Finite time extinction for a damped nonlinear Schr{\"o}dinger equation in the whole space

We consider a nonlinear Schr{\"o}dinger equation set in the whole space with a single power of interaction and an external source. We first establish existence and uniqueness of the solutions and then show, in low space dimension, that the solutions vanish at a finite time. Under a smallness hypothesis of the initial data and some suitable additional assumptions on the external source, we also show that we can choose the upper bound on which time the solutions vanish.

1 Introduction and explanation of the method 1 2 Existence and uniqueness of the solutions 6 3 Finite time extinction and asymptotic behavior 10 1 Introduction and explanation of the method Let us consider the following Schrödinger equation with a nonlinear damping term, iu t + ∆u + a|u| m−1 u = f (t, x), in (0, ∞) × Ω, (1.1) where Ω ⊆ R N is an open subset, a ∈ C, 0 < m < 1 and f : (0, ∞)×Ω −→ C measurable is an external source.When a ∈ R, m 1 and f = 0, equation (1.1) has been intensively studied, especially with Ω = R N (among which existence, uniqueness, blow-up, scattering theory, time decay).The literature is too extensive to give an exhaustive list.See, for instance, the monographs of Cazenave [11], Sulem and Sulem [22], Tao [23] and the references therein.The case a ∈ C is more anecdotic.See, for instance, Bardos and Brezis [3], Lions [16], Tsutsumi [24] and Shimomura [21].Note that except in [16], it is always assumed m > 1.
In this paper, we are looking for solutions which vanishes at a finite time.For many reasons, we have to consider 0 < m < 1.When m = 1, existence is not hard to obtain, since the equation is linear, while the finite time property is not possible (which is a direct consequence of (1.4)).To our knowledge the first paper in this direction is due to Carles and Gallo [9] with a = i, f = 0 and Ω is a compact manifold without boundary.To construct solutions, they regularize the nonlinearity and use a compactness method to pass in the limit.They prove the finite time extinction property for N 3 including the case m = 0.More recently, Carles and Ozawa [10] obtain the existence, uniqueness and finite time extinction for Ω = R N , a ∈ iR + and f = 0. Due to the lack of compactness, they restrict their study to N 2 and add an harmonic confinement in (1.1) for some technical reasons.For the finite time property with N = 2 they also restrict the range of m to 1 2 , 1 and make a smallness assumption of the initial data.In this paper, we work in the whole space and we remove of all these restrictions and extend the previous results to a large class of values of a (see, for instance, Theorems 2.7 and 3.1).Indeed, we shall assume that the complex number a is in a cone of the complex plane.More precisely, The assumption that a belongs to the cone C(m) was considered in a series of papers by Okazawa and Yokota [18,19,20].They studied the asymptotic behavior of the solutions to the complex Ginzburg-Landau equation in a bounded domain with the assumption (1.2) and, sometimes, with m > 1. See also Kita and Shimomura [15] and Hou, Jiang, Li and You [14] where (1.2) is assumed but with (among others restrictive assumptions) m > 1.In all these papers, there is no finite time extinction result.We would also like mention the (very complete) work of Antontsev, Dias and Figueira [1] where they consider the complex Ginzburg-Landau equation, e −iγ u t − ∆u + |u| m−1 u = f (t, x), in (0, ∞) × Ω, (1.3) where Ω is bounded, 0 < m < 1 and − π 2 < γ < π 2 .In particular, e −iγ = ±i.They show spatial localization, waiting time and finite time extinction properties.The case of equation (1.3) with a delayed nonlocal perturbation is studied in the recent paper of Díaz, Padial, Tello and Tello [12].Finally, Hayashi, Li and Naumkin [13] study time decay for a more classical Schrödinger equation (1.1) (a satisfying (1.2), m > 1 and Ω = R N ).
In this paper, we are interested in the finite time extinction of the solution.Formally, this result is not too hard to obtain (the method we explain below for the finite time extinction property is that used in [9,10,7]).Suppose f = 0.It is well known that solutions that vanish in finite time do not exist when m 1 (at least when a ∈ R).Indeed, multiplying (1.1) by iu, integrating by parts and taking the real part, we obtain, To expect a finite time extinction, the mass has to be non increasing and so Im(a) > 0. Now, since m + 1 < 2, we may interpolate L 2 between L m+1 and L p , for some p > 2, and control the L p -norm by a Sobolev norm.Using a Gagliardo-Nirenberg's inequality, for some an explicit constant θ ℓ ∈ (0, 1), if u is bounded in H ℓ then putting together (1.4)-(1.5),we arrive at the ordinary differential equation, with δ = m+1 2θ ℓ , where y(t) = u(t) 2 L 2 .By integration, we then obtain the asymptotic behavior of u with respect to the value of δ.
As a consequence, a sufficient condition to have extinction in finite time is δ < 1 which turns out to be equivalent to N = 1 when ℓ = 1.To increase the space dimension, we assume that u is bounded in H 2 and we deduce that δ < 1 when N 3. Theoretically, we can reach any space dimension if u is bounded in H ℓ for ℓ large enough (actually, if ℓ = N 2 + 1, where N 2 denotes the integer part of N 2 ; see Theorem 2.1 in Bégout and Díaz [7]).But this is not reasonable due to the lack of regularity of the nonlinearity, which is merely Hölder continuous.A reachable goal is to obtain existence and boundedness of the solutions in H 2 .Now, we focus on the construction of a solution to (1.1) in R N with f = 0 (to fix ideas).First of all, we would like to uniformly control u(t) 2 H 1 .Estimate (1.4) partially answers this question.For ∇u(t) 2  L 2 , we multiply (1.1) by i∆u and take the real part.We get, We then expect to have, Regularizing the nonlinearity, integrating by parts and passing to the limit, (1.7) can be proved under assumption (1.2) (Lemma 4.4).Actually, we extended the method found in Carles and Gallo [9], where the situation is simpler since a = i.Assume Ω ⊆ R N .To construct a solution to (1.1), we use theory of the maximal monotone operators in the Hilbert space L2 .We then consider the operator, with the natural domain (1.9) Once (1.9) is proved, it remains to show that R(I + A) = L 2 (Theorem 4.1 and Corollary 4.5).This means that for any F ∈ L 2 , the equation admits a solution belonging to D(A).Existence, uniqueness, a priori estimates and smoothness of the solutions of (1.10) for a large class of values of a (including (1.2)) have been intensively studied in the papers by Bégout and Díaz [4,6].The natural 2 space to look for a solution is H1 0 ∩ L m+1 .When Ω is bounded with a smooth boundary, a bootstrap method yields u ∈ H 2 (Ω).Note that in this case, the condition u m ∈ L 2 (Ω) is automatically verified since u m ∈ L 2 m (Ω) ֒→ L 2 (Ω) and then u ∈ D(A).Although this method works very well, we proposed another one in Bégout and Díaz [7]: we make the sum of two monotone operators, where one of them is maximal monotone (−i∆) and the other one is continuous over L 2 (Ω) (−ia|u| m−1 u).A difficulty appears when Ω is unbounded, say Ω = R N .In this case, we have a natural method would be to multiply (1.10) by −∆u and take the real part.But then we lose the term ∆u 2 L 2 (R N ) .The original idea is to rotate a in the complex plane and stay in the cone C(m) to still have (1.7) (see Lemma 4.2 and the picture p.13).If we can find b ∈ C such that ab ∈ C(m) then multiplying (1.10) by −b∆u, integrating by parts and taking the real part, we arrive at, We see that we must have Im(b) < 0 and so the rotation has to be made in the negative sense.So we exclude the boundary of C(m) located in the first quarter complex plane.Hence Assumption 2.1 below.Note that the sign of Re(b) has no importance since we already have an estimate in H 1 (R N ).
Having a priori estimates, we may construct a solution u ∈ H 2 (R N ) ∩ L 2m (R N ) of (1.10) as a limit of solutions with compact support.The existence of such solutions is provided in Bégout and Díaz [4] (see also Bégout and Díaz [5]).To conclude the explanation of our method, we go back to the proof of (1.9).When a = i, this is very simple since this estimate is equivalent to the monotonicity of the derivative of the convex function defined on R 2 by, (x, y) −→ is entirely treated in Bégout and Díaz [7]: existence, uniqueness and boundedness for any subset Ω ⊆ R N .
We will use the following notations throughout this paper.We denote by z the conjugate of the complex number z, by Re(z) its real part and by Im(z) its imaginary part.Unless if specified, all functions are complex-valued (H 1 (Ω) = H 1 (Ω; C), etc).For 1 p ∞, p ′ is the conjugate of p defined by 1 p + 1 p ′ = 1.For a Banach space X, we denote by X ⋆ its topological dual and by ., .X ⋆ ,X ∈ R the X ⋆ − X duality product.In particular, for any In the same way, we will use the notation u ∈ W 1,p loc [0, ∞); X .As usual, we denote by C auxiliary positive constants, and sometimes, for positive parameters a 1 , . . ., a n , write as C(a 1 , . . ., a n ) to indicate that the constant C depends only on a 1 , . . ., a n and that dependence is continuous (we will use this convention for constants which are not denoted by "C").This paper is organized as follows.In Section 2, we state the mains results about existence, uniqueness and boundness for (1.1) (Theorem 2.4, 2.6 and 2.7).In Section 3, we give the results about the finite time extinction property and the asymptotic behavior (Theorems 3.1, 3.4 and 3.5).The proofs of the existence, uniqueness and boundness are made in Section 4 while those of the finite time extinction property and the asymptotic behavior are given in Section 5.

Existence and uniqueness of the solutions
We consider the following nonlinear Schrödinger equation. (2. 2) The main results in this paper hold with the assumptions below.
Assumption 2.1.We assume that 0 < m < 1 and a ∈ C satisfy, If Re(a) 0 then we assume further that, Here and after, we shall always identify L 2 (R N ) with its topological dual.Let 0 < m < 1 and We recall that (see, for instance, Lemmas A.2 and A.4 in Bégout and Díaz [7]), This justifies the notion of solution below (and especially 4)).
Let us consider the following assertions.
We shall say that u is a strong solution if u is an H 2 -solution or an H 1 -solution.We shall say that u is an We shall say that u is an L 2 -solution or a weak solution of (2.1)-(2.2) is there exists a pair, such that for any n ∈ N, u n is an H 2 -solution of (2.1) where the right-hand side of (2.1) is f n , and if for any T > 0, and if u satisfies (2.2).

1) Any strong or weak solution belongs to
2) It is obvious that an H 2 -solution is also an H 1 -solution and a weak solution.But it is not clear that an H 1 -solution is a weak solution, without a continuous dependence of the solution with respect to the initial data.Such a result will be established with the additional assumptions (2.3)-(2.4) on a (see Lemma 4.6 below).Note also that Assertion 2) of Definition 2.2 is not an additional assumption for the H 1 -solutions.
3) Any H 2 -solution (respectively, any , for almost every t > 0. Indeed, this is a direct consequence of Definition 2.2 and (2.13).

4) If u is a weak solution then
Bégout and Díaz [7]).Indeed, using the notation of Definition 2.2 and (2.11), this comes from (2.10) and the uniform convergences, for any T > 0. In particular, u solves (2.1) in Theorem 2.4 (Existence and uniqueness of L 2 -solutions).Let Assumption 2.1 be fulfilled and Then for any u 0 ∈ L 2 (R N ), there exists a unique weak solution u to (2.1)-(2.2).In addition, ) By interpolation, we infer that for any p then by (2.19), (2.20) and again by interpolation, we have for any p ∈ (m + 1, 2), where for each n ∈ N, u n is the weak solution of (2.1) with u n (0) = ϕ n and f n instead of f.Theorem 2.6 (Existence and uniqueness of H 1 -solutions).Let Assumption 2.1 be fulfilled and Then for any u 0 ∈ H 1 (R N ), there exists a unique H 1 -solution u to (2.1)-(2.2).Furthermore, u is also a weak solution and satisfies the following properties.
for any t 0. 3) ; R and we have, for almost every t > 0.
Remark 2.10.Using a radically different method than the one we propose here, we may show that all the results of this section remain valid if we replace R N with an unbounded domain Ω = R N .This will be the subject of a future work.

Finite time extinction and asymptotic behavior
Following the method by Carles and Gallo [9] (also used by Carles and Ozawa [10]) and Bégout and Díaz [7], we are able to prove the finite time extinction and asymptotic behavior results.
) and assume that one of the following hypotheses holds.
Let ℓ be the exponant in u 0 ∈ H ℓ (R N ).We have the following results.
a) There exists a finite time T ⋆ T 0 such that, Furthermore, where C = C(Im(a), N, m, ℓ).
Remark 3.3.In the case of our nonlinearity, Theorem 3.1 is an improvement of the result of Carles and Ozawa [10] in the sense they obtain the same conclusion as in a) but with a presence harmonic confinement in (2.1), Re(a) = 0, f = 0, N ∈ {1, 2} and nonlinearities are also considered in [10].
Let u be the unique strong solution of (2.1)-(2.2).Finally, assume that there exists T 0 0 such that, for almost every t > T 0 , f (t) = 0.

Then we have for any
) and let u be the unique weak solution of (2.1)-(2.2).If

Proofs of the existence and uniqueness theorems
Since we have to prove existence in the whole space, the method is radically different than that used in Bégout and Díaz [7].
In addition, where M = M (|a|, Arg(a), b 0 , λ).Furthermore, if F is compactly supported then so is u.Finally, let Here and after, Arg(a) ∈ (0, π) denotes the principal value of the argument of a.
The proof of the theorem relies on the following lemmas.Proof.Let θ a = Arg(a) ∈ (0, π), since Im(a) > 0. We look for b = e −iθ b , where 0 We then have ab = i|a| and the conclusion is clear.Im(z) We define the mapping for any measurable function u : R N −→ C, which we still denote by g, by g(u)(x) = g(u(x)).Then for ) and g is bounded on bounded sets. (4.8) for any u, v ∈ L m+1 (R N ).
Proof.Property (4.8) is an obvious consequence of (2.11) which implies the integrability property in the lemma.By Lemma 2.2 of Liskevich and Perel ′ muter [17], we have The lemma is proved.
Proof of Theorem 4.1.Let Assumption 2.1 be fulfilled, λ, b 0 > 0 and F ∈ L 2 (R N ).Let g be as in Lemma 4.3.We want to solve, We proceed with the proof in five steps. Step ) are solutions of (u F ) and (v G ), respectively, then estimate (4.3) holds true.
We multiply by iϕ, for ϕ ∈ D(R N ), the equation satisfied by u − v, we integrate by parts and we take the real part.By density of Estimate (4.3) then comes from (4.12), (4.9) and Cauchy-Schwarz's inequality.
Step 2: A second estimate.If u is a solution to (4.1) then u ∈ L m+1 (R N ) and satisfies (4.2).
Step 3: Compactness of the solution.
Finally, for each n ∈ N, denote by u n the unique solution to (4.1), where the right-hand side is F n instead of F (Steps 4 and 3).By Steps 1 and 2, (u Hence u is a solution to (4.1).This concludes the proof of the lemma.
Then A is maximal monotone on L 2 (R N ) (and so m-accretive) with dense domain.
Proof.Let X = H 1 (R N ) ∩ L m+1 (R N ) and let u, v be as in the lemma.Continuity comes from (2.8) and Definition 2.2.Estimate (4.25) being stable by passing to the limit in C [0, T ]; L 2 (R N ) × L 1 (0, T ); L 2 (R N ) , for any T > 0, it is sufficient to establish it for the H 2 -solutions.And since an H 2 -solution is an H 1 solution, we may assume that u, v are H 1 solution.Making the difference between the two equations, it follows from 3) of Remark 2.3 that we can take the X ⋆ − X duality product of the result with i(u − v).With help of (A.3) of Lemma A.5 in Bégout and Díaz [7], (2.14), (4.9) and Cauchy-Schwarz's inequality, we then arrive at, almost everywhere on (0, ∞).Integrating over (s, t), one obtains (4.25).
Proof of Theorem 2.6.Uniqueness comes from Lemma 4.6.Let Finally, let g be defined as in Lemma 4.3 and for each n ∈ N, let u n the unique H 2 -solution of (2.1) such that u n (0) = ϕ n , be given by Theorem 2.7.By Lemma 4.6, we have for any T > 0 and n, p ∈ N, It follows that for any T > 0, (u n ) n∈N is a Cauchy sequence in C [0, T ]; L 2 (R N ) .As a consequence, there exists u ∈ C [0, ∞); L 2 (R N ) such that for any T > 0, which gives with (4.11) and Cauchy-Schwarz's inequality, By integration, we obtain for any t > 0 and any n ∈ N,

Theorem 4 . 1 .
Let Assumption 2.1 be fulfilled and let λ, b 0 > 0. Then for any F ∈ L 2 (R N ), there exists a unique solution u to,
Lemma 4.6.Let Assumption 2.1 be fulfilled and f, g ∈ L 1 loc [0, ∞); L 2 (R N ) .If u and v are strong solutions or weak solutions of