UPPER AND LOWER TIME DECAY BOUNDS FOR SOLUTIONS OF DISSIPATIVE NONLINEAR SCHR¨ODINGER EQUATIONS

. We study the upper and lower time decay bounds for solutions of dissipative nonlinear Schr¨odinger equations i∂ in space dimensions n = 1 , 2 or 3, where λ = λ 1 + iλ 2 , λ j ∈ R , j = 1 , 2 , λ 2 < 0 and the subcritical order of nonlinearity p = where µ > 0 is small enough.

1. Introduction and main results. We consider the initial value problem for the following nonlinear Schrödinger equations i∂ t u + 1 2 ∆u = λ |u| p−1 u, (t, x) ∈ R + × R n , in space dimensions n = 1, 2 or 3, where λ = λ 1 + iλ 2 , λ j ∈ R, j = 1, 2, λ 2 < 0 and the order of nonlinearity is subcritical where µ > 0 is small enough. There are some works concerning the physical applications of (1) (see e.g., [1] and [10]). We note that the condition λ 2 < 0 implies the dissipation property of |u (t, x)| by a nonlinear Ohm's law (see e.g., [1]). In this paper we consider the initial function which yields the sharp time decay estimates of solutions of (1). In papers [6,8,9] and [10], asymptotic behavior of solutions to (1) was studied. However the sharp time decay of solutions was not obtained. It was shown in [9] that the leading term of the large time asymptotic profile is almost equal to |u (t, x)| 1 t (2) which does not belong to any L p space except p = ∞. In order to justify (2), the smallness condition on the coefficient 2 − n (p − 1) was assumed in [9]. This condition implies that p must be close to the critical exponent 1 + 2 n . We also find that there exists a positive constant C such that If inf ξ | u 0 (ξ)| > 0, then the lower bound for solutions follows. The condition inf ξ | u 0 (ξ)| > 0 means that u 0 does not belong to any L p space except p = ∞. Therefore we consider the problem in such a space that the Fourier transform of the initial data does not vanish. Related work can be seen in [11] in which homogeneous weighted L 2 space was considered. However the initial data from [11] do not satisfy the condition inf ξ | u 0 (ξ)| > 0. On the other hand, in the case of n = 1, the result stated in Theorem 1.1 below was obtained in [7] . However Theorem 1.1 is not enough to get the lower bound for time decay. We also note that the final state problem for nonlinear Schrödinger system in two space dimensions with non decaying final data was studied in [5].
We introduce some function spaces and notations. Let L ∞ denote the usual Lebesgue space with the norm φ L ∞ = ess.sup x∈R n |φ (x)| . The homogeneous Sobolev spaceḢ m is defined bẏ is the largest integer less than s. It is known thatḂ s 2,2 =Ḣ s (see [2]). We also denote t = √ 1 + t 2 . We define the dilation operator by and denote M (t) = e i 2t |x| 2 for t = 0. Schrödinger evolution group U (t) is written as where F denotes the Fourier transformation. The inverse Schrödinger evolution group has the form U where F −1 is the inverse Fourier transformation. Different positive constants might be denoted by the same letter C if it does not cause any confusion. The standard generator of Galilei transformation is given by J = U (t) xU (−t) = x + it∇. We have the commutator relation with J and L = i∂ t + 1 2 ∆ such that [L, J ] = 0. To state our main results, we use the function space We now state our results.
Then there exist sufficiently small ρ 0 > 0 and µ 0 > 0 such that the Cauchy problem (1) has a global in time solution u ∈ X for all 0 < ρ ≤ ρ 0 and 0 < µ ≤ µ 0 . Moreover the time decay estimate holds for all t > 0. In the case of n = 1, the solution u is unique.
Next we give the lower bound for the time decay of solutions to (1) and uniqueness of solutions. Theorem 1.3. In addition to the assumptions of Theorem 1.1, we assume that Then there exists a unique global solution u ∈ X of the Cauchy problem (1). Furthermore for any u 0 satisfying conditions, there exists a positive constant β > 0 such that global solution u satisfies the lower bound To explain our strategy, we look for the solution of (1) in the form By a direct calculation, h (t) satisfies the ordinary differential equation We change the dependent variable h = re iw , r = |h| , w = arg h, with r (0) = ρ, w (0) = 0, then we have ir − rw = λt − n 2 (p−1) r p , which gives us the ordinary differential equations The explicit solution is as follows We note that Then we have The right-hand side coincides with H (t, ξ) if we choose, |y + (ξ)| = ρ, θ = 0 and η + (ξ) = 0. Thus the solution of (3) in the form h (t) = r (t) e iw(t) with r (0) = ρ and w (0) = 0 can be represented as (4) It is expected that solutions of (1) behave like (4) if the Fourier transforms of the initial data do not vanish. Our purpose is to prove the results including a particular solution (4).

Local existence.
In the next lemma we obtain the estimates of the remainder terms Lemma 2.1. Let p > max 1, n 2 and n 2 < b < min (2, p) . Then the estimate for n 2 < b ≤ 2. Therefore the first term R 1 is estimated as , since the norm of the homogeneous Sobolev spaceḢ σ is equivalent to that of the homogeneous Besov spaceḂ σ 2,2 (see [2]), we have where 0 ≤ σ < min (2, p) . Hence using (5) with n 2 < b ≤ 2, (7) with σ = b, and (6), we estimate the second term R 2 as follows . This completes the proof of the lemma.
To prove local existence we introduce the function space X T such that We are now in a position to prove the local existence theorem.
. Then for some time T > 0 there exists a solution u ∈ X T for the Cauchy problem (1).
Proof. Let us consider the Cauchy problem (1) with the regularized initial data as follows : where m = 1, 2, ... The local existence of solution u m ∈ X T to the Cauchy problem (8) can be obtained as in book [3]. Let us prove that there exists a time T > 0 such that the estimate We argue by the contradiction. In view of the continuity of the norm X T with respect to T we can find the first time T > 0 such that Consider the integral equation associated with (8) Using the estimate (7) we obtain where 0 ≤ σ < min (2, p) . In view of (6), we have Hence we find Thus, there exists a time T = T (ρ) > 0 such that for all m. This is the desired contradiction. By the compact embedding of L ∞ ∩Ḣ b into L ∞ ∩Ḣ b−ε which follows from the fact that Hölder class of order α denoted by there exists a subsequence u m k which converges inḢ b−ε to some function u ∈Ḣ b−ε . Passing to the limit m k → ∞ in the integral equation (9) we find that u is a solution of (1) (see [12]). This completes the proof of the lemma.
For the one dimensional case n = 1 the nonlinearity have a sufficient regularity when the order of the nonlinearity p is close to 3. Hence we can apply the contraction mapping principle considering to the linearized version with v ∈ X T and v X T < 2ρ. As above we find that there exists a time T = T (ρ) > 0 such that u X T < 2ρ. Also in the same way as above, we find that there exists T = T (ρ) > 0 such that Thus the transformation u = Sv is a contraction mapping from X T into itself and there exists a unique solution u ∈ X T for the Cauchy problem (1) in the case of n = 1. In the case of n = 2, 3 the nonlinearity does not possess sufficient regularity since the order p < 2. Then to prove the uniqueness we need to make some additional assumptions on the lower bound for the solutions.
, where ϕ (t) = FU (−t − θ) u (t) and C > 0 is the constant from the Sobolev embedding inequality (5). Then the solution u is unique.
Proof. By the contrary suppose that there exist two solutions ϕ 1 and ϕ 2 of the integral equation associated with (1) where F (φ) = |φ| p−1 φ. Then we obtain Therefore by (7) we have where 0 ≤ σ < min (2, p) . By (5) and the assumptions on ϕ we find By the condition on the lower bound for the solutions, F (φ) has enough regularity to estimate for n = 2, 3. Then we obtain where n 2 < b < 2. Therefore by the Granwall inequality we get This completes the proof of the lemma.
Then the estimate Proof. By the contrary we may assume that there exists a time T > 0 such that We also have ϕ (0) L ∞ ≤ ρ from the assumptions of Theorem 1.1. We represent the solution of (1) in the form Therefore we obtain where g (t) = O ρ p+1 (t + θ) for j = 1, 2. Let r 1 , r 2 be solutions to (11) with r 1 (0) = r 2 (0) = |ϕ (0)|, then we have This implies the uniqueness of the solution for (11). Since 0 < r (0) ≤ ρ, to find the upper bound for the solution r (t), it is enough to consider the case r (0) = ρ. Define .
By a direct calculation we see that f (t) satisfies Multiplying both sides of (11) by f −p , we obtain By the Young inequality pf −1 r ≤ f −p r p + (p − 1) and by the dissipative condition λ 2 < 0, we find Integrating in time, we get since by the definition of f (t), we see that Let us consider the second term of the right-hand side of (12). We find Therefore for sufficiently small ρ > 0 and A − 1 p−1 . This is the desired contradiction and the lemma is proved.
Lemma 3.2. Assume that the assumptions of Theorem 1.1 hold. Also suppose that the estimate sup is true. Then the following estimate is valid for sufficiently small γ > 0, where b > n 2 , and ϕ (t) = FU (−t − θ) u(t). Proof. We assume that there exists a time T such that By (6) we have We now turn to the integral equation (10) for ϕ and use (13) to get γ for sufficiently small ρ and A −1 . This is a desired contradiction, which completes the proof of the lemma.
We next consider the asymptotic behavior of Lemma 4.2. Let the initial data satisfy the assumptions of Theorem 1.1 and u be the solution constructed in Theorem 1.1. Then there exists a unique η + such that η + L ∞ ≤ Cρ p−1 and the following estimate

Proof. We have by a direct calculation
Integrating in time, we obtain We denote then by Lemma 4.1 and (15), we find that for t > s > 0. Therefore there exists a unique η + such that We also have by (15) and (16) with t = 0 Then by (16) we find When 1 < p ≤ 2, we have for small ρ > 0 and A −1 , and by (16) we get Therefore by (18) The lemma is proved.