Time Decay for Nonlinear Dissipative Schrödinger Equations in Optical Fields

λ|u| p−1 uof orderp(n) < p ≤ 1+2/n for arbitrarily large initial data, where the lower boundp(n) is a positive root of (n+2)p−6p−n = 0 for n ≥ 2 andp(1) = 1+√2 for n = 1.Ourpurpose is to extend the previous results for higher space dimensions concerningL-time decay and to improve the lower bound ofp under the same dissipative condition onλ ∈ C: Imλ < 0 and |Imλ| > ((p−1)/2√p)|Reλ| as in the previous works.


Introduction and Main Results
We consider the initial value problem for the following nonlinear Schrödinger equations: where  ∈ C, Im  < 0, | Im | > (( − 1)/2√)|Re | and 1 <  ≤ 1 + 2/.From the physical point of view, (1) is applied to investigate the light traveling in optical fibers (see, e.g., [1,2]).Some new nonlinear works concerning the related equations appear in [3,4].We note that Im  < 0 implies the dissipation of |(, )| by nonlinear Ohm's law (see, e.g., [5]).Therefore, the dissipative nonlinearity causes the transportation of the energy.The condition on the coefficient  appeared in the previous papers [1,2,6,7] and so forth.It yields a dissipative property of solutions in one space dimension for arbitrarily large initial data, which was shown in [2].In this paper, we prove L 2 -time decay estimate of solutions in any space dimension  ≥ 1 under the condition of critical or subcritical power order nonlinearity such that where  () = 3 + √ 9 +  2 + 2  + 2 for  ≥ 2,  (1) = 1 + √ 2 for  = 1. (3) In [2], L ∞ -time decay of solutions was studied in one space dimension under the condition on  such that 2.686 ≈ (5 + √ 33)/4 <  ≤ 3.More precisely, in [2] it was shown that the estimates of solutions to (1), hold for any  > 1 in the case of  = 1.For small solutions, in [1] under the conditions such that Im  < 0 and  is close to 1 + 2/, the same L ∞ -time decay as stated above was obtained for 1 ≤  ≤ 3. Below we show that if we restrict our attention to the L 2 -time decay of solutions to (1) for arbitrarily large initial data, then we can reduce the low bound of the power  and consider the problem in higher space dimensions.To investigate the effect of the dissipative nonlinearity, we consider the estimates about the L 2 norm of solutions to (1).In [2] the L 2 -time decay of the solution  to (1) satisfying that lim →∞ ‖()‖ L 2 = 0 was obtained, when  = 1 and (21 + √ 177)/12 <  ≤ 3. We develop our problem to the space dimension  ≥ 1. Moreover the scope of  is extended, when we focus on one space dimension case.Another point in this paper is to prove results directly comparing to the contradiction argument in [2].More precisely, our strategy of the proof is as follows.We first transform our target equation ( 1) into the following nonlinear ordinary equation with remainder terms : where For ,  ∈ , weighted Sobolev space H , is defined by where We write ‖‖ L 2 = ‖‖ and H ,0 = H  for simplicity.Let us introduce some notations.We define the dilation operator by and define  =  (/2)|| 2 for  ̸ = 0. Free evolution group U() is written as where the Fourier transform of  is We also have where the inverse Fourier transform These formulas were used in [8] first to study the asymptotic behavior of solutions to nonlinear Schrödinger equations.We denote by the same letter  various positive constants.
The standard generator of Galilei transformation is given by We have commutation relations with J and To prove Theorem 1, we introduce the function space where In Theorems 1 and 2, we consider the subcritical and critical cases, respectively.Define then we have for  ≥ 0, where 0 ≤  ≤ 2/( + 2) and the identity holds.

Proof of Theorem 1
Under the assumptions we have a dissipative property of solutions.Indeed we have by the usual energy method Nonlinear terms are represented as which is equivalent to Therefore we have for  ≥ 0, where In the same way as in the proof of (29), we have from which it follows that for  ≥ 0. We also have Therefore by (32), (35), and (36), we have an a priori estimate of solutions which implies the global in time existence of solutions to (1) in the function space  1,∞ for Im  < 0 and | Im | > (( − 1)/2√)|Re |.This completes the first part of the proof of the theorem.
Our concern is to estimate the time decay rate in the subcritical case since it is expected that the decay rate of solutions is different from that of solutions to linear problem.Denote  = FU(−), and V = FU(−).Using the Advances in Mathematical Physics factorization formula U() =   F, we multiply both sides of (1) by FU(−) to get where For the remainder term , we have by (37) the following.

Lemma 3.
Let  be a solution of (1) in the function space  1,∞ .
Then the estimate for  ≥ 1 is true, where ≥ 3, 0 <  < 1   = 2,  = 1   = 1. (41) Proof.By Hölder's and Sobolev's inequalities, we get where  = 1 for  = 1, 0 <  < 1 for  = 2, and since we can apply the Sobolev embedding theorem with In the same manner with the same  as above, we obtain In view of (37) the result of the lemma follows.This completes the proof of the lemma.