Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 39, pp. 1–18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu FINITE TIME EXTINCTION FOR A DAMPED NONLINEAR SCHRÖDINGER EQUATION IN THE WHOLE SPACE PASCAL BÉGOUT Communicated by Jesus Ildefonso Diaz Abstract. We consider a nonlinear Schrödinger 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 Let us consider the Schrödinger equation with a nonlinear damping term iut + ∆u+ a|u|m−1u = f(t, x), in (0,∞)× Ω, (1.1) where Ω ⊆ RN 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 = RN (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 article, we are looking for solutions that vanish 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 6 3 including the case m = 0. More recently, Carles and Ozawa [10] obtain the existence, uniqueness and finite time extinction for Ω = RN , a ∈ iR+ and f = 0. Because of the lack of compactness, they restrict their study to N 6 2 and add an harmonic confinement in (1.1) for some technical 2010 Mathematics Subject Classification. 35Q55, 35A01, 35A02, 35B40, 35D30, 35D35. Key words and phrases. Damped Schrödinger equation; existence; uniqueness; finite time extinction; asymptotic behavior. c©2020 Texas State University. Submitted February 24, 2020. Published April 28, 2020. 1 2 P. BÉGOUT EJDE-2020/39 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, a ∈ C(m) := { z ∈ C : Im(z) > 0 and 2 √ m Im(z) > (1−m)|Re(z)| } . (1.2) 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 to mention the (very complete) work of Antontsev, Dias and Figueira [1] where they consider the complex Ginzburg-Landau equation, e−iγut −∆u+ |u|m−1u = 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 6= ±i. They show spatial localization, waiting time and finite time extinction proper- ties. The case of equation (1.3) with a delayed nonlocal perturbation is studied in the recent paper of Dı́az, 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 = RN ). In this article, 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 1 2 d dt ‖u(t)‖2L2 + Im(a)‖u(t)‖m+1 Lm+1 = 0. (1.4) 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 L2 between Lm+1 and Lp, for some p > 2, and control the Lp-norm by a Sobolev norm. Using a Gagliardo-Nirenberg’s inequality, ‖u(t)‖ 2m+1 2θ` L2 6 ‖u(t)‖m+1 Lm+1‖u(t)‖ (m+1)(1−θ`) θ` H` , (1.5) for some an explicit constant θ` ∈ (0, 1), if u is bounded in H` then putting together (1.4)–(1.5), we arrive at the ordinary differential equation, y′ + Cyδ 6 0, (1.6) with δ = m+1 2θ` , where y(t) = ‖u(t)‖2L2 . By integration, we then obtain the asymp- totic behavior of u with respect to the value of δ. • If δ < 1 then y(t)1−δ 6 (y(0)1−δ − Ct)+ and so u vanishes before time T? = C−1y(0)1−δ. • If δ = 1 then y(t) 6 y(0)e−Ct. • If δ > 1 then y(t)δ−1 6 y(0)δ−1(1 + Ct)−1. EJDE-2020/39 EXTINCTION FOR NLS IN THE WHOLE SPACE 3 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 ` = 1. To increase the space dimension, we assume that u is bounded in H2 and we deduce that δ < 1 when N 6 3. Theoretically, we can reach any space dimension if u is bounded in H` for ` large enough (actually, if ` = [ N 2 ] +1, where [ N 2 ] denotes the integer part of N2 ; see Bégout and Dı́az [7, Theorem 2.1 in]). 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 H2. Now, we focus on the construction of a solution to (1.1) in RN with f = 0 (to fix ideas). First of all, we would like to uniformly control ‖u(t)‖2H1 . Estimate (1.4) partially answers this question. For ‖∇u(t)‖2L2 , we multiply (1.1) by i∆u and take the real part. We obtain 1 2 d dt ‖∇u(t)‖2L2 + Re ( ia ∫ RN |u(t)|m−1u(t)∆u(t)dx ) = 0. We then expect to have Re ( ia ∫ RN |u(t)|m−1u(t)∆u(t)dx ) > 0. (1.7) 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 ⊆ RN . 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, Au = −i∆u− ia|u|m−1u, (1.8) with the natural domain D(A) = { u ∈ H1 0 (Ω);um ∈ L2(Ω) and ∆u ∈ L2(Ω) } . It is natural in the sense that it is the smallest domain, in the sense of the inclusion, for which D(A) ⊂ L2. Monotonicity relies on the inequality Re ( − i a ∫ Ω ( |u|m−1u− |v|m−1v ) (u− v)dx ) > 0. (1.9) Once (1.9) is proved, it remains to show that R(I + A) = L2 (Theorem 4.1 and Corollary 4.5). This means that for any F ∈ L2, the equation − i∆u− ia|u|m−1u+ u = F, (1.10) 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 Dı́az [4, 6]. The natural space to look for a solution is H1 0 ∩ Lm+1. (Multiply (1.10) by iu and u, integrate by parts and take the real part.) When Ω is bounded with a smooth boundary, a bootstrap method yields u ∈ H2(Ω). Note that in this case, the condition um ∈ L2(Ω) is automatically verified since um ∈ L 2 m (Ω) ↪→ L2(Ω) and then u ∈ D(A). Although this method works very well, we proposed another one in Bégout and Dı́az [7]: we make the sum of two monotone operators, where one of them is maximal monotone (−i∆) and the other one is continuous over L2(Ω) (−ia|u|m−1u). A difficulty appears when Ω is unbounded, say Ω = RN . In this case, we have D(A) = H2(RN ) ∩ L2m(RN ) and we have to show that a solution u ∈ H1(RN )∩Lm+1(RN ) belongs to L2m(RN ), or equivalently 4 P. BÉGOUT EJDE-2020/39 ∆u ∈ L2(RN ). Having (1.7) in mind, a natural method would be to multiply (1.10) by −∆u and take the real part. But then we lose the term ‖∆u‖2L2(RN ). 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 Figure 1). 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 − Im(b)‖∆u‖2L2(RN ) + Re ( iab ∫ RN |u|m−1u∆udx ) + Re(b)‖∇u‖2L2(RN ) = −Re ( b ∫ RN F∆udx ) . 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 H1(RN ). Having a priori esti- mates, we may construct a solution u ∈ H2(RN ) ∩ L2m(RN ) of (1.10) as a limit of solutions with compact support. The existence of such solutions is provided in Bégout and Dı́az [4] (see also Bégout and Dı́az [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 con- vex function defined on R2 by, (x, y) 7−→ 1 m+1 (x2 + y2)(m+1)/2 (see Bégout and Dı́az [4, Remark 9.3]). But when Re(a) 6= 0 then the imaginary part of the integral in (1.9) is still there. Fortunately, this can be controlled by its real part under assumption (1.2) and a consequence of Liskevich and Perel’muter [17, Lemma 2.2]. Finally, we consider the limit cases m = 0 and m = 1 for the values of a. Since limm↘0 C(m) = {0} × i(0,∞), it seems that no extension of [9, 10] is possible. The other limit case limm↗1 C(m) = R× i(0,∞) is entirely treated in Bégout and Dı́az [7]: existence, uniqueness and boundedness for any subset Ω ⊆ RN . We will use the following notation 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 imagi- nary part. Unless if specified, all functions are complex-valued (H1(Ω) = H1(Ω;C), etc). For 1 6 p 6∞, 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 T ∈ Lp′(Ω) and ϕ ∈ Lp(Ω) with 1 6 p <∞, 〈T, ϕ〉Lp′ (Ω),Lp(Ω) = Re ∫ Ω T (x)ϕ(x)dx. The scalar product in L2(Ω) between two functions u, v is, (u, v)L2(Ω) = Re ∫ Ω u(x)v(x)dx. For a Banach space X and p ∈ [1,∞], u ∈ Lploc ( [0,∞);X ) means that for any T > 0, u|(0,T ) ∈ Lp ( (0, T );X ) . 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 a1, . . . , an, write as C(a1, . . . , an) to indicate that the constant C depends only on a1, . . . , an and that dependence is continuous (we will use this convention for constants which are not denoted by “C”). This article 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. EJDE-2020/39 EXTINCTION FOR NLS IN THE WHOLE SPACE 5 2. Existence and uniqueness of the solutions Let 0 < m < 1, let a ∈ C, let f ∈ L1 loc ( [0,∞);L2(RN ) ) and let u0 ∈ L2(RN ). We consider the following nonlinear Schrödinger equation. i ∂u ∂t + ∆u+ a|u|−(1−m)u = f(t, x), in (0,∞)× RN , (2.1) u(0) = u0, in RN , (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 2 √ m Im(a) > (1−m)|Re(a)|. (2.3) If Re(a) > 0 then we assume further that 2 √ m Im(a) > (1−m) Re(a). (2.4) Here and after, we shall always identify L2(RN ) with its topological dual. Let 0 < m < 1 and let X = H ∩ Lm+1(RN ), where H = L2(RN ) or H = H1(RN ). We recall that (see, for instance, Bégout and Dı́az [7, Lemmas A.2 and A.4]): X? = H? + L m+1 m (RN ), (2.5) D(RN ) ↪→ X ↪→ Lm+1(RN ) with both dense embeddings, (2.6) L m+1 m (RN ) ↪→ X? ↪→ D ′(RN ), with both dense embeddings, (2.7) Lm+1 loc ( [0,∞);X ) ∩W 1,m+1 m loc ( [0,∞);X? ) ↪→ C ( [0,∞);L2(RN ) ) . (2.8) This justifies the notion of solution below (and especially (4)). Definition 2.2. Let 0 < m < 1, let a ∈ C, let f ∈ L1 loc ( [0,∞);L2(RN ) ) and let u0 ∈ L2(RN ). Let us consider the following assertions. (1) u ∈ Lm+1 loc ( [0,∞);H1(RN )∩Lm+1(RN ) ) ∩W 1,m+1 m loc ( [0,∞);H?+L m+1 m (RN ) ) , (2) For almost every t > 0, ∆u(t) ∈ H?. (3) u satisfies (2.1) in D ′ ( (0,∞)× RN ) . (4) u(0) = u0. We shall say that u is a strong solution if u is an H2-solution or an H1-solution. We shall say that u is an H2-solution of (2.1)–(2.2) ( respectively, an H1-solution of (2.1)–(2.2) ) , if u satisfies the Assertions (1)–(4) with H = L2(RN ) ( respectively, with H = H1(RN ) ) . We shall say that u is an L2-solution or a weak solution of (2.1)–(2.2) is there exists a pair, (fn, un)n∈N ⊂ L1 loc ( [0,∞);L2(RN ) ) × C ( [0,∞);L2(RN ) ) , (2.9) such that for any n ∈ N, un is an H2-solution of (2.1) where the right-hand side of (2.1) is fn, and if fn L1((0,T );L2(RN ))−−−−−−−−−−−→ n→∞ f and un C([0,T ];L2(RN ))−−−−−−−−−−→ n→∞ u, (2.10) for any T > 0, and if u satisfies (2.2). 6 P. BÉGOUT EJDE-2020/39 Remark 2.3. Let 0 < m < 1. Set for any z ∈ C, g(z) = |z|−(1−m)z (g(0) = 0). We define the mapping for any measurable function u : RN → C, which we still denote by g, by g(u)(x) = g(u(x)). Let X be as in the beginning of this section (see (2.5)–(2.8)). From (2.6), (2.7) and the basic estimate, ∀(z1, z2) ∈ C2, |g(z1)− g(z2)| 6 C|z1 − z2|m, (2.11) (see, for instance, Bégout and Dı́az [7, Lemma A.1]), we deduce easily that g ∈ C ( Lm+1(RN );L m+1 m (RN ) ) and g is bounded on bounded sets, (2.12) g ∈ C(X;X?) and g is bounded on bounded sets. (2.13) By (2.6)–(2.7) and (2.12)–(2.13), it follows that 〈g(u), v〉X?,X = 〈g(u), v〉 L m+1 m (RN ),Lm+1(RN ) = Re ∫ RN g(u)vdx, (2.14) for any u, v ∈ X. Now, let us collect some basic information about the solutions. (1) Any strong or weak solution belongs to C ( [0,∞);L2(RN ) ) and Asser- tion (4)) makes sense in L2(RN ) (by (2.8)). (2) It is obvious that an H2-solution is also an H1-solution and a weak solu- tion. But it is not clear that an H1-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 H1-solutions. (3) Any H2-solution (respectively, any H1-solution) satisfies (2.1) in L2(RN )+ L m+1 m (RN ) ( respectively, in H−1(RN )+L m+1 m (RN ) ) , 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 u ∈ W 1,1 loc ( [0,∞);Y ? ) and it solves (2.1) in Y ?, for almost every t > 0, where Y = H2(RN ) ∩ L 2 2−m (RN ) and Y ? = H−2(RN ) + L 2 m (RN ) ↪→ D ′(RN ) (by Bégout and Dı́az [7, Lemma A.2]). Indeed, using the notation of Definition 2.2 and (2.11), this comes from (2.10) and the uniform convergences, ∆un C([0,T ];H−2(RN ))−−−−−−−−−−−−→ n→∞ ∆u, (2.15) g(un) C([0,T ];L 2 m (RN ))−−−−−−−−−−−→ n→∞ g(u), (2.16) for any T > 0. In particular, u solves (2.1) in D ′ ( (0,∞)× RN ) . Theorem 2.4 (Existence and uniqueness of L2-solutions). Let Assumption 2.1 be fulfilled and let f ∈ L1 loc ( [0,∞);L2(RN ) ) . Then for any u0 ∈ L2(RN ), there exists a unique weak solution u to (2.1)–(2.2). In addition, u ∈ Lm+1 loc ( [0,∞);Lm+1(RN ) ) , (2.17) 1 2 ‖u(t)‖2L2(RN ) + Im(a) ∫ t s ‖u(σ)‖m+1 Lm+1(RN ) dσ 6 1 2 ‖u(s)‖2L2(RN ) + Im ∫ t s ∫ RN f(σ, x)u(σ, x) dxdσ, (2.18) EJDE-2020/39 EXTINCTION FOR NLS IN THE WHOLE SPACE 7 for any t > s > 0. Finally, if v is a weak solution of (2.1) with v(0) = v0 ∈ L2(RN ) and g ∈ L1 loc([0,∞);L2(RN )) instead of f in (2.1), then ‖u(t)− v(t)‖L2(RN ) 6 ‖u(s)− v(s)‖L2(RN ) + ∫ t s ‖f(σ)− g(σ)‖L2(RN )dσ, (2.19) for any t > s > 0. Remark 2.5. Let Assumption 2.1 be fulfilled. It follows from (2.18) and Hölder’s and Young’s inequalities that if f ∈ L1 ( (0,∞);L2(RN ) ) , then u ∈ L∞ ( (0,∞);L2(RN ) ) ∩ Lm+1 ( (0,∞);Lm+1(RN ) ) . By interpolation, we infer that for any p ∈ [m+ 1, 2), u ∈ Cb ( [0,∞);L2(RN ) ) ∩ L p(1−m) 2−p ( (0,∞);Lp(RN ) ) . (2.20) If, in addition, (ϕn)n∈N ⊂ L2(RN ), (fn)n∈N ⊂ L1 ( (0,∞);L2(RN ) ) , and ϕn L2(RN )−−−−−→ n→∞ u0, fn L1((0,∞);L2(RN ))−−−−−−−−−−−−→ n→∞ f . Then by (2.19), (2.20) and again by interpolation, for any p ∈ (m+ 1, 2), we have un Cb([0,∞);L2(RN ))∩L p(1−m) 2−p ((0,∞);Lp(RN ))−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ n→∞ u, where for each n ∈ N, un is the weak solution of (2.1) with un(0) = ϕn and fn instead of f . Theorem 2.6 (Existence and uniqueness of H1-solutions). Let Assumption 2.1 be fulfilled and let f ∈ W 1,1 loc ( [0,∞);H1(RN ) ) . Then for any u0 ∈ H1(RN ), there exists a unique H1-solution u to (2.1)–(2.2). Furthermore, u is also a weak solution and satisfies the following properties. (1) u ∈ C ( [0,∞);L2(RN ) ) ∩ C1 ( [0,∞);Y ? ) and u satisfies (2.1) in Y ?, for any t > 0, where Y ? = H−2(RN ) + L 2 m (RN ). (2) u ∈ Cw ( [0,∞);H1(RN ) ) ∩W 1,∞ loc ( [0,∞);H−1(RN ) + L 2 m (RN ) ) , and ‖∇u(t)‖L2(RN ) 6 ‖∇u0‖L2(RN ) + ∫ t 0 ‖∇f(s)‖L2(RN )ds, (2.21) for any t > 0. (3) The map t 7−→ ‖u(t)‖2L2(RN ) belongs to W 1,1 loc ( [0,∞);R ) and we have 1 2 d dt ‖u(t)‖2L2(RN ) + Im(a)‖u(t)‖m+1 Lm+1(RN ) = Im ∫ RN f(t, x)u(t, x) dx, (2.22) for almost every t > 0. Theorem 2.7 (Existence and uniqueness of H2-solutions). Let Assumption 2.1 be fulfilled and let f ∈W 1,1 loc ( [0,∞);L2(RN ) ) . Then for any u0 ∈ H2(RN )∩L2m(RN ), there exists a unique H2-solution u to (2.1)–(2.2). Furthermore, u satisfies (2.1) in L2(RN ), for almost every t > 0, and the following properties. (1) u ∈ C ( [0,∞);H1(RN ) ∩ Lm+1(RN ) ) ∩ C1 ( [0,∞);H−1(RN ) + L m+1 m (RN ) ) and u satisfies (2.1) in H−1(RN ) + L m+1 m (RN ), for any t > 0. 8 P. BÉGOUT EJDE-2020/39 (2) u ∈W 1,∞ loc ( [0,∞);L2(RN ) ) ∩ L∞loc ( [0,∞);H2(RN ) ∩ L2m(RN ) ) , and ‖u(t)− u(s)‖L2(RN ) 6 ‖ut‖L∞((s,t);L2(RN ))|t− s|, (2.23) ‖∇u(t)−∇u(s)‖L2(RN ) 6M |t− s| 1 2 , (2.24) ‖ut‖L∞((0,t);L2(RN )) 6 ‖∆u0 + a|u0|m−1u0 − f(0)‖L2(RN ) + ∫ t 0 ‖f ′(σ)‖L2(RN )dσ, (2.25) for any t > s > 0, where M2 = 2‖ut‖L∞((s,t);L2(RN ))‖∆u‖L∞((s,t);L2(RN )). (3) The map t 7−→ ‖u(t)‖2L2(RN ) belongs to C1 ( [0,∞);R ) and (2.22) holds for any t > 0. (4) If f ∈W 1,1 ( (0,∞);L2(RN ) ) , then u ∈Cb ( [0,∞);H1(RN ) ) ∩ L∞ ( (0,∞);H2(RN ) ∩ L2m(RN ) ) ∩W 1,∞((0,∞);L2(RN ) ) . Remark 2.8. Since f ∈ W 1,1 loc ( [0,∞);L2(RN ) ) ↪→ C ( [0,∞);L2(RN ) ) (see, for instance, in Bégout and Dı́az [7, 1) of Lemma A.4]), estimate (2.25) with f(0) makes sense. Remark 2.9. We recall that if u ∈ L2(RN ) with ∆u ∈ L2(RN ) then u ∈ H2(RN ). Furthermore, if ‖u‖2H2,2(RN ) = ‖u‖2L2(RN ) + ‖∆u‖2L2(RN ), then ‖ · ‖H2,2(RN ) and ‖ ·‖H2(RN ) are equivalent norms. Indeed, this is so because of the Fourier transform and Plancherel’s formula. Finally, note that ‖∇u‖2L2(RN ) 6 ‖u‖L2(RN )‖∆u‖L2(RN ) 6 ‖u‖2L2(RN ) + ‖∆u‖2L2(RN ), (2.26) for any u ∈ H2(RN ). 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 RN with an unbounded domain Ω 6= RN . This will be the subject of a future work. 3. Finite time extinction and asymptotic behavior Following the method by Carles and Gallo [9] (also used by Carles and Ozawa [10]) and Bégout and Dı́az [7], we are able to prove the finite time extinction and as- ymptotic behavior results. Theorem 3.1. Let Assumption 2.1 be fulfilled with N ∈ {1, 2, 3}, also let f ∈ W 1,1 ( (0,∞);L2(RN ) ) , let u0 ∈ H1(RN ), and assume that one of the following hypotheses holds. (1) N = 1 and f ∈W 1,1 ( (0,∞);H1(R) ) . (2) N ∈ {1, 2, 3} and u0 ∈ H2(RN ) ∩ L2m(RN ). Let u be the unique strong solution of (2.1)–(2.2). Finally, assume that there exists T0 > 0 such that for almost every t > T0 we have f(t) = 0. Let ` be the exponent in u0 ∈ H`(RN ). We have the following results. (a) There exists a finite time T? > T0 such that ∀t > T?, ‖u(t)‖L2(RN ) = 0. (3.1) Furthermore, T? 6 C‖u‖ N(1−m) 2` L∞((0,∞);H`(RN )) ‖u(T0)‖ (1−m)(2`−N) 2` L2(RN ) + T0, (3.2) EJDE-2020/39 EXTINCTION FOR NLS IN THE WHOLE SPACE 9 where C = C(Im(a), N,m, `). (b) There exists ε? = ε?(|a|, N,m) satisfying the following property. Let δ = (2`+N)+m(2`−N) 4` ∈ ( 1 2 , 1). If f ∈W 1,1 ( (0,∞);H1(RN ) ) ,( ‖u0‖H1(RN ) + ‖f‖L1((0,∞);H1(RN )) )1−m 6 ε? min { 1, T0 } , if N = 1,( ‖u0‖mH2(RN ) + ‖f‖mW 1,1((0,∞);H1(RN )) )1−m 6 ε? min { 1, T0 } , if N ∈ {2, 3}, and if for almost every t > 0, ‖f(t)‖2L2(RN ) 6 ε? ( T0 − t ) 2δ−1 1−δ + , (3.3) then (3.1) holds with T? = T0. Remark 3.2. If (N, `) ∈ {(1, 1), (2, 2)} then 2δ−1 1−δ = 2 1+m 1−m , if (N, `) = (1, 2) then 2δ−1 1−δ = 2 1+3m 3(1−m) and if (N, `) = (3, 2) then 2δ−1 1−δ = 2 3+m 1−m . Note that if N = 1 and u0 ∈ H2(RN ) then there are two possible choices for 2δ−1 1−δ in (3.3): 2 1+m 1−m or 2 1+3m 3(1−m) . Since for t near T0, T0 − t < 1 then the choice the less restrictive is that for which 2δ−1 1−δ is the smallest as possible, that is 2 1+3m 3(1−m) . 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 ( u0 ∈ H1(R) ∩F (H1(R)) ) , if N = 1 and ( u0 ∈ H2(R2) ∩F (H2(R2)), ‖u0‖L2(R2) small enough and 1 2 6 m < 1 ) , if N = 2. In fact F (H1(R)) ↪→ L2m(R) and F (H2(R2)) ↪→ L2m(R2), for any 1/3 < m 6 1. Additional nonlinearities are also considered in [10]. Theorem 3.4. Let Assumption 2.1 be fulfilled with N > 4. Let f ∈ W 1,1 loc ( [0,∞);L2(RN ) ) and let u0 ∈ H1(RN ). Suppose further that f ∈ W 1,1 loc ( [0,∞);H1(RN ) ) or u0 ∈ H2(RN ). Let u be the unique strong solution of (2.1)–(2.2). Finally, assume that there exists T0 > 0 such that for almost every t > T0 we have f(t) = 0. Then for any t > T0, we have ‖u(t)‖L2(RN ) 6 ‖u(T0)‖L2(RN )e −C(t−T0), if N = 4 and u0 ∈ H2(RN ), ‖u(t)‖L2(RN ) 6 ‖u(T0)‖L2(RN )( 1 + C‖u(T0)‖ (1−m)(N−2`) 2` L2(RN ) (t− T0) ) 2` (1−m)(N−2`) , if N > 5 or u0 ∈ H1(RN ), where C = C(‖u‖L∞((0,∞);H`(RN )), Im(a), N,m, `). Theorem 3.5. Let Assumption 2.1 be fulfilled, let f ∈ L1 loc ( [0,∞);L2(RN ) ) , let u0 ∈ L2(RN ) and let u be the unique weak solution of (2.1)–(2.2). If f ∈ L1 ( (0,∞);L2(RN ) ) , then limt↗∞ ‖u(t)‖L2(RN ) = 0. 4. Proofs of the existence and uniqueness theorems Since we have to prove existence in the whole space, the method is radically different than the one used in Bégout and Dı́az [7]. 10 P. BÉGOUT EJDE-2020/39 Theorem 4.1. Let Assumption 2.1 be fulfilled and let λ, b0 > 0. Then for any F ∈ L2(RN ), there exists a unique solution u to u ∈ H2(RN ) ∩ L2m(RN ), −λ∆u− aλ|u|−(1−m)u− ib0u = F, in L2(RN ). (4.1) In addition, ‖u‖2H2(RN ) + ‖u‖m+1 Lm+1(RN ) + ‖u‖2mL2m(RN ) 6M‖F‖ 2 L2(RN ), (4.2) where M = M(|a|,Arg(a), b0, λ). Furthermore, if F is compactly supported then so is u. Finally, let G ∈ L2(RN ). If v is a solution to (4.1) with G instead of F then, ‖u− v‖L2(RN ) 6 1 b0 ‖F −G‖L2(RN ). (4.3) 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. Lemma 4.2. Let Assumption 2.1 be fulfilled. Then there exists b ∈ C, with |b| = 1, satisfying the following properties Re(b) > 0, Im(b) < 0, (4.4) 2 √ m Im(ab) > (1−m) Re(ab) > 0. (4.5) In addition, b = b(Arg(a)). In particular, ab satisfies (2.3) and (2.4) of Assumption 2.1. Proof. Let θa = Arg(a) ∈ (0, π), since Im(a) > 0. We look for b = e−iθb , where 0 < θb < π 2 . Case 1: Re(a) < 0. If follows that, π/2 < θa < π. We choose θb = θa − π 2 . We then have ab = i|a| and the conclusion is clear. Case 2: Re(a) > 0. If follows that, 0 < θa 6 π.2 and by (2.4), one has 2 √ m sin(θa) > (1−m) cos(θa) > 0. (4.6) By continuity and (4.6), there exists θb ∈ (0, θa) such that 2 √ m sin(θa − θb) > (1−m) cos(θa − θb) > 0. (4.7) Then, 0 < θa − θb < π 2 , ab = |a|ei(θa−θb) and again the conclusion is clear. We summarize the proof with the picture 1. � Lemma 4.3. Let 0 < m < 1. Set for any z ∈ C, g(z) = |z|−(1−m)z (g(0) = 0). We define the mapping for any measurable function u : RN → C, which we still denote by g, by g(u)(x) = g(u(x)). Then for any p ∈ [1,∞), g ∈ C ( Lp(RN );L p m (RN ) ) and g is bounded on bounded sets. (4.8) Let a ∈ C with Im(a) > 0 satisfying (2.3). Then ( g(u) − g(v) ) (u− v) ∈ L1(RN ), and Re ( − i a ∫ RN ( g(u)− g(v) ) (u− v)dx ) > 0, (4.9) for any u, v ∈ Lm+1(RN ). EJDE-2020/39 EXTINCTION FOR NLS IN THE WHOLE SPACE 11 . .0 1 i a = |a|eiθa b = e−iθb ab + −θb ←− −θb Re(z) Im(z) Im(z)= 1−m 2 √ m |Re(z)| θb = θa − π 2 Case 1: Re(a) < 0 . . . 0 1 i a b = e−iθb ab + −θb ←− −θb Re(z) Im(z) Im(z)= 1−m 2 √ m |Re(z)| 0 < θb � 1 Case 2: Re(a) > 0 Figure 1. Summary of the proof of Lemma 4.2 Proof. Property (4.8) is an obvious consequence of (2.11) which implies the inte- grability property in the lemma. By Liskevich and Perel’muter [17, Lemma 2.2], we have 2 √ m ∣∣∣ Im((g(z1)− g(z2) )( z1 − z2 ))∣∣∣ 6 (1−m) Re (( g(z1)− g(z2) )( z1 − z2 )) , (4.10) for any (z1, z2) ∈ C2. Let u, v ∈ Lm+1(RN ). By (4.10), we have Re ( − i a ∫ RN ( g(u)− g(v) ) (u− v)dx ) = Im(a) Re ∫ RN ( g(u)− g(v) )( u− v ) dx+ Re(a) Im ∫ RN ( g(u)− g(v) )( u− v ) dx > ( Im(a)− |Re(a)|1−m 2 √ m ) Re ∫ RN ( g(u)− g(v) )( u− v ) dx > 0. The proof is complete. � Lemma 4.4 ([7]). Let 0 < m < 1 and let a ∈ C with Im(a) > 0 satisfying (2.3). Let g be as in Lemma 4.3. Then g(u)∆u ∈ L1(RN ) and Re ( ia ∫ RN g(u)∆udx ) > 0, (4.11) for any u, v ∈ H2(RN ) ∩ L2m(RN ). For a proof of the above lemma, see Bégout and Dı́az [7, Lemma 6.3]. Proof of Theorem 4.1. Let Assumption 2.1 be fulfilled, λ, b0 > 0 and F ∈ L2(RN ). Let g be as in Lemma 4.3. We want to solve − λ∆u− aλg(u)− ib0u = F, in H−1(RN ) + L m+1 m (RN ). (4.12) We proceed with this proof in five steps. Step 1: A first estimate. Let G ∈ L2(RN ). If u, v ∈ H2 loc(RN ) ∩ H1(RN ) ∩ Lm+1(RN ) are solutions of (uF ) and (vG), respectively, then estimate (4.3) holds. We multiply by iϕ, for ϕ ∈ D(RN ), the equation satisfied by u− v, we integrate by 12 P. BÉGOUT EJDE-2020/39 parts and we take the real part. By density of D(RN ) in H1(RN )∩Lm+1(RN ) and (4.8), ( g(u)− g(v) ) (u− v) ∈ L1(RN ) and we may choose ϕ = u− v. It follows that λRe ( − ia ∫ RN ( g(u)− g(v) ) (u− v)dx ) + b0‖u− v‖2L2(RN ) = − Im (∫ RN (F −G)(u− v)dx ) . (4.13) Estimate (4.3) then comes from (4.13), (4.9) and Cauchy-Schwarz’s inequality. Step 2: A second estimate. If u is a solution to (4.1) then u ∈ Lm+1(RN ) and satisfies (4.2). Since 2m < m+ 1 < 2, then L2m(RN ) ∩ L2(RN ) ⊂ Lm+1(RN ). By Bégout and Dı́az [6, Theorem 2.9], ‖u‖2H1(RN ) + ‖u‖m+1 Lm+1(RN ) 6M(|a|, b0, λ)‖F‖2L2(RN ). (4.14) Let b ∈ C be given by Lemma 4.2. We multiply the equation in (4.1) by −ib∆u, integrate by parts and take the real part. We obtain − λ Im(b)‖∆u‖2L2(RN ) + λRe ( iab ∫ RN g(u)∆udx ) + b0 Re(b)‖∇u‖2L2(RN ) = Im ( b ∫ RN F∆udx ) . (4.15) By (4.5), we apply Lemma 4.4. Using (4.4), (4.11) and applying Cauchy-Schwarz’s inequality in (4.15), one obtains ‖∆u‖L2(RN ) 6 |b| λ| Im(b)| ‖F‖L2(RN ). (4.16) Now, since by Plancherel’s formula, ‖u‖Ḣ2(RN ) 6 C‖|ξ|2û‖L2(RN ) 6 C‖∆u‖L2(RN ), putting together (4.14) and (4.16), one obtains (4.2). Step 3: Compactness of the solution. If suppF is compact and if u ∈ H1(RN ) ∩ Lm+1(RN ) is a solution to (4.12) then suppu is compact. This follows from Bégout and Dı́az [4, Theorem 3.6]. Step 4: Existence and uniqueness. There is a unique solution u ∈ H2 loc(RN )∩ H1(RN ) ∩ Lm+1(RN ) to (4.12). By Bégout and Dı́az [6, Theorem 2.8], equation (4.12) admits a solution u ∈ H1(RN )∩Lm+1(RN ). By Bégout and Dı́az [4, Propo- sition 4.5], u ∈ H2 loc(RN ). Finally, by Step 1 this solution is unique. Step 5: Conclusion. Estimates (4.2)–(4.3), uniqueness and compactness property come from Steps 1–3, once the existence of a solution to (4.1) is proved. Let u ∈ H2 loc(RN )∩H1(RN )∩Lm+1(RN ) the solution of (4.12) be given by Step 4. Let (Fn)n∈N ⊂ D(RN ) be such that Fn L2(RN )−−−−−→ n→∞ F . Finally, for each n ∈ N, denote by un the unique solution to (4.1), where the right-hand side is Fn instead of F (Steps 3 and 4). By Steps 1 and 2, (un)n∈N is bounded in H2(RN ) and un L2(RN )−−−−−→ n→∞ u. It follows that u ∈ H2(RN ) and, from the equation in (4.1), g(u) ∈ L2(RN ). Hence u is a solution to (4.1). This concludes the proof. � Corollary 4.5. Let Assumption 2.1 be fulfilled. Let us define the (nonlinear) op- erator on L2(RN ). D(A) = H2(RN ) ∩ L2m(RN ), EJDE-2020/39 EXTINCTION FOR NLS IN THE WHOLE SPACE 13 ∀u ∈ D(A), Au = −i∆u− ia|u|−(1−m)u, Then A is maximal monotone on L2(RN ) (and so m-accretive) with dense domain. Proof. The density is obvious. For any λ > 0, I + λA is bijective from D(A) onto L2(RN ) and (I+λA)−1 is a contraction (Theorem 4.1). It follows that A is maximal monotone (Brezis [8, Proposition 2.2, p.23]). � Proof of Theorem 2.7. Let g be as in Lemma 4.3. We first recall that by Re- mark 2.8, f ∈ C ( [0,∞);L2(RN ) ) . (4.17) By Corollary 4.5 and Barbu [2, Theorem 2.2, p.131], there exists a unique u ∈ W 1,∞ loc ( [0,∞);L2(RN ) ) satisfying u(t) ∈ H2(RN ) ∩ L2m(RN ) and (2.1) in L2(RN ), for almost every t > 0, u(0) = u0 and (2.25). This last estimate yields (2.23). Since u ∈W 1,∞ loc ( [0,∞);L2(RN ) ) , it follows from Bégout and Dı́az [7, Lemma A.5] that the map M : t 7→ 1 2‖u(t)‖2L2(RN ) belongs to W 1,∞ loc ( [0,∞);R ) and M ′(t) =( u(t), ut(t) ) L2(RN ) , for almost every t > 0. Multiplying (2.1) by iu, integrating by parts over RN and taking the real part, we obtain (2.22), for almost every t > 0. We deduce easily from (2.22), (4.17) and Hölder’s inequality that u ∈ L∞loc ( [0,∞);Lm+1(RN ) ) . Multiplying again (2.1) by u, integrating by parts and taking the real part, we obtain ‖∇u(t)‖2L2(RN ) 6 |Re(a)|‖u(t)‖m+1 Lm+1(RN ) + ( ‖ut(t)‖L2(RN ) + ‖f(t)‖L2(RN ) ) ‖u(t)‖L2(RN ), for almost every t > 0. It follows that u ∈ L∞loc ( [0,∞);H1(RN ) ) . We infer that u is an H2-solution. Let b ∈ C be given by Lemma 4.2. We multiply (2.1) by iabg(u), integrate and take the real part. We obtain Re ( ab ∫ RN utg(u)dx ) + Re ( iab ∫ RN g(u)∆udx ) + |a|2 Re(ib)‖g(u)‖2L2(RN ) = Re ( iab ∫ RN fg(u)dx ) . (4.18) By Lemma 4.2, we have (4.11). This implies Re ( iab ∫ RN g(u)∆udx ) = Re ( iab ∫ RN g(u)∆udx ) > 0, (4.19) and (4.18) becomes |a|| Im(b)| ‖u‖2mL2m(RN ) 6 ∫ RN |(ut + if)g(u)|dx, (4.20) since Re(ib) = − Im(b) > 0, by (4.4). By Cauchy-Schwarz’s and Young’s inequali- ties, we obtain∫ RN |(ut+if)g(u)|dx 6 1 2|a|| Im(b)| ‖ut+if‖2L2(RN )+ |a|| Im(b)| 2 ‖u‖2mL2m(RN ). (4.21) Putting together (4.20) and (4.21), we arrive at ‖u(t)‖2mL2m(RN ) 6 1 |a|2| Im(b)|2 ( ‖ut(t)‖L2(RN ) + ‖f(t)‖L2(RN ) )2 , (4.22) 14 P. BÉGOUT EJDE-2020/39 for almost every t > 0. Multiplying again (2.1) by ib∆u, using (4.19) and proceeding as above, we arrive at ‖∆u(t)‖L2(RN ) 6 1 | Im(b)| ( ‖ut(t)‖L2(RN ) + ‖f(t)‖L2(RN ) ) , (4.23) for almost every t > 0. By (4.17), (4.22), (4.23), Remark 2.9 and Hölder’s inequality (recalling that 2m < m+ 1 < 2), we obtain u ∈ L∞loc ( [0,∞);H2(RN ) ) ∩ L∞loc ( [0,∞);L2m(RN ) ) , (4.24) u ∈ C ( [0,∞);L2(RN ) ) ∩ L∞loc ( [0,∞);L2m(RN ) ) ↪→ C ( [0,∞);Lm+1(RN ) ) . (4.25) Recalling that u ∈ W 1,∞ loc ( [0,∞);L2(RN ) ) , by (4.24) and the embedding 3) of Lemma A.4, we have u ∈ C ( [0,∞);H1(RN ) ) . We then deduce Property (1), with help of (2.13), (4.17) and (2.1). With (2.26), (2.23) and (4.24), we obtain (2.24) and Property (2) is proved. Property (3) comes from (2.22), (4.17) and (4.25). Finally, Property (4) follows easily from Remarks 2.5, 2.8 and 2.9, (2.25), (4.22) and (4.23). This concludes the proof of the theorem. � Lemma 4.6. Let Assumption 2.1 be fulfilled and f, g ∈ L1 loc ( [0,∞);L2(RN ) ) . If u and v are strong solutions or weak solutions of iut + ∆u+ a|u|−(1−m)u = f1, ivt + ∆v + a|v|−(1−m)v = f2, respectively, then u, v ∈ C ( [0,∞);L2(Ω) ) and ‖u(t)− v(t)‖L2(Ω) 6 ‖u(s)− v(s)‖L2(Ω) + ∫ t s ‖f1(σ)− f2(σ)‖L2(Ω)dσ, (4.26) for any t > s > 0. Proof. Let X = H1(RN ) ∩ Lm+1(RN ) and let u, v be as in the lemma. Continuity comes from (2.8) and Definition 2.2. Estimate (4.26) being stable by passing to the limit in C ( [0, T ];L2(RN ) ) × L1 ( (0, T );L2(RN ) ) , for any T > 0, it is sufficient to establish it for the H2-solutions. And since an H2-solution is an H1 solution, we may assume that u, v are H1 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 Bégout and Dı́az [7, (A.3) of Lemma A.5], (2.14), (4.9) and Cauchy-Schwarz’s inequality, we then arrive at 1 2 d dt ‖u(·)− v(·)‖2L2(Ω) 6 ‖f1 − f2‖L2(Ω)‖u− v‖L2(Ω), almost everywhere on (0,∞). Integrating over (s, t), one obtains (4.26). � Proof of Theorem 2.4. Existence, estimate (2.19) and uniqueness comes from den- sity of H2(RN ) ×W 1,1 loc ([0,∞);L2(RN )) in L2(RN ) × L1 loc([0,∞);L2(RN )), Theo- rem 2.7, Lemma 4.6 and completeness of C ( [0, T ];L2(RN ) ) , for any T > 0. Finally, estimates (2.17)–(2.18) are due to Bégout and Dı́az [7, Proposition 2.3]. This com- pletes the proof. � Proof of Theorem 2.6. Uniqueness comes from Lemma 4.6. Let f ∈ W 1,1 loc ([0,∞);H1(RN )) and let u0 ∈ H1(RN ). Let (ϕn)n∈N ⊂ D(RN ) be such that ϕn H1(RN )−−−−−→ n→∞ u0. Finally, let g be defined as in Lemma 4.3 and for each EJDE-2020/39 EXTINCTION FOR NLS IN THE WHOLE SPACE 15 n ∈ N, let un the unique H2-solution of (2.1) such that un(0) = ϕn, be given by Theorem 2.7. By Lemma 4.6, we have for any T > 0 and n, p ∈ N, ‖un‖C([0,T ];L2(RN )) 6 ‖ϕn‖L2(RN ) + ∫ T 0 ‖f(t)‖L2(RN )dt, ‖un − up‖L∞((0,∞);L2(RN )) 6 ‖ϕn − ϕp‖L2(RN ), (4.27) It follows that for any T > 0, (un)n∈N is a Cauchy sequence in C ( [0, T ];L2(RN ) ) . As a consequence, there exists u ∈ C ( [0,∞);L2(RN ) ) such that for any T > 0, un C([0,T ];L2(RN ))−−−−−−−−−−→ n→∞ u. (4.28) By definition, it follows from (4.28) that u is a weak solution of (2.1)–(2.2). By Theorem 2.7, we can take the L2-scalar product of (2.1) with −i∆un and it follows from Bégout and Dı́az [7, (A.4)] that for any n ∈ N and almost every s > 0, 1 2 d dt ‖∇un(s)‖2L2(RN ) + Re ( ia ∫ RN g(un(s))∆un(s)dx ) = ( ∇f(s), i∇un(s) ) L2(RN ) . which with (4.11) and Cauchy-Schwarz’s inequality gives 1 2 d dt ‖∇un(s)‖2L2(RN ) 6 ‖∇f(s)‖L2(RN )‖∇un(s)‖L2(RN ). By integration, for any t > 0 and any n ∈ N, we obtain ‖∇un(t)‖L2(RN ) 6 ‖∇ϕn‖L2(RN ) + ∫ t 0 ‖∇f(s)‖L2(RN )ds. (4.29) By the Sobolev embedding (see, for instance, Bégout and Dı́az [7, 1) of Lemma A.4 ]), W 1,1 loc ( [0,∞);L2(RN ) ) ↪→ C ( [0,∞);L2(RN ) ) , (4.30) (4.27), (4.29), (4.8) and (2.1), we infer that (un)n∈N is bounded in L∞ ( (0, T );H1(RN ) ) ∩W 1,∞((0, T );Z? ) , (4.31) for any T > 0, where Z? = H−1(RN ) + L 2 m (RN ) is the topological dual space of Z = H1(RN ) ∩ L 2 2−m (RN ). Note that Z? is reflexive (Bégout and Dı́az [7, Lemma A.2 ]) and since H1(RN ) ↪→ Z?, it follows from (4.28), (4.31), (2.15) and Cazenave [11, Proposition 1.1.2, p.2, and (ii) of Remark 1.3.13, p.12] that u ∈ Cw ( [0,∞);H1(RN ) ) ∩W 1,∞ loc ( [0,∞);Z? ) , (4.32) ∆u ∈ C ( [0,∞);H−2(RN ) ) , (4.33) un(t) ⇀ u(t), in H1 w(RN ), as n→∞, (4.34) for any t > 0. After integration of (2.22), we see with help of (4.27) that for any T > 0, (un)n∈N is bounded in Lm+1 ( (0, T );Lm+1(RN ) ) ∼= Lm+1 ( (0, T ) × RN ) , which is reflexive. With (4.28) We infer that u ∈ Lm+1 loc ( [0,∞);Lm+1(RN ) ) . (4.35) By (4) of Remark 2.3, (4.30), (4.32), (4.35) and (2.1), it follows that u satisfies (1) of Definition 2.2 and then u is an H1-solution. By (3) of Remark 2.3, we can take the X −X? duality product with iu, where X = H1(RN ) ∩ Lm+1(RN ). Applying Lemma A.5 of Bégout and Dı́az [7] and (2.14), Property (3) follows. Estimate (2.21) comes from (4.34), (4.29) and the weak lower semicontinuity of the norm. 16 P. BÉGOUT EJDE-2020/39 Finally, smoothness of the solution in Properties (1) and (2) follows easily from (4.30), (4.32), (4.33), (4.8) and the equation (2.1). This concludes the proof. � 5. Proofs of finite time extinction and asymptotic behavior theorems Proof of Theorem 3.1. Apply Theorems 2.6, 2.7 and use the general theorem of finite time extinction in [7, Theorem 2.1 and Remark 4.8]. Nevertheless, to make the proof more understandable, we briefly explain how to obtain (3.1)–(3.2). Let ` = 1, if u0 ∈ H1(RN ) and ` = 2, if u0 ∈ H2(RN ). Assume that for some T0 > 0, f(t) = 0, for almost every t > T0. It follows from Theorems 2.6, 2.7 and Remark 2.5 that u ∈ L∞ ( (0,∞);H`(RN ) ) . By Gagliardo-Nirenberg’s inequality and (2.22), we have ‖u(t)‖ (2`+N)+m(2`−N) 2` L2(RN ) 6 C‖u‖ N(1−m) 2` L∞((0,∞);H`(RN )) ‖u(t)‖m+1 Lm+1(RN ) , d dt ‖u(t)‖2L2(RN ) + 2 Im(a)‖u(t)‖m+1 Lm+1(RN ) = 0, for almost every t > T0. It follows that y′(t) + Cy(t)δ 6 0, (5.1) for almost every t > T0, where y(t) = ‖u(t)‖2L2(RN ) and δ = (2`+N)+m(2`−N) 4` . By our assumption on `, we have δ ∈ (0, 1) if N 6 3. Hence (3.1)–(3.2) by integration. � Proof of Theorem 3.4. Let ` = 1, if u0 ∈ H1(RN ) and ` = 2, if u0 ∈ H2(RN ). By Theorems 2.6, 2.7 and Remark 2.5, u ∈ L∞ ( (0,∞);H`(RN ) ) . Repeating the proof of Theorem 3.1, we obtain obtain (5.1). According to the different cases as in the theorem, we have δ = 1 or δ > 1. The results then follow by integration (see also (1.6) and the lines below). For more details, see [7, 3) of Remark 2.4]. � Proof of Theorem 3.5. By Remark 2.5, we may assume that f ∈ D ( [0,∞);L2(RN ) ) and u0 ∈ H2(RN ). Let [0, T0] ⊃ supp f . By (2.22), d dt‖u(t)‖2L2(RN ) 6 0, for any t > T0. It follows that limt↗∞ ‖u(t)‖L2(RN ) = `0, for some `0 ∈ [0,∞). Let q ∈ (2,∞) with (N − 2)q < 2N . By Hölder’s inequality and Sobolev’s embedding H1(RN ) ↪→ Lq(RN ), there exists θ ∈ (0, 1) such that `0 6 ‖u(t)‖L2(RN ) 6 ‖u(t)‖θLm+1(RN )‖u(t)‖1−θ Lq(RN ) 6 C‖u(t)‖θLm+1(RN )‖u‖ 1−θ L∞((0,∞);H1(RN )) , for any t > T0. Still by (2.22), we obtain d dt ‖u(t)‖2L2(RN ) 6 −C` m+1 θ 0 6 0, for any t > T0. Hence `0 = 0. � Acknowledgements. The author is grateful to Professor J. I. Dı́az for the useful discussions about this article. EJDE-2020/39 EXTINCTION FOR NLS IN THE WHOLE SPACE 17 References [1] S. Antontsev, J.-P. Dias, M. Figueira; Complex Ginzburg-Landau equation with absorption: existence, uniqueness and localization properties. J. Math. Fluid Mech., 16(2):211–223, 2014. [2] V. Barbu; Nonlinear semigroups and differential equations in Banach spaces. Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. [3] C. Bardos, H. Brezis; Sur une classe de problèmes d’évolution non linéaires. J. Differential Equations, 6:345–394, 1969. [4] P. Bégout, J. I. Dı́az; Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – The stationary case. Ann. Inst. H. Poincaré Anal. Non Linéaire, 29(1):35–58, 2012. [5] P. Bégout, J. I. Dı́az; A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity. Discrete Contin. Dyn. Syst., 34(9):3371–3382, 2014. [6] P. Bégout and J. I. Dı́az; Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity. Rev. R. Acad. Cienc. Exactas F́ıs. Nat. Ser. A Math. RACSAM, 109(1):43–63, 2015. [7] P. Bégout, J. I. Dı́az; Finite time extinction for the strongly damped nonlinear Schrödinger equation in bounded domains. J. Differential Equations, 268(7):4029–4058, 2020. [8] H. Brezis; Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam, 1973. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). [9] R. Carles, C. Gallo; Finite time extinction by nonlinear damping for the Schrödinger equation. Comm. Partial Differential Equations, 36(6):961–975, 2011. [10] R. Carles, T. Ozawa; Finite time extinction for nonlinear Schrödinger equation in 1D and 2D. Comm. Partial Differential Equations, 40(5):897–917, 2015. [11] T. Cazenave; Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 2003. [12] J. I. Dı́az, J. F. Padial, J. I. Tello, L. Tello; Complex Ginzburg-Landau equations with a delayed nonlocal perturbation. To appear in Electronic Journal of Differential Equations. [13] N. Hayashi, C. Li, P. I. Naumkin; Time decay for nonlinear dissipative Schrödinger equations in optical fields. Adv. Math. Phys., pages Art. ID 3702738, 7, 2016. [14] Y. Hou, J. Jiang, F. Li, B. You; Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with p-Laplacian. Discrete Contin. Dyn. Syst. Ser. B, 19(6):1801–1814, 2014. [15] N. Kita, A. Shimomura; Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data. J. Math. Soc. Japan, 61(1):39–64, 2009. [16] J.-L. Lions; Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod; Gauthier-Villars, Paris, 1969. [17] V. A. Liskevich, M. A. Perel’muter; Analyticity of sub-Markovian semigroups. Proc. Amer. Math. Soc., 123(4):1097–1104, 1995. [18] N. Okazawa, T. Yokota; Monotonicity method for the complex Ginzburg-Landau equation, including smoothing effect. Nonlinear Anal., 47(1):79–88, 2001. [19] N. Okazawa, T. Yokota; Global existence and smoothing effect for the complex Ginzburg- Landau equation with p-Laplacian. J. Differential Equations, 182(2):541–576, 2002. [20] N. Okazawa, T. Yokota; Monotonicity method applied to the complex Ginzburg-Landau and related equations. J. Math. Anal. Appl., 267(1):247–263, 2002. [21] A. Shimomura; Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities. Comm. Partial Differential Equations, 31(7-9):1407–1423, 2006. [22] C. Sulem, P.-L. Sulem; The nonlinear Schrödinger equation, volume 139 of Applied Mathe- matical Sciences. Springer-Verlag, New York, 1999. Self-focusing and wave collapse. [23] T. Tao; Nonlinear dispersive equations, volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. 18 P. BÉGOUT EJDE-2020/39 [24] M. Tsutsumi; On global solutions to the initial-boundary value problem for the damped nonlinear Schrödinger equations. J. Math. Anal. Appl., 145(2):328–341, 1990. Pascal Bégout Institut de Mathématiques de Toulouse, Université Toulouse I Capitole, 1, Esplanade de l’Université. 31080 Toulouse Cedex 6, France Email address: Pascal.Begout@math.cnrs.fr 1. Introduction and explanation of the method 2. Existence and uniqueness of the solutions 3. Finite time extinction and asymptotic behavior 4. Proofs of the existence and uniqueness theorems 5. Proofs of finite time extinction and asymptotic behavior theorems Acknowledgements References