On the $L^{2}$-critical nonlinear Schrödinger Equation with a nonlinear damping.

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

Mohamad Darwich

To cite this version:

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

2013. �hal-00712285v4�

ON THE L -CRITICAL NONLINEAR SCHR ¨ ODINGER EQUATION WITH A NONLINEAR DAMPING.

DARWICH MOHAMAD.

Abstract. We consider the Cauchy problem for the L

-critical nonlin- ear Schr¨ odinger equation with a nonlinear damping. According to the power of the damping term, we prove the global existence or the exis- tence of finite time blowup dynamics with the log-log blow-up speed for k∇ u ( t )k

.

1. 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 ):

iu t + ∆u + | u |

u + ia | u | ^p u = 0, (t, x) ∈ [0, ∞ [ ×R ^d , d = 1, 2, 3, 4.

u(0) = u ₀ ∈ H ¹ ( R ^d ) (1.1)

with initial data u(0) = u ₀ ∈ H ¹ ( R ^d ) where a > 0 is the coefficient of fric- tion and p ≥ 1. Note that if we replace + | u |

u by −| u |

u , (1.1) becomes the defocusing NLS equation.

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 < _d−2 ⁴ , 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

t→T k u(t) k H

(R

) = ∞ .

Let us notice that for a = 0 (1.1) becomes the L ² -critical nonlinear Schr¨ odinger equation:

iu t + ∆u + | u |

u = 0

u(0) = u ₀ ∈ H ¹ ( R ^d ) (1.2)

Key words and phrases. Damped Nonlinear Schr¨ odinger Equation, Blow-up, Global existence.

1

For u ₀ ∈ H ¹ , a sharp criterion for global existence for (1.2) has been exhib- ited by Weinstein [25]: Let Q be the unique positive solution to

∆Q + Q | Q |

= Q. (1.3)

For k u ₀ k L

< k Q k L

, 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:

∀ u ∈ H ¹ , E(u) ≥ 1 2 (

Z

|∇ u | ² )

1 −

R | u | ² R | Q | ²

.

On the other hand, there exists explicit solutions with k u ₀ k L

= k Q k L

that blow up in finite time in the regime _T ¹ _−t .

In the series of papers [15, 23], Merle and Raphael have studied the blowup for (1.2) with k Q k L

< k u ₀ k L

< k Q k L

+ δ, δ small and have proven the existence of the blowup regime corresponding to the log-log law:

k u(t) k H

(R

) ∼

log | log(T − t) | T − t

. (1.4)

In [6], Darwich has proved in case of the linear damping (p = 0), the global existence in H ¹ for k u ₀ k L

≤ k Q k L

, 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 per- sists in the case of the nonlinear damping for p < _d ⁴ ( 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:

Theorem 1.1. Let u ₀ in H ¹ ( R ^d ) with d = 1, 2, 3, 4:

(1) if _d−2 ⁴ > p ≥ ⁴ _d , then the solution of (1.1) is global in H ¹ .

(2) if 1 ≤ p < ⁴ _d and 1 ≤ p ≤ 2, then there exists 0 < α < k Q k L

such that for any u ₀ ∈ H ¹ with k u ₀ k L

< 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 k u ₀ k L

= k Q k L

+ 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

k u ₀ k L

≤ k Q k L

such that the solution u of (1.1) blow up in finite

time with this law:

1

(T − t) ^β−ǫ . k∇ u(t) k L

(R

) . 1 (T − t) ^β+ǫ , for any β ∈ ]0, _pd ² [ and 0 < ǫ < ^2−βpd _8+pd .

Remark 1.1. Note that, part (4) of Theorem 1.1 prove in particular that we dont have the blowup in the regime log-log for any p ∈ [1, ⁴ _d [ and in the regime ¹ _t for d = 1 and 1 ≤ p < 2, for initial data with critical or subcritical mass.

In the ”critical” case p = _d ⁴ , we have more precisely :

Theorem 1.2. Let p = ⁴ _d , then the initial-value probem (1.1) is globally well posed in H ^s (R ^d ), s ≥ 0. Moreover, there exist unique u ₊ in L ² such that

k u(., t) − e ^it∆ u ₊ k L

−→ 0, t −→ + ∞ , (1.5) where e ^it∆ is the free evolution.

Remark 1.2. Theorem 1.2 and part (1) and (2) of Theorem 1.1 still hold in the defocusing case.

Remark 1.3. Note that if u(t, .) 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.

2. Proof of part (4) of Theorem 1.1

Special solutions play a fundamental role for the description of the dynam- ics of (NLS). They are the solitary waves of the form u(t, x) = exp(it)Q(x), where Q solves:

∆Q + Q | Q |

= Q. (2.1)

The pseudo-conformal transformation applied to the stationary solution e ^it Q(x) yields an explicit solution for (NLS)

S(t, x) = 1

| t |

Q( x t )e ⁻ⁱ

2 4t

+

which blows up at T = 0.

Note that

k S(t) k L

= k Q k L

and k∇ S(t) k L

∼ 1

t (2.2)

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 k∇ u(t) k L

≥ C(u ₀ )

√ T − t . (2.3)

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.

For this we need the following Theorem ( see [10]) :

Theorem 2.1. Let (v _n ) _n be a bounded family of H ¹ ( R ^d ), such that:

lim sup

n→+∞ k∇ v _n k L

(R

) ≤ M and lim sup

n→+∞ k v _n k _L

≥ m. (2.4) Then, there exists (x _n ) _n ⊂ R ^d such that:

v _n ( · + x _n ) ⇀ V weakly, with k V k L

(R

) ≥ ( _d+4 ^d )

^m

d2 +1

+1

M

k Q k L

(R

) . Let us recall the following quantities:

L ² -norm : k u(t, x) k L

= Z

| u(t, x) | ² dx.

Energy : E(u(t, x)) = ¹ ₂ k∇ u k ² L

− _4+2d ^d k u k

4 d

+2 L

. Kinetic momentum : P (u(t)) = Im(

Z

∇ uu(t, x)).

Remark 2.1. It is easy to prove that if u is a solution of (1.1) on [0, T [, then for all t ∈ [0, T [ it holds

d

dt k u(t) k L

= − 2a Z

| u | ^p+2 , (2.5)

d

dt E(u(t)) = − a( k u

∇ u k ² L

− C _p k u k

^+2+p

L

) (2.6) and

d

dt P (u(t)) = − 2aIm Z

u | u | ^p ∇ u . (2.7) where C p = ^4+2d+pd _4+2d .

Now we are ready to prove part (4) of Theorem 1.1:

Suppose that there exist an initial data u 0 with k u 0 k L

≤ k Q k L

, such that the corresponding solution u(t) blows up with the following law

1

(T − t) ^β−ǫ . k∇ u(t) k L

(R

) . 1 (T − t) ^β+ǫ , where β; 0 < β < β(p, d) = _pd ² and 0 < ǫ < ^2−βpd _8+pd .

Recall that

E(u(t)) = E(u ₀ ) − a Z t

0

K(u(τ ))dτ, t ∈ [0, T [, (2.8)

where K(u(t)) = ( k u

∇ u k ² _L

− C _p k u k

^+2+p

L

).

By Gagliardo-Nirenberg inequality and (2.5), we have:

E(u(t)) . E(u 0 ) + Z t

0 k u k

4 d

+p−

L

k∇ u k ²⁺

pd 2

L

. E(u 0 ) + Z t

0 k∇ u k ²⁺

pd 2

L

Since the choise of ǫ, we obtain that

0 6 lim

t→T

Z t

0 k∇ u(τ ) k ²⁺

pd 2

L

dτ k∇ u(t) k ² L

(R

)

. lim

t→T (T − t)

(2−βpd−ǫ(8+pd)) = 0, (2.9) let

ρ(t) = k∇ Q k L

(R

)

k∇ u(t) k L

(R

)

and v(t, x) = ρ

u(t, ρx)

(t _k ) _k a sequence such that t _k → T and ρ _k = ρ(t _k ), v _k = v(t _k , .). The family (v k ) k satisfies

k v k k L

(R

) ≤ k u ₀ k L

(R

) ≤ k Q k L

(R

) and k∇ v k k L

(R

) = k∇ Q k L

(R

) . Remark that lim

k−→+∞ E(v _k ) = 0, because:

0 ≤ 1 2 (

Z

|∇ v _k | ² )

1 −

R | v k | ² R | Q | ²

2

≤ E(v _k ) = ρ ² _k E(u ₀ ) − aρ ² _k Z t

0

K(u(τ ))dτ

≤ ρ ² _k E(u 0 ) + 1 k∇ u(t _k ) k ² L

(R

)

Z t

0 k∇ u(τ ) k ²⁺

pd 2

L

dτ then using (2.9), the energy of v _k tends to 0. Which yields

k v k k

4 d

+2

L

→ d + 2

d k∇ Q k ² L

(R

) . (2.10) The family (v _k ) _k satisfies the hypotheses of Theorem 2.1 with

m

⁺² = d + 2

d k∇ Q k ² L

(R

) and M = k∇ Q k L

(R

) ,

thus there exists a family (x k ) k ⊂ R and a profile V ∈ H ¹ ( R ) with k V k L

(R

) ≥ k Q k L

(R

) , such that,

ρ

d 2

k u(t k , ρ k · +x k ) ⇀ V ∈ H ¹ weakly. (2.11) Using (2.11), ∀ A ≥ 0

lim inf

n→+∞

Z

B(0,A)

ρ ^d _n | u(t n , ρ n x + x n ) | ² dx ≥ Z

B(0,A) | V | ² dx.

But lim _{n→+∞ ρ} ¹

n

= + ∞ thus _ρ ¹

n

> A, ρ n A < 1. This gives immediately:

lim inf

n→+∞ sup

y∈R

Z

|x−y|≤1 | u(t _n , x) | ² dx ≥ lim inf

n→+∞

Z

|x−x

|≤ρ

A | u(t _n , x) | ² dx ≥ Z

|x|≤A | V | ² dx.

This it is true for all A > 0 thus :

lim inf

n→+∞ sup

y∈R

Z

|x−y|≤1 | u(t n , x) | ² dx ≥ Z

Q ² , (2.12)

then

k u ₀ k L

> lim inf

n→+∞ sup

y∈R

Z

|x−y|≤1 | u(t _n , x) | ² dx > k Q k L

. This gives the proof, the fact that k u ₀ k L

≤ k Q k L

.

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.

Theorem 3.1. Let p ≥ 1 for d = 1, 2 or 1 ≤ p ≤ _d−2 ⁴ for d ≥ 3, then the initial-value probem (1.1) is locally well posed in H ¹ ( R ^d )(If p < _d−2 ⁴ the minimal time of the existence depends on k u ₀ k H

.) .

Proof: See [3] page 93 Theorem 4.4.1.

To prove the following proposition, we will proceed in the same way as in the section 3.1 in [21].

Proposition 3.1. Let u be a solution of (1.1) and _d−2 ⁴ > p > ⁴ _d then

k∇ u(t) k L

(R

) ≤ k∇ u(0) k L

(R

) e ^a

−4t pd−4)

.

Proof : Multiply Eq. (1.1) by ∆u , integrate and take the imaginary part, this gives

1 2

d dt

Z

|∇ u | ² dx+a Z

| u | ^p |∇ u | ² +a ℜ Z

u ∇| u | ^p ∇ udx = − 4 d ℑ

Z

u ∇ u ℜ (u ∇ u) | u |

⁻² . (3.1) In the l.h.s, a simple calculation shows that the third term rewrites in the form ^p ₄ R

| u | ^p−2 ( ∇| u | ² ) ² .Equation (3.1) becomes:

1 2

d dt

Z

|∇ u | ² dx + a Z

| u | ^p |∇ u | ² + a p 4

Z

| u | ^p−2 ( ∇| u | ² ) ² ≤ 2 d

Z

| u |

|∇ u | ² . (3.2) To estimate the r.h.s of (3.2), we rewrite it as ( p > ⁴ _d )

Z

| u |

|∇ u | ² = Z

| u |

|∇ u |

|∇ u | ²⁻

. Now by H¨ older inequality we obtain that

Z

| u |

|∇ u | ² ≤ ( Z

| u | ^p |∇ u | ² )

( Z

|∇ u | ² ) ¹⁻

. Then inequality (3.2) takes the form:

d

dt w(t) + 2av(t) ≤ 4

d v(t)

w(t) ¹⁻

.

where w(t) = Z

|∇ u | ² and v(t) = Z

| u | ^p |∇ u | ² .

Using Young’s inequality ab ≤ ǫa ^q + Cǫ ⁻

b ^q

, ¹ _q + _q ¹

= 1, with q = ^pd ₄ and ǫ = ^ad ₂ we obtain :

d

dt w(t) ≤ a ⁻

1 pd

4−1

w(t).

This ensures that:

w(t) ≤ w(0)e ^a

4t pd−4

. 2

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 0 ≤ j < m. If u is any functions in C ₀ ^m ( R ^d ), then

k D ^j u k ^L

≤ C k D ^m u k ^a r k u k ^1−a q

where

1 s = j

d + a( 1 r − m

d ) + (1 − a) 1 q , for all a in the interval

j

m ≤ a ≤ 1,

where C is a constant depending only on d, m, j,q,r and a.

As a direct consequence we get :

Lemma 3.2. Let 1 ≤ p ≤ 2 and v ∈ C ₀ ^∞ (R ^d ) then:

Z

| v |

^+2+p ≤ C(

Z

|∇ ( | v |

) | ² ) × ( Z

| v | ² )

. where c > 0 depending only on d and p.

Proof: Take s =

₁₊ ^+2+p

, q = ₁₊ ²

r = 2, j = 0 and m = 1,then by Lemma 3.1 we obtain that:

| u |

L

4d+2+p 1+p

2

≤ C |∇ u |

4+2p 8d+4+2p

L

| u |

8 8 d d+4+2p

L

2 1+p

2

.

Taking u = | v | ¹⁺

, we obtain our lemma.

Now we can prove the following proposition:

Proposition 3.2. Let 1 ≤ p ≤ 2. There exists 0 < α = α(p, d) < k Q k L

, such that for any u ₀ ∈ H ¹ with k u ₀ k L

< α, it holds

d

dt E(u(t)) ≤ 0, ∀ t > 0.

Proof: We can whrite that:

d

dt E(u(t)) = a C p

Z

| u |

^+p+2 − 4 (p + 2) ²

Z

|∇ ( | u |

) | ² , then by Lemma 3.2 we obtain that:

d

dt E(u(t)) ≤ a(

Z

|∇ ( | u |

) | ² )(C p C(

Z

| u | ² )

− 4 (p + 2) ² ) Choosing α

< _(p+2) ⁴

1

C

C , and using that k u(t) k L

≤ k u ₀ k L

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 :

∀ u ∈ H ¹ , E(u) ≥ 1 2 (

Z

|∇ u | ² )

1 −

R | u | ² R | Q | ²

.

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 k u k _L

([0,T];L

(R

)) = + ∞ .

Proposition 3.3. Let u be the unique maximal solution of (1.1) in [0, T ^∗ );

if T ^∗ < ∞ , then k u k L

([0,T ],L

) = ∞ where σ = ⁴ _d + 2.

To prove this claim, denoting by S( · ) the free evolution of the linear Schr¨ odinger equation and defining the notion of admissible pair in the follow- ing 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 u ₀ ∈ L ² (R ^d ) and T ∈ (0, ∞ ] are such that k S(.)u 0 k L

([0,T ],L

) < δ, there ex- ists a unique solution u ∈ C([0, T ], L ² (R ^d )) ∩ L ^σ ([0, T ], L ^σ (R ⁿ )) of (1.1).

In addition, u ∈ L ^q ([0, T ], L ^r (R ^d )) for every admissible pair (q, r); for t ∈ [0, T ].Finally, u depends continuously in C([0, T ], L ² (R ^d )) ∩ L ^σ ([0, T ], L ^σ (R ^d )) on u ₀ ∈ L ² (R ⁿ ). If u ₀ ∈ H ¹ (R ^d ), then u ∈ C([0, T ], H ¹ (R ^d )).

See [4] for the proof.

We need also the following lemma ( see [4]):

Lemma 3.3. Let T ∈ (0, ∞ ], let σ = ⁴ _d + 2, and let (q, r) be an admis- sible pair. Then, whenever u ∈ L ^σ ([0, T ], L ^σ ( R ^d )), it follows that F (u) =

− i Z t

0

S(t − s)( | u |

u + ia | u | ^p u)ds ∈ C([0, T ], H ⁻¹ ( R ^d )) ∩ L ^q (0, T, L ^r ( R ^d )).

Furthermore, there exists K, independent of T, such that

k F v − F u k L

(]0,T[,L

) < K( k u k

4 d

L

(]0,T [,L

(R

)) + k v k

4 d

L

(]0,T [,L

(R

)) ) k u − v k L

(]0,T [,L

(R

))

(3.3)

for every u, v ∈ L ^σ (]0, T [, L ^σ ( R ^d )).

Proof of Proposition 3.3:

Let u ₀ ∈ L ² ( R ^d ). Observe that k S(.)u ₀ k L

(0,T.L

) −→ 0 as T −→ 0. Thus for sufficiently small T , the hypotheses of Proposition 3.4 are satisfied. Ap- plying iteratively this proposition, we can construct the maximal solution u ∈ C([0, T ^∗ ), L ² ( R ^d ))) ∩ L ^σ ([0, T ^∗ ), L ^σ ( R ^d )) of (1.1). We proceed by con- tradiction, assuming that T ^∗ < ∞ , and k u k L

(]0,T [,L

) < ∞ . Let t ∈ [0, T ^∗ ).

For every s ∈ [0, T ^∗ − t) we have

S(s)u(t) = u(t + s) − F (u(t+ · ))(s).

From (3.3), we thus obtain

k S(.)u(t) k L

([0,T

−t),L

(R

)) ≤ k u k L

(]t,T

[,L

) + K( k u k L

(]t,T

[,L

) )

⁺¹ Therefore, for t fixed close enough to T ^∗ , it follows that

k S(.)u(t) k L

([0,T

−t),L

(R

)) ≤ δ.

Applying Proposition 3.4, we find that u can be extended after T ^∗ , which contradicts the maximality.

Corollary 3.1. For p = ⁴ _d , the solution of (1.1) is global.

Proof Multiply equation (1.1) by u, and take the imaginary part to obtain:

d

dt k u(t) k ² L

+ 2a k u k

4 d+2

L

= 0.

Hence ∀ t ∈ R +

k u k _L

[0,t[L

(R

) ≤ 1

2a k u ₀ k ² L

.

The global existence follows then directly from Propostion 3.3. Now to finish the proof of Theorem 1.2, we will prove the scattering:

Let v(t) = e ^−it∆ u(t) := S( − t)u(t) then v(t) = u ₀ + i

Z t 0

S( − s)( | u(s) |

u(s) + ia | u(s) |

u)ds.

Therefore for 0 < t < τ , v(t) − v(τ ) = i

Z t τ

S( − s)( | u(s) |

u(s) + ia | u(s) |

u)ds.

It follows from Strichartz’s estimates (see the proof of Lemma 4.2) that:

k v(t) − v(τ ) k L

= k i Z t

τ

S( − s)( | u(s) |

u(s)+ia | u(s) |

u)ds k L

≤ C k u k

⁺¹

L

([t,τ ]×R

) . 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.

4. 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 , k h 0 k H

≤ 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:

u(t, x) = 1

λ

(t) (Q _b(t) + ǫ)(t, x − x(t)

λ(t) )e ^iγ(t) , (4.1) for some geometrical parameters (b(t), λ(t), x(t), γ(t)) ∈ (0, ∞ ) × (0, ∞ ) × R ^d × R , here λ(t) ∼ _k∇u(t)k ¹

L2

, 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 _λ ¹

(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:

E(u(t)) . log(λ(t))λ(t) ⁻

, P (u(t)) . log(λ(t))λ(t) ¹⁻

.

Let us recall that a fonction u :[0, T ] 7−→ H ¹ follows the log-log regime if the following uniform controls on the decomposition (4.1) hold on [0, T ]:

• Control of b(t)

b(t) > 0, b(t) < 10b(0). (4.2)

• Control of λ:

λ(t) ≤ e ^−e

π 100b(t)

(4.3) and the monotonicity of λ:

λ(t ₂ ) ≤ 3

2 λ(t ₁ ), ∀ 0 ≤ t ₁ ≤ t ₂ ≤ T. (4.4)

Let k ₀ ≤ k ₊ be integers and T ⁺ ∈ [0, T ] such that 1

2 ^k

≤ λ(0) ≤ 1 2 ^k

⁻¹ , 1

2 ^k

≤ λ(T ⁺ ) ≤ 1

2 ^k

⁻¹ (4.5) and for k 0 ≤ k ≤ k + , let t k be a time such that

λ(t k ) = 1

2 ^k , (4.6)

then we assume the control of the doubling time interval:

t _k+1 − t _k ≤ kλ ² (t _k ). (4.7)

• control of the excess of mass:

Z

|∇ ǫ(t) | ² + Z

| ǫ(t) | ² e ^−|y| ≤ Γ _b(t)

. (4.8)

4.1. Control of the energy and the kinetic momentum in the log- log regime. We recall the Strichartz estimates. An ordered pair (q, r) is called admissible if ² _q + ^d _r = ^d ₂ , 2 < q ≤ ∞ . We define the Strichartz norm of functions u : [0, T ] × R ^d 7−→ C by:

k u k S

([0,T ]×R

) = sup

(q,r)admissible k u k L

L

([0,T]×R

) (4.9) and

k u k S

([0,T]×R

) = sup

(q,r)admissible k∇ u k L

L

([0,T ]×R

) (4.10) We will sometimes abbreviate S ⁱ ([0, T ] × R ² ) with S _T ⁱ or S ⁱ [0, T ], i = 1, 2.

Let us denote the H¨ older dual exponent of q by q ^′ so that ¹ _q + _q ¹

= 1. The Strichartz estimates may be expressed as:

k u k S

≤ k u ₀ k L

+ k (i∂ _t + ∆)u k _L

t

L

(4.11) 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:

| E(u(t)) | ≤ C(log(λ(t))λ(t) ⁻

), (4.12)

| P (u(t)) | ≤ C(log(λ(t))λ(t) ¹⁻

). (4.13) To prove this lemma, we shall need the following one:

Lemma 4.2. Let u be a solution of (1.1) emanating for u ₀ in H ¹ . Then u

∈ C([0, ∆T ], H ¹ ) where ∆T = k u ₀ k

d−4 d

L

k u ₀ k ⁻² H

, and we have the following control

k u k S

[t,t+∆T ] ≤ 2 k u ₀ k L

, k u k S

[t,t+∆T ] ≤ 2 k u ₀ k H

(R

) .

Proof of Lemma 4.1: According to (4.7) each interval [t _k , t _k+1 ], can be divided into k intervals, [τ _k ^j , τ _k ^j+1 ] such that the estimates of the previous lemma are true. From (2.6) and the Gagliardo-Nirenberg inequality, we obtain that:

d

dt E(u(t)) . k u k

^+p−

pd 2

L

k∇ u k ²⁺

pd 2

L

Using (2.5) this gives Z τ

τ

d

dt E(u(t))dt ≤ C Z τ

τ

k∇ u(t) k ²⁺

pd 2

L

, then by Lemma 4.2, we obtain that:

Z τ

τ

d

dt E(u(t))dt ≤ C(τ _k

− τ _k

)λ ⁻²⁻

(τ _k

) Note that τ _k ^j+1 − τ _k ^j ∼ λ ² (τ _k ^j ) ∼ λ ² (t k ), then

Z τ

τ

d

dt E(u(t))dt ≤ Cλ ⁻

(t k ) Summing from j = 1 to J k ≤ CK, we obtain that:

J

X

j=1

Z τ

τ

d

dt E(u(t))dt ≤ Ckλ ⁻

(t k ) Now taking T ⁺ = T and summing from K 0 to K ⁺ , we obtain:

Z T

0

d

dt E(u(t))dt ≤ CK ⁺ λ ⁻

(T ⁺ ) . Clog(λ(T ))λ ⁻

(T ).

Note that log (λ(T ))λ ⁻

(T ) is small with to respect _λ ¹

because p < ⁴ _d . Now we prove (4.13): From (2.7) we have :

| d

dt P (u(t)) | ≤ Z

| u ||∇ u || u | ^p By Gagliardo-Nirenberg inequality we have:

k u k ^2p+2 _L

≤ k u k ^2p+2−dp _L

k∇ u k ^dp _L

then d

dt P (u(t)) ≤ ( Z

| u | ^2(p+1) )

k∇ u k L

≤ C k∇ u k ¹⁺

pd 2

L

. Then:

Z τ

τ

d

dt P (u(t)) ≤ C(τ _k ^j+1 − τ _k ^j )

∇ u(τ _k ^j )

1+

L

≤ C k∇ u(t _k ) k ⁻¹⁺

pd 2

L

Summing successively into j and k we obtain that:

Z T

0

d

dt P (u(t)) . log(λ(T ⁺ ))λ ¹⁻

(T ⁺ ).

Remark that this quantity is small with to respect _λ ¹ because p < ⁴ _d . 2

References

[1] P. Antonelli and C. Sparber. Global well-posedness for cubic NLS with nonlinear damping. Comm. Partial Differential Equations, 35 (2010) 48324845.

[2] H. Berestycki and P.-L. Lions. Nonlinear scalar field equations. II. Existence of infin- itely many solutions. Arch. Rational Mech. Anal., 82(1983):347–375.

[3] T. Cazenave. Semilinear Schr¨ odinger equations, volume 10 of Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 2003.

[4] T. Cazenave and F. Weissler. Some remarks on the nonlinear Schrdinger equation in the subcritical case. New methods and results in nonlinear field equations (Bielefeld, 1987), 5969, Lecture Notes in Phys., 347, Springer, Berlin, 1989.

[5] J. Colliander and P. Raphael. Rough blowup solutions to the L

critical NLS. Math.

Ann., 345(2009):307–366.

[6] M. Darwich. Blowup for the Damped L

critical nonlinear Shr¨ odinger equations. Ad- vances in Differential Equations. volume 17, Numbers 3-4 (2012),337-367.

[7] G. Fibich and F. Merle. Self-focusing on bounded domains. Phys. D, 155(2001):132–

158.

[8] G. Fibich and M. Klein. Nonlinear-damping continuation of the nonlinear Schrdinger equation-a numerical study. Physica D, 241 (2012), 519-527.

[9] A. Friedman Partial Differential Equations.

[10] T. Hmidi and S. Keraani. Blowup theory for the critical nonlinear Schr¨ odinger equa- tions revisited. Int. Math. Res. Not., 46(2005):2815–2828.

[11] T. Kato. On nonlinear Schr¨ odinger equations Ann. Inst. H. Poincar´e Phys. Th´eor., 46(1987):113–129.

[12] M.K Kwong. Uniqueness of positive solutions of ∆ u − u + u

= 0 in R

. Arch.

Rational Mech. Anal., 105(1989):243–266.

[13] P.-L. Lions. The concentration-compactness principle in the calculus of variations.

The locally compact case. II. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 1(1984):223–

283.

[14] F. Merle. Determination of blow-up solutions with minimal mass for nonlinear Schr

¨

oinger equations with critical power, Duke Math. J. 69:2, (1993), 427-454.

[15] F. Merle and P. Raphael. Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schr¨ odinger equation. In Journ´ees “´ Equations aux D´eriv´ees Partielles” (Forges-les-Eaux, 2002), pages Exp. No. XII, 5. Univ. Nantes, Nantes, 2002.

[16] F. Merle and P. Raphael. Sharp upper bound on the blow-up rate for the critical nonlinear Schr¨ odinger equation. Geom. Funct. Anal., 13(2003):591–642.

[17] F. Merle and P. Raphael. On universality of blow-up profile for L

critical nonlinear Schr¨ odinger equation. Invent. Math., 156(2004):565–672.

[18] F. Merle and P. Raphael. Profiles and quantization of the blow up mass for critical nonlinear Schr¨ odinger equation. Comm. Math. Phys., 253(2005):675–704.

[19] F. Merle and P. Raphael. On a sharp lower bound on the blow-up rate for the L

critical nonlinear Schr¨ odinger equation. J. Amer. Math. Soc., 19(2006):37–90 (elec- tronic).

[20] M. Ohta and G. Todorova. Remarks on global existence and blowup for damped non- linear Schr¨ odinger equations. Discrete Contin. Dyn. Syst., 23(2009):1313–1325.

[21] T. Passota, C. Sulemb and P.L. Sulem. Linear versus nonlinear dissipation for critical NLS equation. Physica D, 203 (2005) 167184

[22] F. Planchon and P. Rapha¨el. Existence and stability of the log-log blow-up dynamics for the L

-critical nonlinear Schr¨ odinger equation in a domain. Ann. Henri Poincar´e, 8(2007):1177–1219.

[23] P. Raphael. Stability of the log-log bound for blow up solutions to the critical non linear Schr¨ odinger equation. Math. Ann., 331(2005):577–609.

[24] M. Tsutsumi. Nonexistence of global solutions to the Cauchy problem for the damped

nonlinear Schr¨ odinger equations. SIAM J. Math. Anal., 15(1984):357–366.

[25] M.I. Weinstein. Nonlinear Schr¨ odinger equations and sharp interpolation estimates.

Comm. Math. Phys., 87(1982/83):567–576.

M. Darwich: Universit´ e Franc ¸ois rabelais de Tours, Laboratoire de Math´ ematiques et Physique Th´ eorique, UMR-CNRS 6083, Parc de Grandmont, 37200 Tours,

France

E-mail address: Mohamad.Darwich@lmpt.univ-tours.fr