Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 101, pp. 1–23. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GLOBAL WELL-POSEDNESS FOR KLEIN-GORDON-HARTREE AND FRACTIONAL HARTREE EQUATIONS ON MODULATION SPACES DIVYANG G. BHIMANI Abstract. We study the Cauchy problems for the Klein-Gordon (HNLKG), wave (HNLW), and Schrödinger (HNLS) equations with cubic convolution (of Hartree type) nonlinearity. Some global well-posedness and scattering are ob- tained for the (HNLKG) and (HNLS) with small Cauchy data in some modula- tion spaces. Global well-posedness for fractional Schrödinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modula- tion spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. As a consequence, we get local and global well-posedness and scattering in larger than usual Lp-Sobolev spaces. 1. Introduction and statement of results 1.1. Klein-Gordon-Hartree and wave-Hartree equations. We study the Cauchy problem for the Klein-Gordon and wave equations with Hartree type nonliearity utt + (I −∆)u = (V ∗ |u|2)u, u(0) = u0, ut(0) = u1 (1.1) and utt −∆u = (V ∗ |u|2), u(0) = u0, ut(0) = u1, (1.2) where u(t, x) is a complex valued function of (t, x) ∈ R × Rd, i = √ −1, ut = ∂ ∂t , utt = ∂2 ∂2t , I is the identity operator, ∆ is the Laplace operator, u0 and u1 are complex valued functions of x ∈ Rd, ∗ denotes the convolution in Rd, and V is of the type V (x) = λ |x|γ , λ ∈ R, x ∈ Rd, 0 < γ < d. (1.3) The stationary equation −∆u+(V ∗|u|2)u = σu is obtained by looking for separated solutions of (1.1) and (1.2), where u = eiλtu(x)(σ = λ2 − 1 and σ = λ2). In the case V (x) = |x|−1, the stationary equations were proposed by Hartree as a model for the helium atom. Thus the homogeneous kernel of the form (1.3) is known as Hartree potential. A class of a “nonlocal” nonlinearity that we call “Hartree type” occurs in the modeling of quantum semiconductor devices. Menzala-Strauss [21] studied the well-posedness and asymptotic behavior of equations (1.1) and (1.2). Mochizuki [26] and Hidano [16] studied scattering theory 2010 Mathematics Subject Classification. 35L71, 35Q55, 42B35, 35A01. Key words and phrases. Klein-Gordon-Hartree equation; fractional Hartree equation; wave-Hartree equation; well-posedness; modulation spaces; small initial data. ©2021. This work is licensed under a CC BY 4.0 license. Submitted April 30, 2021. Published December 21, 2021. 1 2 D. G. BHIMANI EJDE-2021/101 in the energy space (see also [10, 28]). Recently Miao-Zhang [23, 25] and Miao- Zhang-Zheng [24] studied global well-posedness and scattering theory for equations (1.1) and (1.2) below energy space. We remark that all previous authors have stud- ied equations (1.1) and (1.2) on L2-based Sobolev spaces. Mainly because generally Klein-Gordon G(t) = eit(I−∆)1/2 and wave W (t) = eit(−∆)1/2 semigroups fails to be bounded on Lp(Rd) if p 6= 2. Hence we cannot expect to solve equations (1.1) and (1.2) in Lp(Rd)(p 6= 2)-spaces. The question arises if it is possible to remove L2 constraint and consider equations (1.1) and (1.2) in function spaces which are not L2 based. This question has inspired to study equations (1.1) and (1.2) in other func- tion spaces (e.g., modulation spaces Mp,q(Rd), see Definition 2.1 below) arising in harmonic analysis. Pioneering steps in this direction were taken by Wang-Lifeng- Boling [31], Wang-Hudzik [29] and Bényi-Gröchenig-Okoudjou-Rogers [1]. In fact, in [29] it is proved that Klein-Gordon equation with power type nonlinarity is globally well-posed with small Cauchy data in M2,1(Rd). In [1, 31] it is proved that the Fourier multiplier operator with multiplier eit|ξ| α (α ∈ [0, 2]) is bounded on Mp,q(Rd) (1 ≤ p, q ≤ ∞). (The cases α = 1 and α = 2 occurs in the time evolution of the free wave and Schrödinger equations respectively.) Many authors [2, 6, 9, 12, 17, 27, 32] have studied Klein-Gordon and wave equations with power type nonliterary in modulation spaces. However, there is not much progress con- cerning well-posedness and scattering theory for the equations (1.1) and (1.2) in modulation spaces. Taking these considerations into account, we are inspired to study equations (1.1) and (1.2) with Cauchy data in modulation spaces. To sate results, we set up notation. Set 2σ(p) = (d+ 2)( 1 2 − 1 p ) (2 < p <∞, d ∈ N), 1/p+ 1/p′ = 1. We call pair (p, r) is Klein-Gordon admissible if there exists another exponent β such that 1 β + 2 r = 1, 1 3 ≤ 1 β ≤ d d+ 2 ∧ d( 1 2 − 1 p ), 1 4 ≤ p < 1 2 − 1 3d . (1.4) We remark that if pair (p, r) is Klein-Gordon admissible, then 3 ≤ r < ∞ and rd( 1 2 − 1 p ) > 1. Theorem 1.1 (Global well-posedness). Let 2 < p < 3, 1 p + γ d − 1 = 1 2p′ , s ∈ R, and pair (p, r) is Klein-Gordon admissible. Assume that (u0, u1) ∈Mp′,1 s+2σ(p)(R d)×Mp′,1 s+2σ(p)−1(Rd) and there exists a small δ > 0 such that ‖u0‖Mp′,1 s+2σ(p) + ‖u1‖Mp′,1 s+2σ(p)−1 ≤ δ. Then (1.1) has a unique global solution u ∈ C(R,Mp,1 s (Rd)) ∩ C1(R,Mp,1 s−1(Rd)) ∩ Lr(R,Mp,1 s (Rd)). One also has the bound ‖u‖Lr(R,Mp,1 s (Rd)) . ‖u0‖Mp′,1 s+2σ(p) + ‖u1‖Mp′,1 s+2σ(p)−1 . Noticing Lps(Rd) ⊂ Mp,1(Rd) for s > d and taking s = −2σ(p) (see Theorem 2.4 below), Theorem 1.1 reveals that we can control initial Cauchy data beyond EJDE-2021/101 GLOBAL CAUCHY PROBLEMS FOR HNLKG, HNLW AND HNLS 3 Lps-Sobolev spaces. To prove Theorem 1.1 we use some algebraic properties (see Proposition 2.5 below) and the integrability of time decay terms for Klein-Gordon semigroup: ‖G(t)f‖Mp,q s . (1 + |t|)−dθ(1/2−1/p)‖f‖ Mp′,q s+θ2σ(p) , where s ∈ R, 2 ≤ p ≤ ∞, 1 ≤ q < ∞, θ ∈ [0, 1] (see Proposition 2.9 below). We remark that there is no singularity at t = 0 and but preserve the same decay as in the below Lp − Lp′ estimate of G(t). This is a special characteristic of modulation spaces. Recall standard Lp − Lp′ estimate of G(t); ‖G(t)‖Lp 2σ(p) ≤ C|t|−d(1/2−1/p)‖f‖Lp′ , 2 ≤ p <∞ and since |t|−d(1/2−1/p) is not integrable, we do not know whether we can use the similar argument under Lp, Besov, or Sobolev spaces. Theorem 1.1 reveals that we have Lrt (R,Mp,1 s ) bound for the solution of (1.1) if the initial data is small enough. This implies we obtain scattering. Specifically, we have the following result. Corollary 1.2 (Scattering). Let u0 ∈ Mp,1 s (Rd), u1 ∈ Mp,1 s−1(Rd), and let u is the global solution to (1.1) such that ‖u‖Lrt (R,Mp,1 s ) ≤ M for some constant M > 0. Then there exist v±1 , v ± 2 ∈Mp,1 s (Rd) such that v± = G(t)v±1 +G(t)v±2 are solutions to the free Klein-Gordon equation utt + (I −∆)u = 0 and ‖u(t)− v±‖Mp,1 s → 0 as t→ ±∞. It remains open question to obtain the global well-posedness for equations (1.1) and (1.2) and for the large data in modulation spaces. However, we can obtain local existence with persistency of solutions. Specifically, we have the following theorem. Theorem 1.3 (Local wellposedness). Let V is given by (1.3) and X = Mp,q(Rd) (1 ≤ p ≤ 2, 1 ≤ q < 2d d+γ ) or Mp,1 s (Rd) (1 < p <∞, s ∈ R, 1 p + γ d − 1 = 1 p+ε , ε > 0). Assume that u0, u1 ∈ X. Then (1) there exists T ∗ = T ∗(‖u0‖X , ‖u1‖X) such that (1.1) has a unique solution u ∈ C([0, T ∗), X). Moreover, if T ∗ <∞, then lim supt→T∗ ‖u(·, t)‖X =∞. (2) there exists T ∗ = T ∗(‖u0‖X , ‖u1‖X) such that (1.2) has a unique solution u ∈ C([0, T ∗), X). Moreover, if T ∗ <∞, then lim supt→T∗ ‖u(·, t)‖X =∞. Up to now we cannot know if equations (1.1) and (1.2) are locally well posed in Lp(Rd), but by Theorem 1.3, in Mp,1(Rd) ⊂ Lp(Rd) (see Lemma 2.3 (2) below). Mp,1 s1 (Rd) (p ≥ 2, some s1 ∈ R) contains a class of data which are out of control of Hs(Rd). Notice that taking s1 = −d/2, it follows that Hs(Rd) = L2 s(Rd) ( M2,1 s1 (Rd) ( Mp,1 s1 (Rd) for any s > 0 (see Theorem 2.4), Theorem 1.3 reveals that we can get local well-posedness for (1.1) and (1.2) below energy spaces and in any dimension. Remark 1.4. The analogue of Theorem 1.3 holds for the generalized equations (1.1) and (1.2), that is, Klein-Gordon and wave equations with nonlinearity (V ∗ |u|2k)u (k ∈ N) when X = Mp,1 s (Rd). 1.2. Fractional Hartree equation. We study fractional Schrödinger equation with cubic convolution nonlinearity i∂tu− (−∆)α/2u = (V ∗ |u|2)u, u(x, 0) = u0(x) (1.5) 4 D. G. BHIMANI EJDE-2021/101 where u : Rt × Rdx → C, u0 : Rd → C, V is defined by (1.3), and α > 0. The fractional Laplacian is defined as F [(−∆)α/2u](ξ) = |ξ|αFu(ξ) where F denotes the Fourier transform. Equation (1.5) is known as the fractional Hartree equation. Equation (1.5) describes the dynamics of Bose-Einstein conden- sate, in which all particles are in the same state u(t, x). There is an extensive study of (1.5) with Cauchy data in Sobolev spaces, e.g., [22, 11, 7] and the references therein. Recently, for 0 < γ < min{α, d/2}, Bhimani [4] proved global well-posedness for (1.5) in Mp,q(Rd) (1 ≤ p ≤ 2, 1 ≤ q < 2d/(d + γ)) when α = 2, d ≥ 1, and with radial Cauchy data when d ≥ 2, 2d 2d−1 < α < 2 (cf. [3, 19]). Manna [20] proved small data global well-posedness for (1.5) with the potential V ∈ M1,∞(Rd). On the other hand, many authors [31, 29, 2, 15, 6] have studied nonlinear Schrödinger equation in modulation spaces. In this paper, using time integrablity of time decay factors of time decay estimate (see Proposition 2.8), we obtain global well-posedness and scattering for small Cauchy data in modulation spaces. To state result, we set up notations. We call pair (p, r) Schrödinger admissible if there exists another exponent β such that 1 β + 2 r = 1, 1 3 ≤ 1 β ≤ 1 ∧ d( 1 2 − 1 p ), 1 4 ≤ p < 1 2 − 1 3d , (1.6) and (p, r) 6= ( 2d d− 2 ,∞ ) . Notice that if pair (p, r) is Schrödinger admissible, then 3 ≤ r ≤ ∞ and rd( 1 2− 1 p ) > 1. We are now ready to state following theorem. Theorem 1.5 (Global well-posedness). Let 2 < p < 3, 1 p + γ d − 1 = 1 2p′ , s ∈ R, α = 2, and (p, r) be a Schrödinger admissible pair. Assume that u0 ∈ Mp′,1 s (Rd) and there exists a small δ > 0 such that ‖u0‖Mp′,1 s ≤ δ. Then (1.5) has a unique global solution u ∈ C(R,Mp,1 s (Rd)) ∩ Lr(R,Mp,1 s (Rd)). One also has the bound ‖u‖Lr(R,Mp,1 s (Rd)) . ‖u0‖Mp′,1 s . In [4, Theorem 1.1] global well-posedness for (1.5) studied with the range of γ < min{d/2, 2}. Notice that Theorem 1.5 covers range of γ > d/2 as γ d = 1 + p−3 2p and γ/d > 1/2⇔ p > 3/2. Corollary 1.6 (Scattering). Let u0 ∈Mp,1 s (Rd) and let u is the global solution to (1.5) with initial u(0) = u0 such that ‖u‖Lrt (R,Mp,1 s ) ≤M for some constant M > 0 and r < ∞. Then there exist solutions eit∆u± to the free Schrödinger equation iut + ∆u = 0 such that ‖u(t)− eit∆u±‖Mp,1 s → 0 as t→ ±∞. EJDE-2021/101 GLOBAL CAUCHY PROBLEMS FOR HNLKG, HNLW AND HNLS 5 Remark 1.7. Taking Proposition 2.8 into account, the method of proof of Theorem 1.5 may further be applied to equation (1.5) with α > 2 to obtain the global well- posedness for the small data in modulation spaces. Theorem 1.8 (Global well-posedness). Let V ∈ M∞,1(Rd) and 1 2 < α ≤ 2. As- sume that u0 ∈ Mp,q(Rd)(1 ≤ p, q ≤ 2). Then there exists a unique global solution of (1.5) such that u ∈ C(R,Mp,q(Rd)). In [19, Theorem 1.2] it is proved that (1.5) with potential V ∈ M∞,1(Rd) and α = 2 is globally well-posed in Mp,q(Rd)(1 ≤ q ≤ p ≤ 2). Notice that Theorem 1.8 generalize this result for (1.5) with 1 2 < α < 2. Up to now we cannot know (1.5) is locally well-posed in Lp(Rd) but, by Theorem 1.9, in Mp,1(Rd). Local well-posedness for (1.5) are studied by many authors in Sobolev spaces. Modulation spaces enjoy lower derivative regularity (see Proposi- tion 2.4 below) and we can solve (1.5) with the lower regularity assumption for the Cauchy data. Theorem 1.9 (Local well-posedness). Let V is given by (1.3), 1/2 < α ≤ 2 and u0 ∈ Mp,1 s (Rd) (1 < p < ∞, s ∈ R, 1 p + γ d − 1 = 1 p+ε , ε > 0). Then there exists T ∗ = T ∗(‖u0‖Mp,1 s ) such that (1.5) has a unique solution u ∈ C([0, T ∗),Mp,1 s (Rd)). Moreover, if T ∗ <∞, then lim supt→T∗ ‖u(·, t)‖Mp,1 s =∞. Remark 1.10. (1) The analogue of Theorem 1.9 holds for the generalized equation (1.5) and (1.2), that is, fractional Schrödinger equation with nonlinearity (V ∗ |u|2k)u (k ∈ N) when X = Mp,1 s (Rd). (2) We have obtain local well-posedness for generalized equations (1.1), (1.2) and (1.5) with potential V ∈ FLq(Rd) (1 < q < ∞) or M∞,1(Rd) or V ∈M1,∞(Rd). See Theorems 6.1 and Remark 6.2 below. The remainder of this paper is organized as follows. In Section 2, we introduce notations and preliminaries which will be used in the sequel. In Section 3, we prove some Strichartz type estimates and boundedness of Hartree nonlinearity in modulation spaces. In Section 4, we prove Theorems 1.1, 1.3 and Corollary 1.2. In Section 5, we prove Theorems 1.5, 1.8 and 1.9, and Corollary 1.6. In Section 6, we give sketch proof of Remark 1.10 (2). 2. Preliminaries 2.1. Notation. The notation A . B means A ≤ cB for a some constant c > 0, whereas A � B means c−1A ≤ B ≤ cA for some c ≥ 1 and a ∧ b = min{a, b}. The symbol A1 ↪→ A2 denotes the continuous embedding of the topological linear space A1 into A2. The Lp(Rd) norm is denoted by ‖f‖Lp = (∫ Rd |f(x)|pdx )1/p (1 ≤ p <∞), the L∞(Rd) norm is ‖f‖L∞ = ess.supx∈Rd |f(x)|. For 1 ≤ p ≤ ∞, p′ denotes the Hölder conjugate of p, that is, 1/p+ 1/p′ = 1. We use Lrt (I,X) to denote the space time norm ‖u‖Lrt (I,X) = (∫ I ‖u‖rXdt )1/r , 6 D. G. BHIMANI EJDE-2021/101 where I ⊂ R is an interval and X is a Banach space. The Schwartz space is denoted by S(Rd) (with it’s usual topology), and the space of tempered distributions is denoted by S ′(Rd). For x = (x1, . . . , xd), y = (y1, . . . , yd) ∈ Rd, we put x · y =∑d i=1 xiyi. Let F : S(Rd)→ S(Rd) be the Fourier transform defined by Ff(w) = f̂(w) = ∫ Rd f(t)e−2πit·wdt, w ∈ Rd. Then F is a bijection and the inverse Fourier transform is given by F−1f(x) = f∨(x) = ∫ Rd f(w) e2πix·wdw, x ∈ Rd, and this Fourier transform can be uniquely extended to F : S ′(Rd)→ S ′(Rd). The Fourier-Lebesgue spaces FLp(Rd) is defined by FLp(Rd) = { f ∈ S ′(Rd) : ‖f‖FLp := ‖f̂‖Lp <∞ } . The standard Sobolev spaces W s,p(Rd) (1 < p < ∞, s ≥ 0) have a different char- acter according to whether s is integer or not. Namely, for s integer, they consist of Lp-functions with derivatives in Lp up to order s, hence coincide with the Lps- Sobolev spaces (also known as Bessel potential spaces), defined for s ∈ R by Lps(Rd) = { f ∈ S ′(Rd) : ‖f‖Lps := ‖F−1[〈·〉sF(f)]‖Lp <∞ } , where 〈ξ〉s = (1 + |ξ|2)s/2 (ξ ∈ Rd). Note that Lps1(Rd) ↪→ Lps2(Rd) if s2 ≤ s1. 2.2. Modulation spaces. Feichtinger [13] introduced a class of Banach spaces, the so called modulation spaces, which allow a measurement of space variable and Fourier transform variable of a function or distribution on Rd simultaneously using the short-time Fourier transform(STFT). The STFT of a function f with respect to a window function g ∈ S(Rd) is defined by Vgf(x,w) = ∫ Rd f(t)g(t− x)e−2πiw·tdt, (x,w) ∈ R2d whenever the integral exists. For x, y ∈ Rd the translation operator Tx and the modulation operator My are defined by Txf(t) = f(t−x) and Myf(t) = e2πiy·tf(t). In terms of these operators the STFT may be expressed as Vgf(x, y) = 〈f,MyTxg〉 where 〈f, g〉 denotes the inner product for L2 functions, or the action of the tem- pered distribution f on the Schwartz class function g. Thus V : (f, g) → Vg(f) extends to a bilinear form on S ′(Rd) × S(Rd) and Vg(f) defines a uniformly con- tinuous function on Rd × Rd whenever f ∈ S ′(Rd) and g ∈ S(Rd). Definition 2.1 (modulation spaces). Let 1 ≤ p, q ≤ ∞, s ∈ R and 0 6= g ∈ S(Rd). The weighted modulation space Mp,q s (Rd) is defined to be the space of all tempered distributions f for which the following norm is finite: ‖f‖Mp,q s = (∫ Rd (∫ Rd |Vgf(x, y)|pdx )q/p (1 + |y|2)sq/2 dy )1/q , for 1 ≤ p, q < ∞. If p or q is infinite, ‖f‖Mp,q s is defined by replacing the corre- sponding integral by the essential supremum. For s = 0, we write Mp,q 0 (Rd) = Mp,q(Rd). EJDE-2021/101 GLOBAL CAUCHY PROBLEMS FOR HNLKG, HNLW AND HNLS 7 Remark 2.2. The definition of the modulation space given above, is independent of the choice of the particular window function. See [14, Proposition 11.3.2(c)]. Applying the frequency-uniform localization techniques, one can get an equiva- lent definition of modulation spaces [29] as follows. Let Qk be the unit cube with the center at k, so {Qk}k∈Zd constitutes a decomposition of Rd, that is, Rd = ∪k∈ZdQk. Let ρ ∈ S(Rd), ρ : Rd → [0, 1] be a smooth function satisfying ρ(ξ) = 1 if |ξ|∞ ≤ 1 2 and ρ(ξ) = 0 if |ξ|∞ ≥ 1. Let ρk be a translation of ρ, that is, ρk(ξ) = ρ(ξ − k), k ∈ Zd. Denote σk(ξ) = ρk(ξ)∑ l∈Zd ρl(ξ) , k ∈ Zd. Then {σk(ξ)}k∈Zd satisfies the following |σk(ξ)| ≥ c, ∀z ∈ Qk, suppσk ⊂ {ξ : |ξ − k|∞ ≤ 1},∑ k∈Zd σk(ξ) ≡ 1, ∀ξ ∈ Rd, |Dασk(ξ)| ≤ C|α|, ∀ξ ∈ Rd, α ∈ (N ∪ {0})d. The frequency-uniform decomposition operators can be exactly defined by �k = F−1σkF . For 1 ≤ p, q ≤ ∞, s ∈ R, it is known [13] that ‖f‖Mp,q s � ( ∑ k∈Zd ‖�k(f)‖qLp(1 + |k|)sq )1/q , with natural modifications for p, q = ∞. We notice almost orthogonality relation for the frequency-uniform decomposition operators �k = ∑ ‖`‖∞≤1 �k+`�k, k, ` ∈ Zd, where ‖`‖∞ = max{|`i| : `i ∈ Z, i = 1, . . . , d}. Lemma 2.3 ([30, 14, 27]). Let p, q, pi, qi ∈ [1,∞] (i = 1, 2), s, s1, s2 ∈ R. Then (1) Mp1,q1 s1 (Rd) ↪→Mp2,q2 s2 (Rd) whenever p1 ≤ p2 and q1 ≤ q2 and s2 ≤ s1. (2) Mp,q1(Rd) ↪→ Lp(Rd) ↪→ Mp,q2(Rd) holds for q1 ≤ min{p, p′} and q2 ≥ max{p, p′} with 1 p + 1 p′ = 1. (3) Mmin{p′,2},p(Rd) ↪→ FLp(Rd) ↪→Mmax{p′,2},p(Rd), 1 p + 1 p′ = 1. (4) S(Rd) is dense in Mp,q(Rd) if p and q <∞. (5) Mp,p(Rd) ↪→ Lp(Rd) ↪→ Mp,p′(Rd) for 1 ≤ p ≤ 2 and Mp,p′(Rd) ↪→ Lp(Rd) ↪→Mp,p(Rd) for 2 ≤ p ≤ ∞. (6) The Fourier transform F : Mp,p s (Rd)→Mp,p s (Rd) is an isomorphism. (7) The space Mp,q s (Rd) is a Banach space. (8) The space Mp,q s (Rd) is invariant under complex conjugation. Theorem 2.4 ([18, 27]). Let 1 ≤ p, q ≤ ∞, s1, s2 ∈ R, and τ(p, q) = max { 0, d( 1 q − 1 p ), d( 1 q + 1 p − 1) } . 8 D. G. BHIMANI EJDE-2021/101 Then Lps1(Rd) ⊂Mp,q s2 (Rd) if and only if one of the following conditions is satisfied: (i) q ≥ p > 1, s1 ≥ s2 + τ(p, q); (ii) p > q, s1 > s2 + τ(p, q); (iii) p = 1, q =∞, s1 ≥ s2 + τ(1,∞); (iv) p = 1, q 6=∞, s1 > s2 + τ(1, q). Proposition 2.5 (Algebra property [2]). Let m ∈ N, s ≥ 0. Assume that∑m i=1 1 pi = 1 p0 , ∑m i=1 1 qi = m− 1 + 1 q0 with 0 < pi ≤ ∞, 1 ≤ qi ≤ ∞ for 1 ≤ i ≤ m. Then we have ‖ m∏ i=1 ui‖Mp0,q0 s . m∏ i=1 ‖ui‖Mpi,qi s . Proposition 2.6 (isomorphism [13]). Let 0 < p, q ≤ ∞, s, σ ∈ R. Then Jσ : (I − ∆)σ/2 : Mp,q s (Rd) → Mp,q s−σ(Rd) is an isomorphic mapping. (We denote J1 = J .) Lemma 2.7. Let s ∈ R, 1 ≤ p, q < ∞, and Ω be a compact subset of Rd. Then SΩ = {f : f ∈ S(Rd) and supp f̂ ⊂ Ω} is dense in Mp,q s (Rd). For f ∈ S(Rd), we define the fractional Schrödinger propagator eit(−∆)α/2 for t, α ∈ R as follows: U(t)f(x) = eit(−∆)α/2f(x) = ∫ Rd eiπt|ξ| α f̂(ξ)e2πiξ·x dξ. When α = 2, we write U(t) = S(t) = e−it∆ (corresponding to usual Schrödinger equation). The next proposition shows that the uniform boundedness and truncated decay estimates of the Schrödinger propagator eit(−∆)α/2 on modulation spaces. Proposition 2.8 ([8, 29]). (1) Let 1/2 < α ≤ 2, 1 ≤ p, q ≤ ∞. Then ‖U(t)f‖Mp,q ≤ (1+|t|)d| 1 p− 1 2 |‖f‖Mp,q . (2) Let α ≥ 2 and 2 ≤ p, q ≤ ∞. Then ‖U(t)f‖Mp,q ≤ (1+|t|)− 2d α ( 1 2− 1 p )‖f‖Mp′,q . Now we consider the truncated decay estimate and uniform bounded estimates for the Klein-Gordon semigroup G(t). Proposition 2.9 (See [29, Proposition 4.2]). Let G(t) = eit(I−∆)1/2 (t ∈ R). (1) Let s ∈ R, 2 ≤ p ≤ ∞, 1 ≤ q < ∞, θ ∈ [0, 1], and 2σ(p) = (d + 2)( 1 2 − 1 p ). Then we have ‖G(t)f‖Mp,q s . (1 + |t|)−dθ(1/2−1/p)‖f‖ Mp′,q s+θ2σ(p) . (2) Let s ∈ R and 1 ≤ p, q ≤ ∞. Then we have ‖G(t)f‖Mp,q s ≤ C(1 + |t|)d|1/2−1/p|‖f‖Mp,q s . Proposition 2.10 (Uniform boundedness of wave propagator [2]). For σ1(ξ) = sin(2πt|ξ|)/2π|ξ|, σ2(ξ) = cos(2πt|ξ|), and f ∈ S(Rd), we define Hσif(x) = (σif̂)∨(x) (x ∈ Rd, i = 1, 2). Let s ∈ R and 1 ≤ p, q ≤ ∞. Then we have ‖Hσif‖Mp,q s ≤ cd(1 + t2)d/4‖f‖Mp,q s . EJDE-2021/101 GLOBAL CAUCHY PROBLEMS FOR HNLKG, HNLW AND HNLS 9 Proposition 2.11 (Bernstein multiplier theorem [30]). Let L ∈ Z, L > d/2, ∂αxiρ ∈ L 2, i = 1, 2, . . . , d, 0 ≤ α ≤ L. Then ρ is a multiplier on Lp (1 ≤ p ≤ ∞). Moreover there exists a constant C such that ‖ρ‖Mp ≤ C‖ρ‖1−d/2LL2 ( d∑ i=1 ‖∂Lxiρ‖L2 )d/2L . Proposition 2.12 ([30]). Let Ω ⊂ Rd be a compact subset and let 1 ≤ p ≤ ∞, sp = d( 1 p∧1 − 1 2 ). If s > sp, then there exists a C > 0 such that ‖F−1φFφ‖Lp ≤ C‖φ‖Hs‖f‖Lp holds for all f ∈ LpΩ and φ ∈ Hs(Rd) = L2 s(Rd). 3. Nonlinear estimates in Mp,q s (Rd) In this section we prove estimates for Hartree nonlinearity (Corollary 3.3 and Lemmas 3.4 and 3.5) and Strichartz type estimates (Proposition 3.6). We shall apply these to prove main theorems in the following sections. We define fractional integral operator Tγ(0 < γ < d) as follows Tγf(x) = Vγ ∗ f(x) = ± ∫ Rd f(y) |x− y|γ dy, (f ∈ S(Rd), Vγ(x) = ±|x|−γ). It is known Tγ is bounded from Lp(Rd) to Lq(Rd) for some specific p, q and γ. Proposition 3.1 (Hardy-Littlewood-Sobolev inequality). Assume that 0 < γ < d and 1 < p < q <∞ with 1 p + γ d − 1 = 1 q . Then we have ‖Tγf‖Lq ≤ Cd,γ,p‖f‖Lp . We prove an analogue of Hardy-Littlewood-Sobolev inequality in case of modu- lation spaces. Proposition 3.2. Assume that 0 < γ < d, 1 < p1 < p2 <∞ with 1 p1 + γ d − 1 = 1 p2 and 1 ≤ q ≤ ∞, s ≥ 0. Then the map Tγ is bounded from Mp1,q s (Rd) to Mp2,q s (Rd): ‖Tγf‖Mp2,q s . ‖f‖Mp1,q s . Proof. We may rewrite the STFT as Vg(x,w) = e−2πix·w(f ∗ Mwg ∗)(x) where g∗(y) = g(−y). Using Hardy-Littlewood-Sobolev inequality, we obtain ‖Tγf‖Mp2,q s = ‖ ‖Vγ ∗ (f ∗Mwg ∗)‖Lp2 〈w〉s‖Lqw . ‖ ‖f ∗Mwg ∗)‖Lp1 〈w〉s‖Lqw . ‖f‖Mp1,q s . This completes the proof. � Corollary 3.3. Let 1 < p <∞ and 1 p + γ d − 1 = 1 p+ε for some ε > 0. Then ‖(Vγ ∗ |f |2k)f‖Mp,1 s . ‖f‖2k+1 Mp,1 s (k ∈ N). Proof. By Proposition 2.5 and Lemma 2.3(1), we have ‖(Vγ ∗ |f |2k)f‖Mp,1 s . ‖Tγ |f |2k‖M∞,1s ‖f‖Mp,1 s . ‖Tγ |f |2k‖Mp+ε,1 s ‖f‖Mp,1 s , for some ε > 0. By Propositions 3.2 and 2.5, we have ‖Tγ |f |2k‖Mp+ε,1 s . ‖|f |2k‖Mp,1 s . ‖f‖2k Mp,1 s . This completes the proof. � 10 D. G. BHIMANI EJDE-2021/101 Lemma 3.4. Let 1 < p <∞ and 1 p + γ d − 1 = 1 p+ε for some ε > 0. Then we have ‖(Vγ ∗|f |2)f−(Vγ ∗|g|2)g‖Mp,1 s . (‖f‖2 Mp,1 s +‖f‖Mp,1 s ‖g‖Mp,1 s +‖g‖2 Mp,1 s )‖f−g‖Mp,1 s . Proof. Using the ideas of proof as in Corollary 3.3, we obtain ‖(Vγ ∗ |f |2)(f − g)‖Mp,1 s . ‖f‖2 Mp,1 s ‖f − g‖Mp,1 s , and ‖(Vγ ∗ (|f |2 − |g|2))g‖Mp,1 s . ‖|f |2 − |g|2‖Mp,1 s ‖g‖Mp,1 s . ( ‖f‖Mp,1 s ‖g‖Mp,1 s + ‖g‖2 Mp,1 s ) ‖f − g‖Mp,1 s . This together with the following identity (Vγ ∗ |f |2)f − (Vγ ∗ |g|2)g = (Vγ ∗ |f |2)(f − g) + (Vγ ∗ (|f |2 − |g|2))g, gives the desired inequality. � Lemma 3.5. Let 2 < p < 2p′ and 1 p + γ d − 1 = 1 2p′ . Then we have ‖(Vγ∗|f |2)f−(Vγ∗|g|2)g‖ Mp′,1 s . (‖f‖2 Mp,1 s +‖f‖Mp,1 s ‖g‖Mp,q s +‖g‖2 Mp,1 s )‖f−g‖Mp,1 s . Proof. By Proposition 2.5, we have ‖(Vγ ∗ |f |2)(f − g)‖ Mp′,1 s . ‖Vγ ∗ |f |2‖M2p′,1 s ‖f − g‖ M2p′,1 s . ‖|f |2‖Mp,1 s ‖f − g‖Mp,1 s and ‖(Vγ ∗ (|f |2 − |g|2))g‖ Mp′,1 s . ‖Vγ ∗ (|f |2 − |g|2)‖ M2p′,1 s ‖g‖ M2p′,1 s . ‖|f |2 − |g|2‖Mp,1 s ‖g‖Mp,1 s . (‖f‖Mp,1 s ‖g‖Mp,1 s + ‖g‖2 Mp,1 s )‖f − g‖Mp,1 s . � Recall that equation (1.1) have the following equivalent form u(t) = K ′(t)u0 +K(t)u1 − Bf(u), where we denote ω = (I −∆), K(t) = sin tω1/2 ω1/2 , K ′(t) = cos tω1/2, B = ∫ t 0 K(t− τ) · dτ. We prove following Strichartz type estimates in modulation spaces. Proposition 3.6. Let F (u) = (Vγ ∗ |u|2)u, p ∈ (2, 3), 1 p + γ d − 1 = 1 2p′ and pair (p, r) is Klein-Gordon admissible. Then we have∥∥∫ t 0 K(t− τ)F (u(τ))dτ ∥∥ Lrt (R,Mp,1 s ) . ‖F (u)‖ L r/3 t (R,Mp′,1 s ) . ‖u‖3 Lrt (R,Mp,1 s ) . Proof. Since G(t) = eitω 1/2 , we have K(t)ω1/2 = (G(t)−G(−t))/2i. By the general Minkowski inequality, Propositions 2.9 and 2.6, we have∥∥∫ t 0 K(t− τ)F (u(τ))dτ ∥∥ Lrt (R,Mp,1 s ) EJDE-2021/101 GLOBAL CAUCHY PROBLEMS FOR HNLKG, HNLW AND HNLS 11 . ∥∥∫ t 0 ‖K(t− τ)F (u(τ))‖Mp,1 s dτ ∥∥ Lrt (R) . ∥∥∫ t 0 (1 + |t− τ |)−dθ(1/2−1/p)‖F (u)‖ Mp′,1 s+θ2σ(p)−1 dτ ∥∥ Lrt (R) . ∥∥∫ R (1 + |t− τ |)−dθ(1/2−1/p)h(τ)dτ ∥∥ Lrt (R) . ∥∥g ∗ h‖Lrt , where h(τ) = ‖F (u)‖ Mp′,1 s+θ2σ(p)−1 , g(t) = (1 + |t|)−dθ(1/2−1/p) and θ ∈ [0, 1]. We divide Klein-Gordon admissible pairs (see (1.4)) into two cases. Case I: 1 β = d d+2 ∧ d( 1 2 − 1 p ). In this case 1 β < 1 and there exists θ ∈ (0, 1] such that 1 β = θd( 1 2 − 1 p ) = d d+ 2 ∧ d( 1 2 − 1 p ). With this θ, we have θ2σ(p)− 1 ≤ 0. Since pair (p, r) is Klein-Gordon admissible, we have 1 r = 3 r − 1− dθ(1/2− 1/p) 1 and r/3 > 1. With this θ, by Hardy-Littlewood-Sobolev inequality in dimension one, we have∥∥∫ t 0 K(t− τ)F (u(τ))dτ ∥∥ Lrt (R,Mp,1 s ) . ‖g ∗ h‖Lrt (R) . ‖‖F (u)‖ Mp′,1 s ‖Lr/3 = ‖F (u)‖ Lr/3(R,Mp′,1 s ) . Case II: 1 β < d d+2 ∧ d( 1 2 − 1 p ). In this case there exists θ ∈ [0, 1] such that 1 β < θd( 1 2 − 1 p ) ≤ d d+ 2 ∧ d( 1 2 − 1 p ). With this θ, we have βθd( 1 2 − 1 p ) > 1, and θ2σ(p) − 1 ≤ 0. By Young and Hölder inequalities, we have∥∥∫ t 0 K(t− τ)F (u(τ))dτ ∥∥ Lrt (R,Mp,1 s ) . ‖g ∗ h‖Lrt . ‖g‖Lβ‖ ‖F (u)‖ Mp′,1 s ‖Lr/3 . ‖F (u)‖ Lr/3(R,Mp′,1 s ) . By Propositions 2.5 and 3.2 and Lemma 2.3 (1), we have ‖F (u)‖ Lr/3(R,Mp′,1 s ) . (∫ (‖Tγ |u|2‖M2p′,1 s ‖u‖ M2p′,1 s )r/3dt )3/r . (∫ (‖|u|2‖Mp,1 s ‖u‖Mp,1 s )r/3dt )3/r . (∫ ‖u‖r Mp,1 s dt )3/r . ‖u‖3 Lrt (R,Mp,1 s ) . This completes the proof. � 12 D. G. BHIMANI EJDE-2021/101 Lemma 3.7. Let F (u) = (Vγ ∗ |u|2)u, p ∈ (2, 3), 1 p + γ d − 1 = 1 2p′ and pair (p, r) is Klein-Gordon admissible. Then ‖ ∫ t 0 K(t− τ)[F (u(τ))− F (v(τ))]dτ‖Lrt (R,Mp,1 s ) . (‖u‖2 Lrt (R,Mp,1 s ) + ‖u‖Lrt (R,Mp,1 s )‖v‖Lrt (R,Mp,1 s ) + ‖v‖2 Lrt (R,Mp,1 s )‖u− v‖3 Lrt (R,Mp,1 s ) . Proof. By Proposition 3.6, we have ‖ ∫ t 0 K(t− τ)[F (u(τ))− F (v(τ))]dτ‖Lrt (R,Mp,1 s ) . ‖F (u)− F (v))‖ L r/3 t (R,Mp′,1 s ) . By Proposition 2.5, Lemma 2.3(1) and Hölder inequality, we obtain ‖(Vγ ∗ |u|2)(u− v)‖ Lr/3(R,Mp′,1 s ) . ‖u‖2 Lr(R,Mp,1 s ) ‖u− v‖Lr(R,Mp,1 s ) and ‖(Vγ ∗ (|u|2 − |v|2))v‖ Lr/3(R,Mp′,1 s ) . ( ‖u‖Lr(R,Mp,1 s )‖v‖Lr(R,Mp,1 s ) + ‖v‖2 Lr(R,Mp,1 s ) ) ‖u− v‖Lr(R,Mp,1 s ). � Lemma 3.8 ([4]). Let V be given by (1.3), 1 ≤ p ≤ 2, 1 ≤ q < 2d d+γ . Then for any f, g ∈Mp,q(Rd), we have (1) ‖(V ∗ |f |2)f‖Mp,q . ‖f‖3Mp,q . (2) ‖(V ∗|f |2)f−(K ∗|g|2)g‖Mp,q . (‖f‖2Mp,q +‖f‖Mp,q‖g‖Mp,q +‖g‖2Mp,q )‖f−g‖Mp,q . 4. Proofs of theorems 1.1 and 1.3 Proof of Theorem 1.1. Recall that equation (1.1) have the equivalent form u(t) = K ′(t)u0 +K(t)u1 − ∫ t 0 K(t− τ)F (u(τ))dτ =: J (u) where K(t) = sin t(I −∆)1/2 (I −∆)1/2 , K ′(t) = cos t(I −∆)1/2, F (u) = (Vγ ∗ |u|2)u. Denote X = Lr(R,Mp,1 s (Rd)). For δ > 0, put Bδ = {u ∈ X : ‖u‖X ≤ δ} which is the closed ball of radius δ, and centered at the origin in X. Since rd( 1 2 − 1 p ) > 1, we have (1 + |t|)−d( 1 2− 1 p ) ∈ Lr(R). Now by Proposition 2.9, we have ‖K(t)u0‖X . ‖(1 + |t|)−d( 1 2− 1 p )‖u0‖Mp′,1 s+2σ(p) ‖Lr . ‖u0‖Mp′,1 s+2σ(p) . By Propositions 2.9 and 2.6, we have ‖K ′(t)u1‖X . ‖(1 + |t|)−d( 1 2− 1 p )‖u1‖Mp′,1 s+2σ(p)−1 ‖Lr . ‖u1‖Mp′,1 s+2σ(p)−1 . By Proposition 3.6, we have ‖ ∫ t 0 K(t− τ))F (u(τ))dτ‖X . ‖u‖3X . EJDE-2021/101 GLOBAL CAUCHY PROBLEMS FOR HNLKG, HNLW AND HNLS 13 Thus we have ‖J (u)‖X . ‖u0‖Mp′,1 s+2σ(p) + ‖u1‖Mp′,1 s+2σ(p)−1 + ‖u‖3X . By Lemma 3.7, for any u, v ∈ Bδ, we have ‖J u− J v‖X . (‖u‖2X + ‖u‖X‖v‖X + ‖v‖2X)‖u− v‖X . If we assume that δ > 0 is sufficiently small, then J : X → X is a strict contraction. Therefor J has a unique fixed point and we have u ∈ Lr(R,Mp,1 s (Rd)). We shall now verify this u ∈ C(R,Mp,1 s (Rd)) ∩ C1(R,Mp,1 s−1(Rd)) and ‖u‖Lr(R,Mp,1 s (Rd)) . ‖u0‖Mp′,1 s+2σ(p) + ‖u1‖Mp′,1 s+2σ(p)−1 . To prove u ∈ C(R,Mp,1 s (Rd)). It is equivalent to prove that ‖u(tn, ·)− u(t, ·)‖Mp,1 s → 0 (4.1) as tn → t for arbitrary fixed t > 0. We note that ‖u(tn, ·)− u(t, ·)‖Mp,1 s ≤ ‖K ′(tn)u0 −K ′(t)u0‖Mp,1 s + ‖K(tn)u1 −K(t)u1‖Mp,1 s + ‖ ∫ tn 0 K(tn − τ)F (u(τ))− ∫ t 0 K(t− τ)F (u(τ))‖Mp,1 s = I + II + III. Recall that u0, J −1u1 ∈Mp,1 s (Rd) (see Proposition 2.6). For I and II, by density Lemma 2.7, Proposition 2.9, triangle inequality, and since G(t) = eitω 1/2 (ω = I − ∆), we only need to prove that G(t)v ∈ C(R,Mp,1 s (Rd)) for v ∈ SΩ. By Hausdroff-Young inequality, we have ‖�k(G(tn)v −G(t)v)‖Lp . ‖σk(eitn(1+|ξ|2)1/2 − eit(1+|ξ|2)1/2)v̂(ξ)‖Lp′ . ‖(eitn(1+|ξ|2)1/2 − eit(1+|ξ|2)1/2)v̂(ξ)‖Lp′ → 0 as tn → t, by Lebesgue dominated convergence theorem. Since v̂ ∈ SΩ, there exists only finite number of k such that �k(G(tn)v − G(t)v) 6= 0, so we have ‖G(tn)v−G(t)v‖Mp,1 s → 0 as tn → t. It follows that I and II tends to 0 as tn → t. For III, we note that III . ∥∥ ∫ tn 0 K(tn − τ)F (u(τ))dτ − ∫ tn 0 K(t− τ)F (u(τ))dτ ∥∥ Mp,1 s + ∥∥∫ tn 0 K(t− τ)F (u(τ))dτ − ∫ t 0 K(t− τ)F (u(τ))dτ ∥∥ Mp,1 s . ∫ tn 0 ‖(K(tn − τ)−K(t− τ))F (u(τ))‖Mp,1 s dτ + ∫ t tn ‖K(t− τ)F (u(τ))‖Mp,1 s dτ = Ĩ + ĨI. For ‖(K(tn − τ)−K(t− τ))F (u(τ)‖Mp,1 s . ‖F (u(τ))‖ Mp′,1 s . ‖u‖3 Mp,1 s ∈ Lr(R). Since 3 ≤ r, we have Lr[0, t] ⊂ L1[0, t] and so ‖u‖3 Mp,1 s ∈ L1[0, t], hence ‖(K(tn − τ)−K(t− τ))F (u(τ)‖Mp,1 s ∈ L1[0, t]. 14 D. G. BHIMANI EJDE-2021/101 Since ‖(K(tn− τ)−K(t− τ))F (u(τ)‖Mp,1 s → 0 as tn → t, therefore we have Ĩ → 0 as tn → t. Secondly as in the proof of Proposition 3.6, we obtain ĨI . ∫ t tn (1 + |t− τ |)−d(1/2−1/p)‖F (u(τ))‖ Mp′,1 s dτ . ∫ t tn ‖F (u(τ))‖ Mp′,1 s dτ . ∫ t tn ‖u‖3 Mp,1 s dτ → 0 as tn → t as ‖u‖3 Mp,1 s ∈ L1([0, t]). It follows that (4.1) holds. We now prove that ut(t) exists and is continuous in Mp,1 s sense. For u0, J −1u1 ∈ Mp,1 s (Rd) (see Proposition 2.6), and since G(t) = eitω 1/2 (ω = I−∆), we should only deal with the derivative of G(t)ψ(x) for ψ ∈ Mp,1 s (Rd) and ∫ t 0 K(t− τ)F (u(τ))dτ . By Lemma 2.7, for every ε > 0, there exists v ∈ SΩ ∩Mp,1 s (Rd) such that ‖ψ − v‖Mp,1 s < ε. For the derivative of G(t)ψ(x) at t = t3 for ψ ∈Mp,1 s (Rd), we have ∥∥G(t)ψ −G(t3)ψ t− t3 − iω1/2G(t3)ψ ∥∥ Mp,1 s−1 = ∥∥G(t)ψ −G(t3)ψ (t− t3)ω1/2 − iG(t3)ψ ∥∥ Mp,1 s ≤ ∥∥G(t)(ψ − v)−G(t3)(ψ − v) (t− t3)ω1/2 ∥∥ Mp,1 s + ∥∥G(t)(v)−G(t3)(v) (t− t3)ω1/2 − iG(t3)v ∥∥ Mp,1 s + ‖iG(t3)(ψ − v)‖Mp,1 s = IV + V + V I. For V , by the Hausdroff-Young inequality and the Lebesgue dominated conver- gence theorem, we have ‖�k( G(t)(v)−G(t3)(v) (t− t3)ω1/2 − iG(t3)v)‖Lp . ‖σk( eit〈ξ〉 − eit3〈ξ〉 (t− t3)〈ξ〉 − ieit3〈ξ〉)v̂‖Lp′ → 0 as t→ t3. As v ∈ SΩ ∩Mp,1 s (Rd), so there is only the finite number of k such that(G(t)(v)−G(t3)(v) (t− t3)ω1/2 − iG(t3)v ) 6= 0. Thus we get V → 0 as t → t3, that is, (G(t)v(x))t = iω1/2G(t)v(x) in Mp,1 s−1(Rd) for v ∈ SΩ ∩Mp,1 s (Rd). For IV , by the Bernstein multiplier theorem, we have ‖�l (G(t)(ψ − v)−G(t3)(ψ − v) (t− t3)ω1/2 ) ‖Lp . ‖ψ − v‖Lp . Using the almost orthogonality of modulation space, we have IV . ‖ψ−v‖Mp,1 s < ε. For V I, by Proposition 2.9 (2), we have V I = ‖iG(t3)(ψ−v)‖Mp,1 s . ‖ψ−v‖Mp,1 s < ε. Accordingly, for ψ ∈Mp,1 s (Rd), (G(t)ψ)t = iω1/2G(t)ψ in Mp,1 s (Rd). (4.2) EJDE-2021/101 GLOBAL CAUCHY PROBLEMS FOR HNLKG, HNLW AND HNLS 15 For the nonlinear part,∥∥∥∫ t0 K(t− τ)F (u(τ))dτ − ∫ t3 0 K(t3 − τ)F (u(τ))dτ t− t3 − ∫ t3 0 K ′(t3 − τ)F (u)dτ ∥∥∥ Mp,1 s−1 ≤ ∥∥∥∫ t30 (K(t− τ)−K(t3 − τ)F (u(τ))dτ t− t3 − ∫ t3 0 K ′(t3 − τ)F (u)dτ ∥∥∥ Mp,1 s−1 + ∥∥∫ tt3 K(t− τ)F (u(τ))dτ t− t3 ∥∥ Mp,1 s−1 . ∫ t3 0 ∥∥( (K(t− τ)−K(t3 − τ) t− t3 −K ′(t3 − τ))F (u) ∥∥ Mp,1 s−1 dτ + max τ∈[t3,t] ‖K(t− τ)F (u(τ))‖Mp,1 s−1 . If ω(t, x) ∈ C(I,Mp,1 s (Rd)), then we have K(t)ω(t, x) ∈ C(I,Mp,1 s−1(Rd)). In fact taking taking advantage of (4.2) and the Lebesgue dominated convergence theorem, we obtain ‖K(t)ω(t, x)−K(t3)ω(t3, x)‖Mp,1 s−1 ≤ ‖(K(t)−K(t3))ω(t3, x)‖Mp,1 s−1 + ‖K(t)(ω(t, x)− ω(t3, x))‖Mp,1 s−1 → 0 as t→ t3. Recall that F (u) ∈ C(R,Mp,1 s (Rd)) and apply (4.2) and the Lebesgue dominated convergence theorem, we can get(∫ t 0 K(t− τ)F (u(τ))dτ )′ t ∣∣∣ t=t3 = ∫ t3 t=0 K ′(t3 − τ)F (u(τ))dτ in Mp,1 s−1(Rd). Consequently, ut(t) = −J2K(t)u0 +K ′(t)u1 − ∫ t 0 K ′(t− τ)F (u(τ))dτ in Mp,1 s (Rd). Next, the proof of time continuity of ut is similar to u. It only needs to take care of the difference of smoothness and the action of the Bessel potential. Finally, we obtain u ∈ C(R,Mp,1 s (Rd)) ∩ C1(R,Mp,1 s−1(Rd)). � Proof of Corollary 1.2. Let 2v1(t) = u0 + u1 iω1/2 − ∫ t 0 G(−τ)F (u(τ)) iω1/2 dτ, 2v2(t) = u0 − u1 iω1/2 + ∫ t 0 G(−τ)F (u(τ)) iω1/2 dτ. For 0 < s < t, we have v1(t)− v1(s) = − ∫ t s G(−τ)F (u(τ)) iω1/2 dτ. Since the pair (p, r) is Klein-Gordon admissible, there exists β̃ such that 1 β̃ + 3 r = 1, β̃d( 1 2 − 1 p ) > 1. 16 D. G. BHIMANI EJDE-2021/101 By Proposition 3.6 and Hölder’s inequality, we have ‖v1(t)− v1(s)‖Mp,1 s . ∫ t s (1 + |τ |)−d( 1 2− 1 p )‖F (u(τ))‖ Mp′,1 s dτ . ∫ t s (1 + |τ |)−d( 1 2− 1 p )‖u‖3 Mp,1 s dτ . ‖(1 + |τ |)−d( 1 2− 1 p )‖Lβ̃‖‖u‖ 3 Mp,1 s ‖Lr/3([s,t],Mp,1 s ) . ‖u‖3 Lr([s,t],Mp,1 s ) . Since ‖u‖Lr([s,t],Mp,1 s ) ≤M , we have ‖v1(t)− v1(s)‖Mp,1 s . ‖u‖3 Lr([s,t],Mp,1 s ) → 0 as t, s→∞. This implies that v1(t) is Cauchy in Mp,1 s (Rd) as t → ∞. Denote v+ 1 to be the limit: 2v+ 1 = lim t→+∞ 2v1(t) = u0 + u1 iω1/2 − ∫ t 0 G(−τ)F (u(τ)) iω1/2 dτ and 2v−1 = lim t→+∞ 2v1(t) = u0 − u1 iω1/2 + ∫ t 0 G(−τ)F (u(τ)) iω1/2 dτ. Similarly, we obtain v+ 2 (t) = lim t→∞ v2(t) and v−2 (t) = lim t→∞ v2(t). Recall that v± = G(t)v±1 +G(t)v±2 , we note that ‖u(t)− v+‖Mp,1 s = ∥∥∫ ∞ t K(t− τ)F (u(τ))dτ ∥∥ Mp,1 s . ‖(1 + |τ |)−d( 1 2− 1 p )‖Lβ̃‖ ‖u‖ 3 Mp,1 s ‖Lr/3([t,∞],Mp,1 s ) . ‖u‖3 L3([t,∞],Mp,1 s ) → 0 as t→∞. So is v− respectively. In fact, in our proof we also have v+ 1 ∈Mp,1 s (Rd). � Proof of Theorem 1.3. Equation (1.2) can be written in the equivalent form u(·, t) = K̃(t)u0 +K(t)u1 − ∫ t 0 K(t− τ)[(Vγ ∗ |u|2)(τ)u(τ)]dτ =: J (u) (4.3) where K(t) = sin(t √ −4)√ −4 , K̃(t) = cos(t √ −4). By using Proposition 2.10 for the first two inequalities below, and Propositions 3.2 and 3.8 for the last inequality, we can write ‖K̃(t)u0‖X ≤ CT ‖u0‖X , ‖K(t)u1‖X ≤ CT ‖u1‖X ,∥∥ ∫ t 0 K(t− τ)[(Vγ ∗ |u|2)(τ)u(τ)]dτ ∥∥ X ≤ TCT ‖u‖3X , (4.4) where CT is some constant times (1+T 2)d/4, as before. Thus the standard contrac- tion mapping argument can be applied to J to complete the proof. This completes the proof of Theorem 1.3 (2). Taking Propositions 2.9, 3.8 and Corollary 3.3 and EJDE-2021/101 GLOBAL CAUCHY PROBLEMS FOR HNLKG, HNLW AND HNLS 17 Lemma 3.4 into account, the standard contraction mapping argument give the proof of Theorem 1.3 (1). � 5. Proofs of theorems 1.5, 1.8 and 1.9 To prove Theorem 1.8 first we shall prove following Strichartz type estimates for Schrödinger admissible pairs. Proposition 5.1. Let F (u) = (Vγ ∗ |u|2)u, p ∈ (2, 3), 1 p + γ d − 1 = 1 2p′ and pair (p, r) is Schrödinger admissible. Then we have ‖ ∫ t 0 S(t− τ)F (u(τ))dτ‖Lrt (R,Mp,1 s ) . ‖F (u)‖ L r/3 t (R,Mp′,1 s ) . ‖u‖3 Lrt (R,Mp,1 s ) . Proof. By the general Minkowski inequality, Proposition 2.8, we have∥∥ ∫ t 0 S(t− τ)F (u(τ))dτ ∥∥ Lrt (R,Mp,1 s ) . ∥∥∫ t 0 ‖S(t− τ)F (u(τ))‖Mp,1 s dτ ∥∥ Lrt (R) . ∥∥∫ t 0 (1 + |t− τ |)−d(1/2−1/p)‖F (u)‖ Mp′,1 s dτ ∥∥ Lrt (R) . ∥∥∫ R (1 + |t− τ |)−d(1/2−1/p)h(τ)dτ ∥∥ Lrt (R) . ‖g ∗ h‖Lrt , where h(τ) = ‖F (u)‖ Mp′,1 s , g(t) = (1 + |t|)−d(1/2−1/p). We divide Schroödinger admissible pairs (see (1.6)) into several cases. Case I: 1 β < d( 1 2 − 1 p ) ∧ 1. In this case we have dβ( 1 2 − 1 p ) > 1. Using Young inequality and Hölder’s inequality we have∥∥∫ t 0 S(t− τ)F (u(τ))dτ ∥∥ Lrt (R,Mp,1 s ) . ‖g‖Lβ‖F (u)‖ Lr/3(R,Mp′,1 s (Rd)) . ‖u‖3 Lr(R,Mp,1 s (Rd)) . Case II: 1 β = 1∧d( 1 2 − 1 p ), d( 1 2 − 1 p ) > 1. In this case, we can get β = 1 and r =∞. Obviously dβ( 1 2 − 1 p ) > 1, and therefor, we have the desired result by the same way as Case I. Case III: 1 β = 1 ∧ d( 1 2 − 1 p ), d( 1 2 − 1 p ) < 1. In this case we have dβ( 1 2 − 1 p ) = 1. Since pair (p, r) is Schrödinger admissible, we have 1 r = 3 r − 1− d(1/2− 1/p) 1 18 D. G. BHIMANI EJDE-2021/101 and r/3 > 1. By Hardy-Littlewood-Sobolev inequality in one dimension, we have∥∥∫ t 0 K(t− τ)F (u(τ))dτ ∥∥ Lrt (R,Mp,1 s ) . ‖g ∗ h‖Lrt (R) . ‖‖F (u)‖ Mp′,1 s ‖Lr/3 . ‖F (u)‖ Lr/3(R,Mp′,1 s ) . ‖u‖3 Lr(R,Mp,1 s ) . Case IV: 1 β = 1 ∧ d( 1 2 − 1 p ), d( 1 2 − 1 p ) = 1. In this case (p, r) = ( 2d d−2 ,∞) which is not Schrödinger admissible. � Lemma 5.2. Let F (u) = (Vγ ∗ |u|2)u, p ∈ (2, 3), 1 p + γ d − 1 = 1 2p′ and pair (p, r) is Schrödinger admissible. Then∥∥∫ t 0 S(t− τ)[F (u(τ))− F (v(τ))]dτ ∥∥ Lrt (R,Mp,1 s ) . (‖u‖2 Lrt (R,Mp,1 s ) + ‖u‖Lrt (R,Mp,1 s )‖v‖Lrt (R,Mp,1 s ) + ‖v‖2 Lrt (R,Mp,1 s )‖u− v‖3 Lrt (R,Mp,1 s ) . Proof. Using Propositions 2.5 and 5.1, Lemma 2.3 (1) and Hölder inequality, the proof can be produced. We omit the details. � Proof of Theorem 1.5. For α = 2, we may rewrite equation (1.5) in the form u(t) = S(t)u0 − ∫ t 0 S(t− τ)F (u(τ))dτ =: J (u) where S(t) = e−it∆ and F (u) = (Vγ ∗ |u|2)u. Denote X = Lr(R,Mp,1 s (Rd)). For δ > 0, we put Bδ = {u ∈ X : ‖u‖X ≤ δ} which is the closed ball of radius δ, and centered at the origin in X. Since rd( 1 2 − 1 p ) > 1, we have (1+ |t|)−d( 1 2− 1 p ) ∈ Lr(R). Now by Proposition 2.8, we have ‖S(t)u0‖X . ‖(1 + |t|)−d( 1 2− 1 p )‖u0‖Mp′,1 s ‖Lr . ‖u0‖Mp′,1 s . By Proposition 5.1, we have∥∥∫ t 0 S(t− τ))F (u(τ))dτ ∥∥ X . ‖u‖3X . Thus ‖J (u)‖X . ‖u0‖Mp′,1 s + ‖u‖3X . By Lemma 5.2, for any u, v ∈ Bδ, we have ‖J u− J v‖X . (‖u‖2X + ‖u‖X‖v‖X + ‖v‖2X)‖u− v‖X . If we assume that δ > 0 is sufficiently small, then J : X → X is a strict con- traction. Therefor J has a unique fixed point and we have u ∈ Lr(R,Mp,1 s (Rd)) and ‖u‖Lr(R,Mp,1 s (Rd)) . ‖u0‖Mp′,1 s . We want to show that if f ∈ Mp,1 s (Rd) then S(t)f ∈ C(R,Mp,1 s (Rd)). Let t > 0 and tn → t. By Lemma 2.7, Proposition 2.8 and the triangle inequality, we have ‖S(t)f − S(tn)f‖Mp,1 s ≤ ‖S(t)f − S(t)g‖Mp,1 s + ‖S(t)g − S(tn)g‖Mp,1 s + ‖S(tn)f − S(tn)g‖Mp,1 s . EJDE-2021/101 GLOBAL CAUCHY PROBLEMS FOR HNLKG, HNLW AND HNLS 19 We only need to treat the case f ∈ SΩ. Using Lemma 2.12 and the Hausdroff-Young inequality, we have ‖�k(S(tn)− S(t))f‖Lp . ‖(S(tn)− S(t))f‖Lp . ‖(eitn|ξ| 2 − eit|ξ| 2 )f̂‖Lp′ → 0 as tn → t by the Lebesgue dominated convergence theorem. Since f ∈ SΩ, there exist only finite number of k such that �k(S(tn)− S(t))f 6= 0, and thus ‖S(t)f − S(tn)f‖Mp,1 s → 0 as tn → t. We write I = ∫ t 0 S(t− τ)F (u(τ))dτ − ∫ tn 0 S(tn − τ)F (u(τ))dτ = (∫ tn 0 S(t− τ)F (u(τ))dτ − ∫ tn 0 S(tn − τ)F (u(τ))dτ ) + (∫ t 0 S(t− τ)F (u(τ))dτ − ∫ tn 0 S(t− τ)F (u(τ))dτ ) = I1 + I2. For I2, we have ‖I2‖Mp,1 s . ∫ t tn ‖S(t− τ)F (u)(τ)‖Mp,1 s dτ . ∫ t tn (1 + |t− τ |)−d(1/2−1/p)‖F (u)(τ)‖ Mp′,1 s dτ . ∫ t tn ‖u‖3 Mp,1 s dτ . |t− tn|β‖u‖3Lr([0,t],Mp,1 s ) → 0. For I1, we have I1 . ∫ tn 0 ‖S(τ)(S(tn)− S(t))F (u(τ))‖Mp,1 s dτ . ∫ I ‖(S(tn)− S(t))F (u(τ))‖Mp,1 s dτ. We note that ‖(S(tn) − S(t))F (u(τ))‖Mp,1 s . ‖F (u)(τ)‖3 Mp,1 s and recalling r ≥ 3 and u ∈ Lr(R,Mp,1 s (Rd)), we have ‖u(τ)‖3 Mp,1 s ∈ L1[0, t]. Since F (u) ∈ Mp,1 s (Rd), for every τ ∈ [0, t] we have ‖(S(tn)− S(t))F (u(τ))‖Mp,1 s → 0. � Proof of Corollary 1.6. We only prove the statement for u+, since the proof for u− follows similarly. Let us first construct the scattering state u+(0). For t > 0 define v(t) = e−it∆u(t). We will show that v(t) converges in Mp,1 s (Rd) as t → ∞, and define u+ to be the limit. Indeed from Duhamel’s formula we have v(t) = u0 − ∫ t 0 e−iτ∆F (u(τ))dτ (F (u) = (Vγ ∗ |u|2)u). (5.1) Therefore, for 0 < s < t, we have v(t)− v(s) = −i ∫ t s e−iτ∆F (u(τ))dτ. 20 D. G. BHIMANI EJDE-2021/101 Since the pair (p, r) is a Schrödinger admissible, there exists β̃ such that 1 β̃ + 3 r = 1, β̃d( 1 2 − 1 p ) > 1. By Proposition 3.6 and Hölder’s inequality, we have ‖v(t)− v(s)‖Mp,1 s . ∫ t s (1 + |τ |)−d( 1 2− 1 p )‖F (u(τ))‖ Mp′,1 s dτ . ∫ t s (1 + |τ |)−d( 1 2− 1 p )‖u‖3 Mp,1 s dτ . ‖(1 + |τ |)−d( 1 2− 1 p )‖Lβ̃‖ ‖u‖ 3 Mp,1 s ‖Lr/3([s,t],Mp,1 s ) . ‖u‖3 Lr([s,t],Mp,1 s ) . Since ‖u‖Lr(R,Mp,1 s ) ≤M , we have ‖v(t)− v(s)‖Mp,1 s . ‖u‖3 Lr([s,t],Mp,1 s ) → 0 as t, s→∞. This implies that v(t) is Cauchy in Mp,1 s (Rd) as t → ∞. We define u+ to be the limit. In view of (5.1), we see that u+(0) = u0 − ∫ ∞ 0 e−iτ∆F (u(τ))dτ and thus u+(t) = eit∆u0 − ∫ ∞ 0 ei(t−τ)∆F (u(τ))dτ. We note that ‖u(t)− eit∆u+‖Mp,1 s = ‖ ∫ ∞ t S(t− τ)F (u(τ))dτ‖Mp,1 s . ‖(1 + |τ |)−d( 1 2− 1 p )‖Lβ̃‖ ‖u‖ 3 Mp,1 s ‖Lr/3([t,∞],Mp,1 s ) . ‖u‖3 Lr([t,∞],Mp,1 s ) → 0 as t→∞. In fact, in our proof we also have eit∆u0, e it∆u+ ∈Mp,1 s (Rd). � To prove Theorem 1.8 first we recall following result. Lemma 5.3 ([5]). Let V ∈ M∞,1(Rd), and 1 ≤ p, q ≤ 2. For f ∈ Mp,q(Rd), we have ‖(V ∗ |f |2)f‖Mp,q . ‖f‖3Mp,q , and ‖(V ∗ |f |2)f− (V ∗ |g|2)g‖Mp,q . (‖f‖2Mp,q +‖f‖Mp,q‖g‖Mp,q +‖g‖2Mp,q )‖f−g‖Mp,q . Proof of Theorem 1.8. Recall (1.5) can be written in the equivalent form u(·, t) = U(t)u0 − i ∫ t 0 U(t− τ)[(V ∗ |u|2)u] dτ =: J (u). We first prove the local existence on [0, T ) for some T > 0. By Minkowski’s in- equality for integrals, Proposition 2.8 and Lemma 5.3, we obtain ‖ ∫ t 0 U(t− τ)[(V ∗ |u|2(τ))u(τ)] dτ‖Mp,q ≤ cT (1 + |t|)d| 1 p− 1 2 |‖u(t)‖3Mp,p , EJDE-2021/101 GLOBAL CAUCHY PROBLEMS FOR HNLKG, HNLW AND HNLS 21 for some universal constant c. By Proposition 2.8 and the above inequality, we have ‖J u‖C([0,T ],Mp,q) ≤ CT (‖u0‖Mp,q + cT‖u‖3Mp,q ) where CT = (1 + |T |)d| 1 p− 1 2 |. For M > 0, put BT,M = {u ∈ C([0, T ],Mp,q(Rd)) : ‖u‖C([0,T ],Mp,q) ≤M}, which is the closed ball of radius M , centered at the origin in C([0, T ],Mp,q(Rd)). Next, we show that the mapping J takes BT,M into itself for suitable choice of M and small T > 0. Indeed, if we let, M = 2CT ‖u0‖Mp,p and u ∈ BT,M , it follows that ‖J u‖C([0,T ],Mp,p) ≤ M 2 + cCTTM 3. We choose a T such that cCTTM 2 ≤ 1/2, that is, T ≤ T̃ (‖u0‖Mp,p) and as a consequence we have ‖J u‖C([0,T ],Mp,p) ≤ M 2 + M 2 = M, that is, J u ∈ BT,M . By Lemma 5.3, and the arguments as before, we obtain ‖J u− J v‖C([0,T ],Mp,q) ≤ 1 2 ‖u− v‖C([0,T ],Mp,q). Therefore, using Banach’s contraction mapping principle, we conclude that J has a fixed point in BT,M which is a solution of (1.5). Indeed, the solution constructed before is global in time: in view of the conser- vation of L2 norm, Proposition 2.5 and Lemma 2.3, we have ‖u((t)‖Mp,p . CT ( ‖u0‖Mp,q + ∫ t 0 ‖V ∗ |u(τ)|2‖M∞,1‖u(τ)‖Mp,qdτ ) . CT ( ‖u0‖Mp,q + ∫ t 0 ‖V ‖M∞,1‖ |u(t)|2‖M1,∞‖u(τ)‖Mp,qdτ ) . CT ( ‖u0‖Mp,q + ∫ t 0 ‖ |u(t)|2‖L1‖u(τ)‖Mp,qdτ ) . CT ( ‖u0‖Mp,p + ‖u0‖2L2 ∫ t 0 ‖u(τ)‖Mp,pdτ ) and by Gronwall’s inequality, we conclude that ‖u(t)‖Mp,q remains bounded on finite time intervals. This completes the proof. � Proof of Theorem 1.9. Recall (1.5) can be written in the equivalent form u(·, t) = U(t)u0 − i ∫ t 0 U(t− τ)[(V ∗ |u|2)u] dτ =: J (u). By using Proposition 2.8 and Corollary 3.3, we can write ‖U(t)u0‖Mp,1 s ≤ CT ‖u0‖Mp,1 s ,∥∥∫ t 0 U(t− τ)[(Vγ ∗ |u|2)(τ)u(τ)]dτ ∥∥ X ≤ TCT ‖u‖3Mp,1 s , (5.2) where CT is some constant times (1 + T 2)d/4, as before. Thus the standard con- traction mapping argument can be applied to J to complete the proof. � 22 D. G. BHIMANI EJDE-2021/101 6. Local well-posedness with potential V ∈ FLq or M1,∞ or M∞,1 We consider generalized Klein-Gordon equation with Hartree type linearity: utt + (I −∆)u = (V ∗ |u|2k)u, u(0) = u0, ut(0) = u1, k ∈ N. (6.1) When k = 1, equation (6.1) coincides with (1.1). Theorem 6.1 (Local well-posedness). Let i = 0, 1. (1) Let V ∈ FLq(Rd) (1 ≤ q ≤ ∞) and ui ∈M1,1(Rd). Then there exists T ∗ = T ∗(‖ui‖M1,1) such that (6.1) has a unique solution u ∈ C([0, T ∗),M1,1(Rd)). (2) Assume that V ∈ FLq(Rd) with 1 < q < r ≤ 2, and ui ∈ Mp, 2r 2r−1 (Rd). Then there exists T ∗ = T ∗(‖ui‖ M p, 2r 2r−1 ) such that (1.1) has a unique solu- tion u ∈ C([0, T ∗),Mp, 2r 2r−1 (Rd)). (3) Assume that V ∈ M∞,1(Rd) and ui ∈ Mp,q(Rd). Then there exists T ∗ = T ∗(‖ui‖Mp,q ) such that (1.1) has a unique solution u ∈ C([0, T ∗),Mp,q(Rd)). (4) Assume that V ∈ M∞,1(Rd) and ui ∈ Mp,q(Rd) (1 ≤ p, q ≤ 4, 1 ≤ q ≤ 22k−2 22k−2−1 , 1 < k ∈ N). Then there exists T ∗ = T ∗(‖ui‖Mp,q ) such that (6.1) has a unique solution u ∈ C([0, T ∗),Mp,q(Rd)). (5) Assume that V ∈ M1,∞(Rd) and ui ∈ Mp,1(Rd) (1 ≤ p ≤ ∞). Then there exists T ∗ = T ∗(‖ui‖Mp,1) such that (1.1) has a unique solution u ∈ C([0, T ∗),Mp,q(Rd)). Proof. Taking Proposition 2.9 and [5, Lemmas 4.8 and 4.9] and [20, Lemmas 4.2 and 4.3] into account, the standard fixed point argument gives the desired result. We will omit the details. � Remark 6.2. The analogue of Theorem 6.1 is true for equations (1.2) and (1.5). Acknowledgments. D. 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Guo; Klein-Gordon equations on modulation spaces, Abstr. Appl. Anal., (2014), 15 pp. Divyang G. Bhimani Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India Email address: divyang.bhimani@iiserpune.ac.in 1. Introduction and statement of results 1.1. Klein-Gordon-Hartree and wave-Hartree equations 1.2. Fractional Hartree equation 2. Preliminaries 2.1. Notation 2.2. Modulation spaces 3. Nonlinear estimates in Mp,qs(Rd) 4. Proofs of theorems ?? and ?? 5. Proofs of theorems ??, ?? and ?? 6. Local well-posedness with potential V FLq or M1, or M, 1 Acknowledgments References