter 5 of [36] or Chapter 6 of [2] for various properties of these function spaces. We note thatH_q^γ, B^γ_q,H˙_q^γ and ˙B_q^γ are Banach spaces with the normskuk_Hq^γ,kuk_B^γ_q,kukH˙_q^γ and kukB˙_q^γ respectively. In the sequel, we shall use H^γ :=H₂^γ, H˙^γ := ˙H₂^γ. We also have that if 2 ≤ q < ∞, then ˙B_q^γ ⊂ H˙_q^γ with the reverse inclusion for 1< q≤2. In particular, ˙B^γ₂ = ˙H^γ and ˙B₂⁰= ˙H₂⁰=L². Moreover, ifγ >0, thenH_q^γ =L^q∩H˙_q^γ andB_q^γ =L^q∩B˙_q^γ.

Before stating main results, we recall some facts on (NLFS) and (NLFW). By a standard approximation argument, the following quantities are conserved by the flow of (NLFS),

M_s(u) = Z

|u(t, x)|²dx, E_s(u) = Z 1

2|Λ^σ/2u(t, x)|²+ µ

ν+ 1|u(t, x)|^ν+1dx.

Moreover, if we set forλ >0,

uλ(t, x) =λ^−ν−1σ u(λ^−σt, λ⁻¹x),

then the (NLFS) is invariant under this scaling that is forT ∈(0,+∞],

usolves (NLFS) on (−T, T)⇐⇒u_λ solves (NLFS) on (−λ^σT, λ^σT).

We also have

kuλ(0)kH˙^γ =λ^{d2−ν−1σ −γ}kϕkH˙^γ. From this, we define the critical regularity exponent for (NLFS) by

γs=d 2 − σ

ν−1. (1.1)

Similarly, the following energy is conserved under the flow of (NLFW), Ew(v) =

Z 1

2|∂tv(t, x)|²+1

2|Λ^σv(t, x)|²+ µ

ν+ 1|v(t, x)|^ν+1dx, and the (NLFW) is invariant under the following scaling

vλ(t, x) =λ^−ν−12σ v(λ^−σt, λ⁻¹x).

Using

kvλ(0)kH˙^γ=λ^{d2−ν−12σ −γ}kϕkH˙^γ, k∂tvλ(0)kH˙^γ−σ =λ^{d2−ν−12σ −γ}kφkH˙^γ−σ, we define the critical regularity exponent for (NLFW) by

γ_w= d 2 − 2σ

ν−1. (1.2)

By the standard argument (see e.g [28]), it is easy to see that the (NLFS) (resp. (NLFW)) is ill-posed ifϕ∈H˙^γ withγ < γ_s (resp. v₀∈H˙^γ, v₁∈H˙^γ−σ withγ < γ_w). Indeed ifusolves the (NLFS) with initial dataϕ∈H˙^γ and the lifespanT, then the normkuλ(0)kH˙^γ and the lifespan of u_λ go to zero asλ → 0. Thus we can expect the well-posedness results for (NLFS) (resp.

(NLFW)) whenγ≥γ_s (resp. γ≥γ_w).

Throughout this note, a pair (p, q) is said to be admissible if (p, q)∈[2,∞]², (p, q, d)6= (2,∞,2), 2

p+d q ≤d

2. (1.3)

We also denote for (p, q)∈[1,∞]²,

γ_p,q =d 2 −d

q −σ

p. (1.4)

Note that when σ ∈ (0,2)\{1}, then γp,q > 0 for all admissible pairs except (p, q) = (∞,2).

Therefore, it is convenient to separate two cases σ ∈ (0,2)\{1} and σ ∈ [2,∞). Moreover, since we are working in spaces of fractional orderγ,γsor γw, we need the nonlinearityF(z) =

−µ|z|^ν−1z to have enough regularity. When ν is an odd integer, F ∈ C^∞(C,C) (in the real sense). Whenν is not an odd integer, we need the following assumption

dγe,dγseor dγwe ≤ν, (1.5)

where dγe is the smallest integer greater than or equal to γ, similarly for dγseand dγwe. Our first result is the following local well-posedness for (NLFS) in the sub-critical case.

Theorem 1.1. Given σ∈(0,2)\{1} andν >1. Let γ∈[0, d/2) be such that γ >1/2−σ/max(ν−1,4) whend= 1,

γ > d/2−σ/max(ν−1,2) whend≥2, (1.6) and also, if ν is not an odd integer, (1.5). Then for all ϕ∈H^γ, there exist T^∗∈(0,∞] and a unique solution to (NLFS)satisfying

u∈C([0, T^∗), H^γ)∩L^p_loc([0, T^∗), L^∞),

for somep >max(ν−1,4)whend= 1 and somep >max(ν−1,2)when d≥2. Moreover, the following properties hold:

i. If T^∗<∞, thenku(t)kH^γ → ∞ast→T^∗.

ii. u depends continuously on ϕ in the following sense. There exists 0 < T < T^∗ such that if ϕn → ϕ in H^γ and if un denotes the solution of (NLFS) with initial data ϕn, then 0 < T < T^∗(ϕ_n) for all n sufficiently large and u_n is bounded in L^a([0, T], H_b^γ−γa,b) for any admissible pair(a, b)withb <∞. Moreover,un→uinL^a([0, T], H_b^−γa,b)asn→ ∞.

In particular, un →uinC([0, T], H^γ−)for all >0.

Remark 1.2. If we assume thatν >1 is an odd integer or

dγe ≤ν−1 (1.7)

otherwise, then the continuous dependence holds inC([0, T], H^γ) (see Remark 4.1).

As mentioned before, this result improves the one in [22] at the point that Hong-Sire only give the local well-posedness for ν ≥ 2 whend = 1 and ν ≥ 3 when d≥ 2. This result also covers the one in [11] when d = 1 and in [18] when d ≥ 2, where the authors considered the cubic fractional Schr¨odinger equation withσ∈(1,2). Whenσ≥2, we have the following better result which generalizes the caseσ= 2 given in [4].

Theorem 1.3. Given σ≥2 and ν >1. Letγ ∈[0, d/2) be such that γ > γs, and also, ifν is not an odd integer,(1.5). Let(p, q) be the admissible pair defined by

p= 2σ(ν+ 1)

(ν−1)(d−2γ), q= d(ν+ 1)

d+ (ν−1)γ. (1.8)

Then for allϕ∈H^γ, there existT^∗∈(0,∞] and a unique solution to (NLFS) satisfying u∈C([0, T^∗), H^γ)∩L^p_loc([0, T^∗), H_q^γ).

Moreover, the following properties hold:

i. If T^∗<∞, thenku(t)kH˙^γ → ∞ast→T^∗.

ii. u depends continuously on ϕ in the following sense. There exists 0 < T < T^∗ such that if ϕn → ϕ in H^γ and if un denotes the solution of (NLFS) with initial data ϕn, then 0 < T < T^∗(ϕn) for all n sufficiently large and un is bounded in L^a([0, T], H_b^γ) for any admissible pair (a, b) with γa,b = 0 and b < ∞. Moreover, un → u in L^a([0, T], L^b) as n→ ∞. In particular,un→uin C([0, T], H^γ−)for all >0.

Thanks to the conservation of mass, we immediately have the following global well-posedness inL²whenσ≥2.

Corollary 1.4. Let σ ≥2 and ν ∈(1,1 + 2σ/d). Then for all ϕ∈ L², there exists a unique global solution to (NLFS)satisfying u∈C(R, L²)∩L^p_loc(R, L^q), where(p, q)given in(1.8).

Proposition 1.5. Let











σ∈(2/3,1) whend= 1, σ∈(1,2) whend= 2, σ∈(3/2,3) whend= 3, σ∈[2, d) whend≥4,

(1.9)

andν >1be such that σ/2> γs, and also, if ν is not an odd integer,dσ/2e ≤ν. Then for any ϕ∈H^σ/2, the solution to (NLFS) given in Theorem 1.1 and Theorem 1.7 can be extended to the wholeRif one of the following is satisfied:

i. µ= 1.

ii. µ=−1, ν <1 + 2σ/d.

iii. µ=−1, ν = 1 + 2σ/d andkϕkL² is small.

iv. µ=−1 andkϕk_Hσ/2 is small.

We now turn to the local well-posedness and scattering with small data for (NLFS) in the critical case.

Theorem 1.6. Let σ∈(0,2)\{1} and

ν >5 whend= 1,

ν >3 whend≥2 (1.10)

be such thatγs≥0, and also, ifν is not an odd integer,(1.5). Then for allϕ∈H^γs, there exist T^∗∈(0,∞]and a unique solution to (NLFS)satisfying

u∈C([0, T^∗), H^γs)∩L^p_loc([0, T^∗), B_q^γs−γp,q),

where p = 4, q = ∞ when d = 1; 2 < p < ν −1, q = p^? = 2p/(p−2) when d = 2 and p= 2, q= 2^? = 2d/(d−2)when d≥3. Moreover, ifkϕkH˙^γs < εfor someε >0 small enough, thenT^∗=∞and the solution is scattering inH^γs, i.e. there existsϕ⁺∈H^γs such that

t→+∞lim ku(t)−e^itΛσϕ⁺kH^γs = 0.

This theorem is a modification of Theorem 1.2 and Theorem 1.3 in [22] where the au- thors proved the global well-posedness and scattering for small inhomogeneous data. Note that Strichartz estimate is not sufficient to give the local existence in the critical case. It needs a delicate estimate on L^ν−1_t L^∞_x (see Lemma 3.5 in [22]). The range ν ∈ (1,5] when d = 1 and ν ∈(1,3] still remains open, and it requires another technique rather than Strichartz estimate.

The situation becomes better whenσ≥2, and we have the following result.

Theorem 1.7. Letσ≥2andν >1such thatγs≥0, and also, ifν is not an odd integer,(1.5).

Let

p=ν+ 1, q= 2d(ν+ 1)

d(ν+ 1)−2σ. (1.11)

Then for anyϕ∈H^γs, there existT^∗∈(0,∞]and a unique solution to (NLFS)satisfying u∈C([0, T^∗), H^γs)∩L^p_loc([0, T^∗), H_q^γs).

Moreover, if kϕkH˙^γs < ε for some ε > 0 small enough, then T^∗ = ∞ and the solution is scattering in H^γs.

We next give the local well-posedness results for the (NLFW). Let us start with the local well-posedness in the sub-critical case.

Theorem 1.8. Given σ∈(0,∞)\{1} andν >1. Letγ ∈[0, d/2) be as in(1.6) and also, if ν is not an odd integer, (1.5). Then for all (ϕ, φ)∈H^γ ×H^γ−σ, there exist T^∗ ∈(0,∞] and a unique solution to (NLFW)satisfying

v∈C([0, T^∗), H^γ)∩C¹([0, T^∗), H^γ−σ)∩L^p_loc([0, T^∗), L^∞),

for somep >max(ν−1,4)whend= 1 and somep >max(ν−1,2)when d≥2. Moreover, the following properties hold:

i. If T^∗<∞, thenk[v](t)kH^γ → ∞ ast→T^∗.

ii. v depends continuously on(ϕ, φ)in the following sense. There exists0< T < T^∗ such that if (ϕn, φn)→(ϕ, φ)inH^γ×H^γ−σ and if vn denotes the solution of (NLFW)with initial data (ϕn, φn), then 0 < T < T^∗(ϕn, φn) for all n sufficiently large and vn is bounded in L^a([0, T], H_b^γ−γa,b) for any admissible pair (a, b) with b < ∞. Moreover, vn → v in L^a(I, H_b^−γa,b) asn → ∞. In particular, vn →v in C([0, T], H^γ−)∩C¹([0, T], H^γ−σ−) for all >0.

We note that (1.6) is necessary to use the Sobolev embedding, but it produces a gap between γwand 1/2−σ/max(ν−1,4) whend= 1 andd/2−σ/max(ν−1,2) whend≥2. Moreover, if we assume that ν >1 is an odd integer or (1.7) otherwise, then the continuous dependence holds inC([0, T], H^γ)∩C¹([0, T], H^γ−σ).

The following result gives the local well-posedness for (NLFW) in the σ-sub-critical case.

Theorem 1.9. 1. Assume for d= 1,2,3,4, σ∈

0, d d+ 2

, ν ∈ d

d−2σ, 2d−dσ 2d−(d+ 4)σ

i

orσ∈h d d+ 2,d

2

\{1}, ν ∈ d

d−2σ,d+ 2σ d−2σ

; (1.12) ford= 5, ...,11,

σ∈ 0,2

3

, ν ∈ d

d−2σ, 2d−dσ 2d−(d+ 4)σ

i

or σ∈h2 3,d

6

\{1}, ν∈ d

d−2σ, d d−3σ

i

or σ∈hd 6,2

\{1}, ν ∈ d

d−2σ,d+ 2σ d−2σ

; (1.13) and ford≥12,

σ∈ 0,2

3

, ν ∈ d

d−2σ, 2d−dσ 2d−(d+ 4)σ

i

orσ∈h2 3,2

\{1}, ν∈ d

d−2σ, d d−3σ

i

. (1.14) Let (p, q)be an admissible pair defined by

p= 2σν

(d−2σ)ν−d, q= 2ν. (1.15)

Then for all (ϕ, φ) ∈ H˙^σ ×L², there exist T^∗ ∈ (0,∞] and a unique solution to (NLFW) satisfying

v∈C([0, T^∗),H˙^σ)∩C¹([0, T^∗), L²)∩L^p_loc([0, T^∗), L^q).

Moreover, the following properties hold:

i. If T^∗<∞, thenk[v](t)kH˙^σ → ∞ ast→T^∗.

ii. v depends continuously on (ϕ, φ) in the following sense. There exists 0 < T < T^∗ such that if (ϕn, φn) → (ϕ, φ) in H˙^σ ×L² and if vn denotes the solution of (NLFW) with initial data (ϕn, φn), then 0 < T < T^∗(ϕn, φn) for all nsufficiently large and vn →v in C([0, T],H˙^σ)∩C¹([0, T], L²).

2. Let

σ∈h 2,d

2

, ν∈h dσ^∗ d+σ, σ^∗

, (1.16)

whereσ^∗:= (d+ 2σ)/(d−2σ). Let(p, q)be an admissible pair defined by p= 2σ^∗, p= 2dσ^∗

d+σ. (1.17)

Then the same conclusion as in Item 1 holds true.

This theorem and the conservation of energy imply the following global well-posedness for the defocusing (NLFW).

Corollary 1.10. Under the assumptions of Theorem 1.9, for all(ϕ, φ)∈H˙^σ×L², there exists a unique global solution to the defocusing (NLFW)satisfying

v∈C(R,H˙^σ)∩C¹(R, L²)∩L^p_loc(R, L^q), where(p, q) are as in Theorem 1.9.

The next result gives the local well-posedness with small data scattering for (NLFW) in the critical case.

Theorem 1.11. 1. Assume for d≥1 that σ∈h d

d+ 1, d

\{1}, ν ∈h

1 + 4σ d−σ,∞

, (1.18)

and also, if ν is not an odd integer,

dγwe −σ

2 ≤ν−1. (1.19)

Let p, abe defined by

p=(d+σ)(ν−1)

2σ , a= 2(d+σ)

d−σ . (1.20)

Then for all (ϕ, φ)∈H˙^γw ×H˙^γw−σ, there exist T^∗∈(0,∞] and a unique solution to (NLFW) satisfying

v∈C([0, T^∗),H˙^γw)∩C¹([0, T^∗),H˙^γw−σ)∩L^p_loc([0, T^∗), L^p)∩L^a_loc([0, T^∗),H˙^γw−

σ

a 2).

Moreover, if k[v](0)kH˙^γw < ε for some ε > 0 small enough, then T^∗ = ∞ and the solution is scattering in H˙^γw ×H˙^γw−σ, i.e. there exist (ϕ⁺, φ⁺) ∈ H˙^γw ×H˙^γw−σ such that the (weak) solution to the linear fractional wave equation

∂_t²v⁺(t, x) + Λ^2σv⁺(t, x) = 0, (t, x)∈R×R^d,

v⁺(0, x) =ϕ⁺(x), ∂_tv⁺(0, x) =φ⁺(x), x∈R^d, satisfy

t→+∞lim k[v(t)−v⁺(t)]kH˙^γw = 0.

2. Assume for d≥1 that σ∈hd²+ 4d

3d+ 4 ,∞

\{1}, ν ∈h

1 +4σ(d+ 2) d(d+σ),∞ orσ∈h d

d+ 1,d²+ 4d 3d+ 4

\{1}, ν∈h

1 + 4σ(d+ 2)

d(d+σ),1 + 4σ(d+ 2) d²−3dσ+ 4d−4σ

i

. (1.21) Then for all (ϕ, φ)∈H˙^γw ×H˙^γw−σ, there exist T^∗∈(0,∞] and a unique solution to (NLFW) satisfying

v∈C([0, T^∗),H˙^γw)∩C¹([0, T^∗),H˙^γw−σ)∩L^p_loc([0, T^∗), L^p),

where pis as above. Moreover, ifk[v](0)kH˙^γw < ε for some ε >0 small enough, then T^∗=∞ and the solution is scattering inH˙^γw×H˙^γw−σ.

Finally, we have the following local well-posedness and scattering with small data for (NLFW) in theσ-critical case.

Theorem 1.12. Let





 σ∈h

d d+2,^d₂

\{1} whend={1,2,3,4}, σ∈h

d 6,^d₂

\{1} whend≥5,

(1.22)

andν = 1 + 4σ/(d−2σ). Then for all(ϕ, φ)∈H˙^σ×L², there exist T^∗∈(0,∞] and a unique solution to (NLFW) satisfying

v∈C([0, T^∗),H˙^σ)∩C¹([0, T^∗), L²)∩L^ν_loc([0, T^∗), L^2ν).

Moreover, if k[v](0)kH˙^σ < ε for some ε > 0 small enough, then T^∗ = ∞ and the solution is scattering in H˙^σ×L².

The rest of this note is organized as follows. In Section 2, we prove Strichartz estimates for the fractional Schr¨odinger and wave equations. In Section 3, we recall the fractional derivatives of the nonlinearity. Section 4 is devoted to the proofs of the local well-posedness in the sub-critical case and the local well-posedness with small data scattering in the critical case for the (NLFS).

We finally prove the local well-posedness in the sub-critical case and the local well-posedness with small data scattering in the critical case for the (NLFW) in Section 5.

2 Strichartz estimates

In this section, we recall Strichartz estimates for the linear fractional Schr¨odinger and wave equations.

Theorem 2.1 (Strichartz estimates [12]). Let d ≥ 1, σ ∈ (0,∞)\{1}, γ ∈ R and a (weak) solution to the linear fractional Schr¨odinger equation, namely

u(t) =e^itΛσϕ+ Z t

0

e^i(t−s)ΛσF(s)ds, for some dataϕ, F. Then for all (p, q)and(a, b)admissible pairs,

kuk_Lp(R,B˙^γ_q).kϕkH˙^γ+γp,q +kFk

L^a0(R,B˙^{γ+γp,q−γa0,b0 −σ}

b0 ), (2.1)

whereγp,q andγa⁰,b⁰ are as in (1.4). In particular, kuk_Lp(

R,B˙^γ−_q ^γp,q).kϕkH˙^γ+kFk_L1(R,H˙^γ), (2.2) and

kuk_L∞(R,B˙^γp,q₂ )+kuk_Lp(R,B˙_q⁰).kϕkH˙^γp,q +kFk_La0

(R,B˙⁰

b0), (2.3)

provided that

γp,q=γa⁰,b⁰+σ. (2.4)

Here (a, a⁰)is a conjugate pair.

Sketch of proof. We firstly note this theorem is proved if we establish

ke^itΛσP₁ϕk_Lp(R,L^q).kP₁ϕk_L2, (2.5)

Z t 0

e^i(t−s)ΛσP1F(s)ds _Lp(

R,L^q).kP1Fk_La0

(R,L^b0), (2.6) for all (p, q), (a, b) admissible pairs. Indeed, by change of variables, we see that

ke^itΛσP_Nϕk_Lp(R,L^q)=N^{−(d/q+σ/p)}ke^itΛσP₁ϕ_Nk_Lp(R,L^q), kP1ϕNkL² =N^d/2kPNϕkL²,

whereϕN(x) =ϕ(N⁻¹x). The estimate (2.5) implies that

ke^itΛσPNϕkL^p(R,L^q).N^γp,qkPNϕkL², (2.7) for allN∈2^Z. Similarly,

Z t 0

e^i(t−s)ΛσPNF(s)ds _Lp(

R,L^q)=N−(d/q+σ/p+σ)

Z t 0

e^i(t−s)ΛσP1FN(s)ds _Lp(

R,L^q), whereF_N(t, x) =F(N^−σt, N⁻¹x). We also have from (2.6) and the fact

kP1F_Nk_La0

(R,L^b0)=N^(d/b0+σ/a0)kPNFk_La0

(R,L^b0)

that

Z t 0

e^i(t−s)ΛσPNF(s)ds _Lp(

R,L^q).N^{γp,q−γa0,b0−σ}kPNFk_La0(R,L^b0), (2.8) for allN∈2^Z. We see from (2.7) and (2.8) that

N^γkPNuk_Lp(R,L^q).N^γ+γp,qkPNϕk_L2+N^{γ+γp,q−γa0,b0−σ}kPNFk_La0(R,L^b0).

By taking the `²(2^Z) norm both sides and using the Minkowski inequality, we get (2.1). The estimates (2.2) and (2.3) follow easily from (2.1). It remains to prove (2.5) and (2.6). By the T T^∗-criterion (see [25] or [1]), we need to show

kT(t)k_L2→L² .1, (2.9)

kT(t)kL¹→L^∞ .(1 +|t|)^−d/2, (2.10) for allt∈RwhereT(t) :=e^itΛσP1. The energy estimate (2.9) is obvious by using the Plancherel theorem. The dispersive estimate (2.10) follows by the standard stationary phase theorem. The

proof is complete.

Corollary 2.2. Let d ≥ 1, σ ∈ (0,∞)\{1}, γ ∈ R. If u is a (weak) solution to the linear fractional Schr¨odinger equation for some data ϕ, F, then for all(p, q)and(a, b)admissible with q <∞andb <∞satisfying (2.4),

kuk_Lp(

R,H˙_q^γ−γp,q).kϕkH˙^γ+kFk_L1(R,H˙^γ), (2.11) kuk_L∞(R,H˙^γp,q)+kukL^p(R,L^q).kϕkH˙^γp,q +kFk_La0

(R,L^b0). (2.12) Corollary 2.3. Let d≥1, σ∈(0,∞)\{1}, γ≥0 andI a bounded interval. If u is a (weak) solution to the linear fractional Schr¨odinger equation for some data ϕ, F, then for all (p, q) admissible satisfyingq <∞,

kuk_Lp(I,H^γ−γp,q

q ).kϕkH^γ+kFkL¹(I,H^γ). (2.13) Proof. We firstly note that whenγ_p,q ≥0 (or at least σ∈(0,2]\{1}), we can obtain (2.13) for any γ ∈Rand I =R. To see this, we write kuk

L^p(R,Hq^γ−γp,q)=k hΛi^γ−γp,quk_Lp(R,L^q) and use (2.11) withγ=γ_p,q to obtain

kuk_Lp(

R,H^γ−_q ^γp,q).k hΛi^γ−γp,qϕkH˙^γp,q +k hΛi^γ−γp,qFk_L1(R,H˙^γp,q).

This gives the claim sincekvkH˙^γp,q ≤ kvkH^γp,q using thatγp,q ≥0. It remains to treat the case γp,q < 0. By the Minkowski inequality and the unitary of e^itΛσ in L², the estimate (2.13) is proved if we can show forγ≥0,I⊂Ra bounded interval and all (p, q) admissible withq <∞ that

ke^itΛσϕk

L^p(I,H^γ−γp,qq ).kϕkH^γ. (2.14)

Indeed, if we have (2.14), then

Z t 0

e^i(t−s)ΛσF(s)ds

L^p(I,Hq^γ−γp,q)

≤ Z

I

k1[0,t](s)e^i(t−s)ΛσF(s)k_Lp(I,Hγ−γp,q

q )ds

≤ Z

I

ke^i(t−s)ΛσF(s)k_Lp(I,Hγ−γp,q

q )ds

. Z

I

kF(s)kH^γds=kFk_L1(I,H^γ). We now prove (2.14). To do so, we write

hΛi^γ−γp,qe^itΛσϕ=ψ(D)hΛi^γ−γp,qe^itΛσϕ+ (1−ψ)(D)hΛi^γ−γp,qe^itΛσϕ,

for some ψ ∈ C₀^∞(R^d) valued in [0,1] and equal to 1 near the origin. For the first term, the Sobolev embedding implies

kψ(D)hΛi^γ−γp,qe^itΛσϕkL^q.kψ(D)hΛi^γ−γp,qe^itΛσϕk_Hδ,

for someδ > d/2−d/q. Thanks to the support ofψand the unitary property ofe^itΛσ inL², we get

kψ(D)hΛi^γ−γp,qe^itΛσϕk_Lp(I,L^q).kϕk_L2 .kϕkH^γ.

Here the boundedness ofIis crucial to have the first estimate. For the second term, using (2.12), we obtain

k(1−ψ)(D)hΛi^γ−γp,qe^itΛσϕkL^p(I,L^q).k(1−ψ)(D)hΛi^γ−γp,qϕkH˙^γp,q .kϕkH^γ. Combining the two terms, we have (2.14). This completes the proof.

Corollary 2.4. Let d≥1, σ∈(0,∞)\{1}, γ ∈Rand a (weak) solution to the linear fractional wave equation, namely

v(t) = cos(tΛ^σ)ϕ+sin(tΛ^σ) Λ^σ φ+

Z t 0

sin((t−s)Λ^σ)

Λ^σ G(s)ds, for some dataϕ, φ, G. Then for all (p, q)and(a, b)admissible pairs,

k[v]k_Lp(R,B˙_q^γ).k[v](0)kH˙^γ+γp,q +kGk

L^a0(R,B˙^γ+γp,q

−γa0,b0 −2σ

b0 ), (2.15)

where

k[v]k_Lp(R,B˙_q^γ):=kvk_Lp(R,B˙^γ_q)+k∂tvk_Lp(R,B˙q^γ−σ); k[v](0)kH˙^γ+γp,q :=kϕkH˙^γ+γp,q +kφkH˙^{γ+γp,q−σ}. In particular,

k[v]k_Lp(

R,B˙^γ−_q ^γp,q).k[v](0)kH˙^γ+kGk_L1(R,H˙^γ−σ), (2.16) and

k[v]k_L∞(R,B˙₂^γp,q)+k[v]k_Lp(R,B˙⁰_q).k[v](0)kH˙^γp,q +kGk_La0

(R,B˙⁰

b0), (2.17) provided that

γp,q =γa⁰,b⁰+ 2σ. (2.18)

Proof. It follows easily from Theorem 2.1 and the fact that cos(tΛ^σ) = e^itΛσ +e^−itΛσ

2 , sin(tΛ^σ) = e^itΛσ −e^−itΛσ

2i .

As in Corollary 2.2, we have the following usual Strichartz estimates for the fractional wave equation.

Corollary 2.5. Let d ≥ 1, σ ∈ (0,∞)\{1}, γ ∈ R. If v is a (weak) solution to the linear fractional wave equation for some dataϕ, φ, G, then for all(p, q)and(a, b)admissible satisfying q <∞, b <∞and(2.18),

kvkL^p(R,H˙_q^γ−γp,q).k[v](0)kH˙^γ+kGk_L1(R,H˙^γ−σ), (2.19) k[v]k_L∞(R,H˙^γp,q)+kvk_Lp(R,L^q).k[v](0)kH˙^γp,q +kGk_La0

(R,L^b0). (2.20)

The following result, which is similar to Corollary 2.3, gives the local Strichartz estimates for the fractional wave equation.

Corollary 2.6. Let d ≥ 1, σ ∈ (0,∞)\{1}, γ ≥ 0 and I ⊂ R a bounded interval. If v is a (weak) solution to the linear fractional wave equation for some data ϕ, φ, G, then for all (p, q) admissible satisfyingq <∞,

kvkL^p(I,Hq^γ−γp,q).k[v](0)kH^γ+kGk_L1(I,H^γ−σ). (2.21) Proof. The proof is similar to the one of Corollary 2.3. Thanks to the Minkowski inequality, it suffices to prove for all γ ≥0, all I ⊂ R bounded interval and all (p, q) admissible pair with q <∞,

kcos(tΛ^σ)ϕk_Lp(I,Hγ−γp,q

q ).kϕkH^γ, (2.22)

sin(tΛ^σ) Λ^σ φ

_Lp(I,H^γ−γp,q

q ).kφk_H^γ−σ. (2.23)

The estimate (2.22) follows from the ones ofe^±itΛσ. We will give the proof of (2.23). To do this, we write

hΛi^γ−γp,q sin(tΛ^σ)

Λ^σ =ψ(D)hΛi^γ−γp,q sin(tΛ^σ)

Λ^σ + (1−ψ)(D)hΛi^γ−γp,q sin(tΛ^σ) Λ^σ ,

for someψas in the proof of Corollary 2.3. For the first term, the Sobolev embedding and the fact

sin(tΛ^σ) Λ^σ

_L2→L2 ≤ |t|imply

ψ(D)hΛi^γ−γp,q sin(tΛ^σ) Λ^σ φ

_Lq.|t|kψ(D)hΛi^{γ+δ−γp,q}φk_L2, for someδ > d/2−d/q. This gives

ψ(D)hΛi^γ−γp,q sin(tΛ^σ) Λ^σ φ

_Lp(I,Lq).kφkH^γ−σ.

Here we use the fact thatkψ(D)hΛi^{δ+σ−γp,q}k_L2→L² .1. For the second term, we apply (2.14) with the fact sin(tΛ^σ) = (e^itΛσ−e^−itΛσ)/2iand get

(1−ψ)(D)hΛi^γ−γp,q sin(tΛ^σ) Λ^σ φ

_Lp(I,Lq).k(1−ψ)(D)Λ^−σφkH^γ .kφk_Hγ−σ.

Here we use thatk(1−ψ)(D)hΛi^σΛ^−σk_L2→L² .1 by functional calculus. Combining two terms, we have (2.23). The proof is complete.

3 Nonlinear estimates

In this section, we recall some estimates related to the fractional derivatives of nonlinear opera- tors. Let us start with the following Kato-Ponce inequality (or fractional Leibniz rule).

Proposition 3.1. Let γ ≥0,1< r < ∞ and1 < p₁, p₂, q₁, q₂ ≤ ∞satisfying ¹_r = _p¹

1 +_q¹

1 =

1 p2 +_q¹

2. Then there existsC=C(d, γ, r, p1, q1, p2, q2)>0 such that for all u, v∈S, kΛ^γ(uv)kL^r ≤C

kΛ^γukL^p1kvkL^q1 +kukL^p2kΛ^γvkL^q2

, (3.1)

k hΛi^γ(uv)kL^r ≤C

k hΛi^γukL^p1kvkL^q1 +kukL^p2k hΛi^γvkL^q2

. (3.2)

We refer to [21] (and references therein) for the proof of above inequalities and more general results. We also have the following fractional chain rule.

Proposition 3.2. Let F ∈C¹(C,C) andG∈C(C,R⁺)such that F(0) = 0and

|F⁰(θz+ (1−θ)ζ)| ≤µ(θ)(G(z) +G(ζ)), z, ζ ∈C, 0≤θ≤1,

where µ∈L¹((0,1)). Then for γ∈(0,1) and 1< r, p <∞,1 < q ≤ ∞satisfying ¹_r = ¹_p +¹_q, there existsC=C(d, µ, γ, r, p, q)>0 such that for allu∈S,

kΛ^γF(u)kL^r ≤CkF⁰(u)kL^qkΛ^γukL^p, (3.3) k hΛi^γF(u)kL^r ≤CkF⁰(u)kL^qk hΛi^γukL^p. (3.4) We refer to [13] (see also [32]) for the proof of (3.3) and Proposition 5.1 of [35] for (3.4). A direct consequence of the fractional Leibniz rule and the fractional chain rule is the following fractional derivatives estimates.

Corollary 3.3. Let F ∈C^k(C,C), k∈N\{0}. Assume that there is ν≥ksuch that

|DⁱF(z)| ≤C|z|^ν−i, z∈C, i= 1,2, ...., k.

Then for γ ∈ [0, k] and 1 < r, p < ∞, 1 < q ≤ ∞ satisfying ¹_r = ¹_p + ^ν−1_q , there exists C=C(d, ν, γ, r, p, q)>0 such that for all u∈S,

kΛ^γF(u)kL^r ≤Ckuk^ν−1_Lq kΛ^γukL^p, (3.5) k hΛi^γF(u)kL^r ≤Ckuk^ν−1_Lq k hΛi^γukL^p. (3.6) The reader can find the proof of (3.5) in Lemma A.3 of [24]. The one of (3.6) follows from (3.5), the H¨older inequality and the fact that

k hΛi^γukL^r ∼ kukL^r+kΛ^γukL^r,

for 1< r <∞, γ >0. Another consequence of the fractional Leibniz rule given in Proposition 3.1 is the following result.

Corollary 3.4. Let F(z)be a homogeneous polynomial in z, z of degreeν≥1. Then(3.5)and (3.6) hold true for anyγ≥0andr, p, q as in Corollary 3.3.

Corollary 3.5. Let F(z) =|z|^ν−1z withν >1,γ≥0 and1< r, p <∞,1< q≤ ∞ satisfying

1

r =¹_p +^ν−1_q .

i. If ν is an odd integer or ¹ dγe ≤ν otherwise, then there exists C =C(d, ν, γ, r, p, q)>0 such that for all u∈S,

kF(u)kH˙_r^γ ≤Ckuk^ν−1_Lq kukH˙_p^γ.

A similar estimate holds with H˙_r^γ,H˙_p^γ-norms are replaced byH_r^γ, H_p^γ-norms respectively.

ii. If ν is an odd integer ordγe ≤ν−1otherwise, then there existsC=C(d, ν, γ, r, p, q)>0 such that for all u, v∈S,

kF(u)−F(v)kH˙_r^γ ≤C

(kuk^ν−1_Lq +kvk^ν−1_Lq )ku−vkH˙_p^γ

+ (kuk^ν−2_Lq +kvk^ν−2_Lq )(kukH˙_p^γ+kvkH˙_p^γ)ku−vk_Lq

. A similar estimate holds with H˙_r^γ,H˙_p^γ-norms are replaced byH_r^γ, H_p^γ-norms respectively.

Proof. Item 1 is an immediate consequence of Corollary 3.3 and Corollary 3.4. For Item 2, we firstly write

F(u)−F(v) =ν Z 1

0

|v+t(u−v)|^ν−1(u−v)dt,

and use the fractional Leibniz rule given in Proposition 3.1. Then the results follows by applying the fractional derivatives given in Corollary 3.3 and Corollary 3.4.

1see (1.5) for the definition ofd·e.

4 Nonlinear fractional Schr¨ odinger equations

4.1 Local well-posedness in sub-critical cases

In this subsection, we give the proofs of Theorem 1.1, Theorem 1.3 and Proposition 1.5.

Proof of Theorem 1.1. We follow the standard process (see e.g. Chapter 4 of [5] or [3]) by using the fixed point argument in a suitable Banach space. We firstly choosep >max(ν−1,4) when d = 1 and p > max(ν−1,2) when d ≥ 2 such that γ > d/2−σ/p and then choose q∈[2,∞) such that

2 p+d

q ≤ d 2. Step 1. Existence. Let us consider

X :=n

u∈L^∞(I, H^γ)∩L^p(I, H_q^γ−γp,q)| kuk_L^∞_(I,Hγ)+kuk_Lp(I,Hγ−γp,q

q )≤Mo

, equipped with the distance

d(u, v) :=ku−vkL^∞(I,L²)+ku−vk_Lp(I,H⁻γp,q q ),

whereI= [0, T] andM, T >0 to be chosen later. The persistence of regularity (see e.g. Theorem 1.2.5 of [5]) shows that (X, d) is a complete metric space. By the Duhamel formula, it suffices to prove that the functional

Φ(u)(t) =e^itΛσϕ+iµ Z t

0

e^i(t−s)Λσ|u(s)|^ν−1u(s)ds (4.1) is a contraction on (X, d). The local Strichartz estimate (2.13) gives

kΦ(u)kL^∞(I,H^γ)+kΦ(u)k_Lp(I,Hγ−γp,q

q ).kϕkH^γ+kF(u)kL¹(I,H^γ), kΦ(u)−Φ(v)kL^∞(I,L²)+kΦ(u)−Φ(v)k_Lp(I,H−γp,q

q ).kF(u)−F(v)kL¹(I,L²), whereF(u) =|u|^ν−1u. By our assumptions onν, Corollary 3.5 gives

kF(u)k_L1(I,H^γ).kuk^ν−1_Lν−1(I,L^∞)kuk_L∞(I,H^γ).T^1−ν−1p kuk^ν−1_Lp(I,L^∞)kuk_L∞(I,H^γ), (4.2) kF(u)−F(v)k_L1(I,L²).

kuk^ν−1_Lν−1(I,L^∞)+kvk^ν−1_Lν−1(I,L^∞)

ku−vk_L∞(I,L²)

.T^1−ν−1p

kuk^ν−1_Lp(I,L^∞)+kvk^ν−1_Lp(I,L^∞)

ku−vk_L^∞_(I,L2). (4.3) Using thatγ−γ_p,q > d/q, the Sobolev embedding implies L^p(I, Hq^γ−γp,q)⊂L^p(I, L^∞). Thus, we get

kΦ(u)k_L∞(I,H^γ)+kΦ(u)k

L^p(I,H_q^γ−γp,q).kϕk_Hγ+T^1−ν−1p kuk^ν−1

L^p(I,H_q^γ−γp,q)kuk_L∞(I,H^γ), and

d(Φ(u),Φ(v)).T^1−ν−1p kuk^ν−1

L^p(I,H_q^γ−γp,q)+kvk^ν−1

L^p(I,H_q^γ−γp,q)

ku−vkL^∞(I,L²). This shows that for allu, v∈X, there existsC >0 independent ofϕ∈H^γ such that

kΦ(u)k_L∞(I,H^γ)+kΦ(u)k

L^p(I,Hq^γ−γp,q)≤Ckϕk_Hγ+CT^1−ν−1p M^ν, d(Φ(u),Φ(v))≤CT^1−ν−1p M^ν−1d(u, v).

Therefore, if we setM = 2CkϕkH^γ and chooseT >0 small enough so thatCT^1−ν−1p M^ν−1≤ ¹₂, thenX is stable by Φ and Φ is a contraction onX. By the fixed point theorem, there exists a