On the L 2 -critical nonlinear Schrödinger Equation with a nonlinear damping.

. We consider the Cauchy problem for the L 2 -critical nonlinear Schr¨odinger equation with a nonlinear damping. According to the power of the damping term, we prove the global existence or the existence of ﬁnite time blowup dynamics with the log-log blow-up speed for k∇ u ( t ) k L 2 .


Introduction
In this paper, we study the blowup and the global existence of solutions for the focusing NLS equation with a nonlinear damping (N LS ap ): with initial data u(0) = u 0 ∈ H 1 (R d ) where a > 0 is the coefficient of friction and p ≥ 1.Note that if we replace +|u| Equation (1.1) arises in various areas of nonlinear optics, plasma physics and fluid mechanics.Fibich [7] noted that in the nonlinear optics context, the origin of the nonlinear damping is multiphoton absorption.For example, in the case of solids the number p corresponds to the number of photons it takes to make a transition from the valence band to the conduction band.Similar behavior can occur with free atoms, in this case p corresponds to the number of photons needed to make a transition from the ground state to some excited state or to the continuum.
The Cauchy problem for (1.1) was studied by Kato [11] and Cazenave [3] and it is known that if p < 4 d−2 , then the problem is locally well-posed in H 1 (R d ): For any u 0 ∈ H 1 (R d ), there exist T ∈ (0, ∞] and a unique solution u(t) of (1.1) with u(0) = u 0 such that u ∈ C([[0, T ]); H 1 (R d )).Moreover, T is the maximal existence time of the solution u(t) in the sense that if T < ∞ then lim Let us notice that for a = 0 (1.1) becomes the L 2 -critical nonlinear Schrödinger equation: Key words and phrases.Damped Nonlinear Schrödinger Equation, Blow-up, Global existence.
For u 0 ∈ H 1 , a sharp criterion for global existence for (1.2) has been exhibited by Weinstein [25]: Let Q be the unique positive solution to ∆Q + Q|Q| 4 d = Q. (1.3) For u 0 L 2 < Q L 2 , the solution of (1.2) is global in H 1 .This follows from the conservation of the energy and the L 2 norm and the sharp Gagliardo-Nirenberg inequality: .
On the other hand, there exists explicit solutions with u 0 L 2 = Q L 2 that blow up in finite time in the regime 1 T −t .In the series of papers [15,23], Merle and Raphael have studied the blowup for (1.2) with Q L 2 < u 0 L 2 < Q L 2 + δ, δ small and have proven the existence of the blowup regime corresponding to the log-log law: . (1.4) In [6], Darwich has proved in case of the linear damping (p = 0), the global existence in H 1 for u 0 L 2 ≤ Q L 2 , and has showed that the log-log regime is stable by such perturbations (i.e.there exist solutions blows up in finite time with the log-log law).Numerical observations suggest that this finite time blowup phenomena persists in the case of the nonlinear damping for p < 4 d ( see Fibich [7] and [21]).Passot and Sulem [21] have proved that the solutions are global in H 1 (R 2 ) in the case where the power of the damping term is strictly greater to the focusing nonlinearity.The case where the power of the damping term is equal to the focusing nonlinearity, "small damping prevents blow-up ?"was an open question for Sparber and Antonelli in their paper [1] and for Fibich and Klein in their paper [8].Our results can gives an answer for their open problem, at least for the L 2 -critical case.In fact, our aim in this paper is study for each value of (d, p), the existence of blow-up solutions as well as global existence criteria.And know if the regime log-log still stable by such perturbations.Let us now our results: d and 1 ≤ p ≤ 2, then there exists 0 < α < Q L 2 such that for any u 0 ∈ H 1 with u 0 L 2 < α, the emanating solution is global in H 1 .
(3) if 1 ≤ p < 4 d , then there exists δ 0 > 0 such that ∀a > 0 and ∀δ ∈ ]0, δ 0 [, there exists u 0 ∈ H 1 with u 0 L 2 = Q L 2 + δ, such that the solution of (1.1) blows up in finite time in the log-log regime.(4) if 1 ≤ p < 4 d , then there does not exists an intial data u 0 with u 0 L 2 ≤ Q L 2 such that the solution u of (1.1) blow up in finite time with this law: for any β ∈]0, 2 pd [ and 0 < ǫ < ) is a solution of (N LS a,p ) then u(−t, .) is a solution of (N LS −a,p ), then we dont have the scattering in −∞, because this changes the sign of the coefficient of friction.
Acknowledgments.I would like to thank my advisor Prof. Luc Molinet for his rigorous attention to this paper, Dr. Christof Sparber for his remarks and Prof. Baoxiang Wang for having given me the reference of Lemma 3.1.

Proof of part (4) of Theorem 1.1
Special solutions play a fundamental role for the description of the dynamics of (NLS).They are the solitary waves of the form u(t, x) = exp(it)Q(x), where Q solves: The pseudo-conformal transformation applied to the stationary solution e it Q(x) yields an explicit solution for (NLS) It turns out that S(t) is the unique minimal mass blow-up solution in H 1 up to the symmetries of the equation ( see [14]).
A known lower bound ( see [19]) on the blow-up rate for (NLS) is Note that this blow-up rate is the one of S(t) given by (2.2) and log-log given by (1.4).Now, we will prove the part (4) of Theorem 1.1.
Now we are ready to prove part (4) of Theorem 1.1: Suppose that there exist an initial data u 0 with u 0 L 2 ≤ Q L 2 , such that the corresponding solution u(t) blows up with the following law where . By Gagliardo-Nirenberg inequality and (2.5), we have: Since the choise of ǫ, we obtain that 0 lim Remark that lim dτ then using (2.9), the energy of v k tends to 0. Which yields (2.10) The family (v k ) k satisfies the hypotheses of Theorem 2.1 with thus there exists a family ( But lim n→+∞ 1 ρn = +∞ thus 1 ρn > A, ρ n A < 1.This gives immediately: This it is true for all A > 0 thus : This gives the proof, the fact that 3. Global existence. In this section, we prove assertion ( 1) and ( 2) of Theorem 1.1 and Theorem 1.2.To prove part (1), we will prove that the H 1 -norm of u is bounded for any time.To prove part (2), we use generalised Gagliardo-Nirenberg inequalities to show that the energy is non increasing.Finally to prove Theorem 1.2, we etablish an a priori estimate on the critical Strichartz norm.To prove the following proposition, we will proceed in the same way as in the section 3.1 in [21]. .
Proof : Multiply Eq. ( Using Young's inequality ab ≤ ǫa q + Cǫ − 1 q−1 b q ′ , 1 q + 1 q ′ = 1, with q = pd 4 and ǫ = ad 2 we obtain : This ensures that: This show that the H 1 -norm of u is bounded for any time and gives directly the proof of part one of Theorem 1.1 in the case p > 4/d.Now we will prove the global existence for small data, for this we will use the following generalized Gagliardo-Niremberg inequalities (see for instance [9]): Lemma 3.1.Let q, r be any real numbers satisying 1 ≤ q, r ≤ ∞, and let j, m be any integers satisfying for all a in the interval where C is a constant depending only on d, m, j,q,r and a.
As a direct consequence we get : where c > 0 depending only on d and p.
Taking u = |v| 1+ p 2 , we obtain our lemma.Now we can prove the following proposition: Proof: We can whrite that: then by Lemma 3.2 we obtain that: CpC , and using that u(t) L 2 ≤ u 0 L 2 for all t ≥ 0 (see 2.5 below) we get the result.Now the proof of part (2) of Theorem 1.1 follows from the sharp Gagliardo-Nirenberg inequality : . Proposition 3.2 together with the above inequality ensure that the H 1 -norm of u is uniformly bounded in time.This leads to the global existence result for small initial data when 1 ≤ p ≤ 2.
3.1.Critical case ( p = 4 d ).Now we will treat the critical case and prove Theorem 1.2.First let us prove that, if the solution blows up in finite time T , then u To prove this claim, denoting by S(•) the free evolution of the linear Schrödinger equation and defining the notion of admissible pair in the following way : An ordered pair (q, r) is called admissible if 2 q + d r = d 2 , 2 < q ≤ ∞, we will use the following proposition: Proposition 3.4.There exists δ > 0 with the following property.If See [4] for the proof.
We need also the following lemma ( see [4]): , and let (q, r) be an admissible pair.Then, whenever u Furthermore, there exists K, independent of T , such that

Proof of Proposition 3.3:
Let u 0 ∈ L 2 (R d ).Observe that S(.)u 0 L σ (0,T.L σ ) −→ 0 as T −→ 0. Thus for sufficiently small T , the hypotheses of Proposition 3.4 are satisfied.Applying iteratively this proposition, we can construct the maximal solution (1.1).We proceed by contradiction, assuming that T * < ∞, and From (3.3), we thus obtain Therefore, for t fixed close enough to T * , it follows that Applying Proposition 3.4, we find that u can be extended after T * , which contradicts the maximality.Proof Multiply equation (1.1) by u, and take the imaginary part to obtain: The global existence follows then directly from Propostion 3.3.Now to finish the proof of Theorem 1.2, we will prove the scattering: It follows from Strichartz's estimates (see the proof of Lemma 4.2) that: .
The right hand side goes to zero when t, τ −→ +∞, then scattering follows from the Cauchy criterion.This completes the proof of Theorem 1.2.

Blow up solution.
In this section, we will prove the existence of the explosive solutions in the case 1 ≤ p < 4/d.Theorem 4.1.Let 1 ≤ p < 4/d.There exist a set of initial data Ω in H 1 , such that for any 0 < a < a 0 with a 0 = a 0 (p) small enough, the emanating solution u(t) to (1.1) blows up in finite time in the log-log regime.
The set of initial data Ω is the set described in [15] in order to initialize the log-log regime.It is open in H 1 .Using the continuity with regard to the initial data and the parameters, we easily obtain the following corollary: Corollary 4.1.Let 1 ≤ p < 4/d and u 0 ∈ H 1 be an initial data such that the corresponding solution u(t) of (1.2) blows up in the loglog regime.There exist β 0 > 0 and a 0 > 0 such that if v 0 = u 0 + h 0 , h 0 H 1 ≤ β 0 and a ≤ a 0 , the solution v(t) for (1.1) with the initial data v 0 blows up in finite time.
Assertion (3) of Theorem 1.1 now follows directly from this corollary together the results of [15] on the L 2 -critical NLS equation and a scaling argument in order to drop the smallness condition on the damped coefficient a > 0. Now to prove Theorem 4.1, we look for a solution of (1.1) such that for t close enough to blowup time, we shall have the following decomposition: for some geometrical parameters (b(t), λ(t), x(t), γ(t)) ∈ (0, ∞) × (0, ∞) × R d × R, here λ(t) ∼ 1 ∇u(t) L 2 , and the profiles Q b are suitable deformations of Q related to some extra degeneracy of the problem.
Note that we will abbreviated our proof because it is very very close to the case of linear damping (p = 0 see Darwich [6]).Actually, as noticed in [22], we only need to prove that in the log-log regime the L 2 norm does not grow, and the growth of the energy( resp the momentum) is below 1 λ 2 (resp 1 λ ) .In this paper, we will prove that in the log-log regime, the growth of the energy and the momentum are bounded by: Let us recall that a fonction u :[0, T ] −→ H 1 follows the log-log regime if the following uniform controls on the decomposition (4.1) hold on [0, T ]: and the monotonicity of λ: and for k 0 ≤ k ≤ k + , let t k be a time such that then we assume the control of the doubling time interval: • control of the excess of mass: 4.1.Control of the energy and the kinetic momentum in the loglog regime.We recall the Strichartz estimates.An ordered pair (q, r) is called admissible if and We will sometimes abbreviate S i ([0, T ] × R 2 ) with S i T or S i [0, T ], i = 1, 2. Let us denote the Hölder dual exponent of q by q ′ so that 1 q + 1 q ′ = 1.The Strichartz estimates may be expressed as: where (q, r) is any admissible pair.Now we will derive an estimate on the energy, to check that it remains small with respect to λ −2 : Lemma 4.1.Assuming that (4.2)-(4.8)hold, then the energy and kinetic momentum of the solution u to (1.1) are controlled on [0, T ] by: |P (u(t))| ≤ C(log(λ(t))λ(t) Summing from j = 1 to J k ≤ CK, we obtain that: Note that log (λ(T ))λ − pd 2 (T ) is small with to respect By Gagliardo-Nirenberg inequality we have: Remark that this quantity is small with to respect 1 λ because p < 4 d . 2