DISSIPATIVE NONLINEAR SCHR¨ODINGER EQUATIONS FOR LARGE DATA IN ONE SPACE DIMENSION

. In this study, we consider the global Cauchy problem for the nonlinear Schr¨odinger equations with a dissipative nonlinearity in one space di- mension. In particular, we show the global existence, smoothing eﬀect and asymptotic behavior for solutions to the nonlinear Schr¨odinger equations with data which belong to F H γ , 1 / 2 < γ ≤ 1 . In the proof of main theorem, we introduce a priori estimate for H γ -type norm and the condition F H 1 for data relaxed into F H γ , 1 / 2 < γ ≤ 1 .

To be specific, we assume that the nonlinearity F (u) has the following form: There is a large literature on the Cauchy problem for the nonlinear Schrödinger equations (see [3,5,13,14,19,26,27] and references therein).
The critical exponent between long-range and short-range in the scattering theory is If λ > 0, 1 < p ≤ p * and n ≥ 1, the nonlinear Schrödinger equations do not have nontrivial asymptotically free solution in L 2 ( [2,25], see also Chapter 7 of [3]). The nonexistence of nontrivial asymptotically free solution in L 1 has been shown in [17].
In [21], the author has shown the existence of modified scattering state for the nonlinear Schrödinger equations with long-range potential in one space dimension. After this work the study of long-range scattering and asymptotic behavior for solutions to nonlinear Schrödinger equations are developed by many authors (see [4,6,7,9,10,12,15,16,18,23,24,17] and references therein).
In [6,7,9,10,15,16,24,17] (see also [18] and references therein), the authors have studied the asymptotic behavior for solutions to the nonlinear Schrödinger equations. More precisely, in [9] the authors assume Im(λ) = 0 and in [6,7,10,15,16,24,17] the authors assume Im(λ) < 0 (see also [18]). In particular in [10,16], the authors do not need any smallness condition to the H 1 ∩ FH 1 -norm of the data in one space dimension under the condition by using a priori estimate with the following condition for the coefficient λ ∈ C in front of the nonlinear term: In [7], the authors have studied asymptotic behavior for solutions to (1)-(2) for large data with some ε > 0 under the condition Our aim of this study is to consider the global existence and asymptotic behavior for solutions t (1)-(2) for large data φ ∈ FH γ with 1/2 < γ ≤ 1, under the condition and Im(λ) < 0 and where the condition (3) appears in the argument to obtain the time decay estimate (see (9) in Sect.4, below) and we need the condition (4) to prove the a priori estimate.
To prove the global existence and the time decay estimate of solutions, we show the following a priori estimate for solutions with fractional order 0 < γ < 1, which has an important role in this study (see Lemma 3.6 in Sect.3, below) : The proof of this estimate is similar to the proof of L 2 -conservation law like as the following argument. Let G y be the Galilei transform (see [19] and proof of Lemma 3.6 in Sect.3, below), then by the Galilei invariant property, we have Multiplying G y u − u by the above equation and integrating over R with taking imaginary part, we have If λ ∈ C satisfies (4), then (see [16,20] and Lemma 4 in Sect.3 below ) and we obtain the above a priori estimate with the identity (see Proposition 1.37 in [1] for example).
In this study, we do not need any smallness condition and regularity assumption for the data φ ∈ FH γ . The asymptotic behavior of solutions to the Cauchy problem for the dissipative nonlinear Schrödinger equations has been studied by [6,7,10,16,18,23,24,17] (see also references therein).
is a Banach space with the norm ϕ A∩B = max ( ϕ A , ϕ B ) , for two Banach spaces (A, · A ) and (B, · B ). The map F : ϕ → ϕ is the Fourier transform defined by The bracket is denoted by ζ = (1 + |ζ| 2 ) 1/2 , ζ ∈ R. Let s > 0. The Sobolev space and the homogeneous Sobolev space are defined by H s shortly. As is well known, if 0 < γ < 1 then there exists a constant C γ > 0 such that for any ϕ ∈Ḣ γ (see Proposition 1.37 in [1]). The free Schrödinger propagator is defined by We often use the notation where the phase modulation operator M (t) : ϕ → e ix 2 /2t ϕ and the dilations D(t) : The generator of Galilei transform is defined by For t = 0, J(t) is also represented as (see [8]) Let γ > 0. We define the following Sobolev type space and which has another representation We introduce the following function spaces 2. Main result. We introduce the exponent p(γ) defined by We state our man result: Then (1)-(2) has a unique global solution u ∈ X loc (R + ) L 4 loc (R + ; A γ ∞ ) satisfying the following properties: 2. There exist C > 0 such that as t → +∞ for p(γ) < p ≤ 3 and some α > 0.
Remark 1. We do not need the smallness condition on the weighted L 2 -norm to the data φ ∈ FH γ with 1/2 < γ ≤ 1. Also we do not need any regularity assumption on the data φ ∈ FH γ , however the solutions belong to C(R) (smoothing effect). For example the function φ R = Rχ {|x|≤1} , R > 0 which belongs to FH γ , with large FH γ -norm and which is not continuous on R, where the characteristic function Remark 2. The solutions u ∈ X loc (R + ) satisfy u ∈ L r (R + ; L ∞ ) , with appropriate 1 ≤ r < ∞ because the L ∞ -norm of solutions: u(t) L ∞ have time decay estimate above for large t ≥ 1 and u L ∞ ∈ L 4 loc (R + ; L ∞ ).
Remark 3. We see that which says that, if γ = 1/2, then it is difficult to obtain the time decay estimate above, because we need the assumption p(γ) < p < 3 to show the time decay estimate. Also we see that and which is the same to the condition appears in the previous studies [10,16]. Therefore p(γ) is regarded as a extension of p(1) and which has monotone decreasing property p(γ 1 ) < p(γ 0 ) for 1/2 ≤ γ 0 < γ 1 ≤ 1. Because {|J| γ (t)u k (t)} k≥1 is bounded in L 2 and L 2 is a reflexive Banach space, there exist sub sequence {|J| γ (t)u kj (t)} j≥1 , which converges weakly in L 2 and is the usual L 2 -scalar product. Therefore for t ∈ I. This completes the proof.
with some constant C > 0.
Proof. By using the identities for t ∈ R.
Proof. Let t = 0. By and the Leibniz estimate in the Appendix of [14], we have: This completes the proof.
We define the following two quantities:
Lemma 3.6. Let 1/2 < γ ≤ 1. We put I = [0, T ], T > 0 and we assume that Let u ∈ X(I) be solutions to (1)- (2). Then The desired result for the case of γ = 1 has been obtained in [16]. We assume that 1/2 < γ < 1. Let u ∈ X(I) be solutions to (1). We define the translation and the Galilei transform by τ y u(t) = u(t, · + y) and for y ∈ R. By the Galilei invariant property of (1), we have where we have used the another representation Therefore, as in the similar way of [22], we have where we have used the Lemma 3.5 and the following identity Therefore by the above estimate and the identity (5), we have Also we see that This completes the proof.

4.
Proof of Theorem 2.1. We consider the map Φ : u → Φu defined by in the complete metric space (X r (I), d) with X r (I) = u ∈ X(I); u X(I) ≤ r , for interval I = [0, T ]. We have the following estimates By the Banach fixed point theorem with r = 2C φ F H γ , CT 1− p−1 4 r p−1 = 1 2 we have the desired unique local solution u ∈ X(I).
By the Strichartz estimate, we have By a standard continuation argument with a priori estimate . In the last part of the proof of Theorem 2.1, we follow the argument studied in [9,10,11]. By the relation and by using the formulas Therefore we obtain where Setting v = FU −1 u and multiplying both sides of (6) by v we have for 1/2 < γ ≤ 1 and 0 < 2k + 1/2 < γ by Lemma 4 and Lemma 6. Then by ∂ t |v| 2 = 2|v|∂ t |v|, we have In the Case: p = 3. If p = 3, then Multiplying both sides of (8) by (log t) 3/2 and by the identity we have By the Young inequality: ab ≤ 2 3 a 3/2 + 1 3 b 3 , we have 3 2 Hence |v(t)| ≤ (log 2) 3/2 |v(2)| + C φ Therefore, we have and there exists a unique function ψ + ∈ L 2 ∩ L ∞ such that where 0 < α < k.