REMARKS ON GLOBAL EXISTENCE AND BLOWUP FOR DAMPED NONLINEAR SCHRÖDINGER EQUATIONS

We consider the Cauchy problem for the damped nonlinear Schrödinger equations, and prove some blowup and global existence results which depend on the size of the damping coefficient. We also discuss the L concentration phenomenon of blowup solutions in the critical case.

1. Introduction. In this paper, we study global existence and blowup of solutions to the Cauchy problem for the damped nonlinear Schrödinger equation: with initial data u(0) = u 0 ∈ H 1 (R n ), where a ≥ 0, p > 1, and p < 1 + 4/(n − 2) if n ≥ 3. The equation (1.1) arises in various areas of nonlinear optics, plasma physics and fluid mechanics, and has been studied by many mathematicians and physicists (see, e.g., [4,10,25,26] and references therein). It is known that the Cauchy problem for (1.1) is locally well-posed in H 1 (R n ) (see Kato [12] and also Cazenave [2,Section 4.4]): For any u 0 ∈ H 1 (R n ), there exist T * a (u 0 ) ∈ (0, ∞] and a unique solution u(t) of (1.1) with u(0) = u 0 such that u ∈ C([0, T * a (u 0 )); H 1 (R n )). Moreover, T * a (u 0 ) is the maximal existence time of the solution u(t) in the sense that if T * a (u 0 ) < ∞ then lim t→T * a (u0) u(t) H 1 = ∞. First, we recall some known results for the case a = 0 (see [2,23] for more information). When p < 1 + 4/n, we have T * 0 (u 0 ) = ∞ for any u 0 ∈ H 1 (R n ). When p ≥ 1 + 4/n, we have T * 0 (u 0 ) = ∞ if the initial data u 0 is sufficiently small in H 1 (R n ), and T * 0 (u 0 ) < ∞ if u 0 ∈ Σ and E(u 0 ) < 0, where we put Σ = {v ∈ H 1 (R n ) : xv ∈ L 2 (R n )} and the energy E is defined by (1.2) (see [6,9,24,29] and also Remark 1.3 below). The global existence result follows from the local well-posedness in H 1 (R n ), the conservation laws of energy E(u) and charge u 2 L 2 , and the Gagliardo-Nirenberg inequality: The blowup result is based on the virial identity: where Next, we consider the case a > 0 in (1.1). In this case, we have but the energy E(u(t)) is no longer conserved nor decreasing. In fact, we have (see M. Tsutsumi [25]). It is proved in [25] that when p > 1 + 4/n, we have where we put The proof in [25] is based on the following two identities: where P is the functional defined by (1.4). For the case p < 1 + 4/n, we have T * a (u 0 ) = ∞ for all u 0 ∈ H 1 (R n ) and for all a ≥ 0. Indeed, for this case, we have the following blowup alternative: if [27], and also Sections 4.6 and 5.2 of [2]). This alternative and (1.5) imply that T * a (u 0 ) = ∞ for all u 0 ∈ H 1 (R n ) (for the long time behavior of global solutions for (1.1) with external force f (x), see, e.g., [5,11,14]). Therefore, we consider the case p ≥ 1 + 4/n only. Now, we state our main results.
Note that 1 < p 0 (n) < 1 + 4/n. The proof in [3] is based on the Strichartz estimates but not on the energy method. Since the energy E(u(t)) is not conserved nor decreasing when a > 0, the energy method is not useful for the proof of Theorem 1, and we will apply the argument in the proofs of Proposition 2.4 and Corollary 2.5 of [3]. The rest of the paper is organized as follows. In Section 2, we give the proof of Theorem 1 along the argument in Cazenave and Weissler [3]. In Section 3, we prove Theorem 2 by modifying the proof of M. Tsutsumi [25]. In Section 4, we construct some invariant sets under the flow of (1.1), which are independent of the damping coefficient a ≥ 0, and prove a global existence result for solutions with initial data in these invariant sets. In Section 5, assuming the existence of blowup solutions, we study the L 2 concentration phenomenon for blowup solutions of (1.1) for the case p = 1 + 4/n and a > 0.
2. Proof of Theorem 1. In this section, we prove Theorem 1. Let U a (t) be the propagator for the linear equation: The Cauchy problem for (1.1) with u(0) = u 0 ∈ H 1 (R n ) is equivalent to the integral equation (see [2,12]): The following result is a key moment in the proof of Theorem 1.
We begin with several space-time estimates, which will be used in the proof of Proposition 3. These estimates go back to Strichartz [22], and have been widely used to study nonlinear Schrödinger equations (see, e.g., [7,12,2,23]). Remark that when a > 0, U a (t) are not unitary on L 2 (R n ), and the expression (2.4) below plays an important role, especially in the proof of Lemma 6.
Lemma 4. Let 0 < T ≤ ∞ and 2 ≤ r ≤ ∞. For 0 < t < T and τ ∈ R, we have where C depends only on n and r, and r ′ denotes the Hölder conjugate exponent of r.
Proof. By the decay estimates for U 0 (t) and by (2.4), we have This completes the proof.
Lemma 5. Let 0 < T ≤ ∞ and 0 < t < T . For any admissible pair (q, r), we have where C depends only on n and r.

REMARKS ON EXISTENCE & BLOWUP FOR SCHRÖDINGER EQUATIONS 5
Proof. For 0 < t < T and τ ∈ R, it follows from Lemma 4 that Thus, the result follows from the Riesz potential inequalities.
where C depends only on n and r. Proof By the Hölder inequality and Lemma 5, we have for t ∈ (0, T ). This completes the proof.
Proof. For 0 < t < T , it follows from Lemma 4 that Thus, the result follows from the Riesz potential inequalities.
Now we are in a position to prove Proposition 3.
Proof of Theorem 2.

Invariant sets and global existence.
In this section, we construct some invariant sets under the flow of (1.1), which are independent of the damping coefficient a ≥ 0, and prove that the solutions of (1.1) are global if the initial data belong to the invariant sets. Let 1 < p < 1 + 4/(n − 2). For ω > 0, we put Note that since we have It is well-known that d(ω) > 0 and it is attained by a positive solution ϕ ∈ H 1 (R n ) of the stationary problem for (1.1) with a = 0: (see, e.g., [19]). Although the energy E(u(t)) is not monotone for the case a > 0 in general, we have the following global existence result.
Remark 4.2 For the case a = 0 and p > 1 + 4/n, it is known that the sets are invariant under the flow of (1.1) with a = 0, where ω > 0, S ω and P are defined by (4.1) and (1.4) respectively, and (see [1,2]). Note that and it is known that d 1 (ω) > 0 and it is attained by a positive solution of (4.4) (see [1,2]). Since the positive solution of (4.4) is unique, up to translations (see [13]), we see that d 1 (ω) = d(ω), where d(ω) is defined by (4.2). Moreover, it is also known that we have [1,2]). We do not know whether A ω can be replaced by B + ω in Theorem 8 for the case a > 0, nor whether the conditions (i), (ii) and (iii) can be replaced by u 0 ∈ B − ω in Theorem 2. 5. L 2 concentration for critical case. In this section, we study the L 2 concentration phenomenon for blowup solutions of (1.1) for the case p = 1+4/n and a > 0. For the case a = 0, this phenomenon has been studied by many authors (see, e.g., [15,16,17,28,30,31]). Because of Remark 1.2, we assume the existence of blowup solutions of (1.1) in the case p = 1 + 4/n and a > 0.
Theorem 10. Let p = 1 + 4/n, n ≥ 2 and a > 0. Assume that u 0 ∈ H 1 (R n ) is radially symmetric, and suppose that the solution u(t) of (1.1) with u(0) = u 0 blows up in finite time T ∈ (0, ∞). Then, for any function ρ(t) satisfying ρ(t) → ∞ as t → T , we have Remark 5.1 By (1.5) and the blowup alternative, we have lim t→T ∇u(t) L 2 = ∞ in Theorem 10. In particular, taking ρ(t) = R ∇u(t) L 2 in (5.2), for any R > 0 we have lim sup Remark 5.2 For the case a = 0, it is well-known that lim sup t→T can be replaced by lim inf t→T in (5.2) (see Y. Tsutsumi [28]). For related results on the Zakharov system, we refer to Glangetas and Merle [8, Theorem 1].

Remark 5.3
We do not consider the case where the initial data u 0 is not radially symmetric (see, e.g., [16,17,18,31] for the case a = 0). We first prove a simple lemma.
Lemma 11. Let T ∈ (0, ∞), and assume that a function F : [0, T ) → (0, ∞) is continuous, and lim t→T F (t) = ∞. Then, there exists a sequence {t k } such that t k → T and Proof. We follow the argument in the proof of Lemma 2.1 of [4]. We put If lim t→T G(t) < ∞, then (5.3) holds for any sequence {t k } satisfying t k → T . So, we assume that lim t→T G(t) = ∞. Then, since lim t→T log G(t) = ∞, we see that which shows that there exists a sequence {t k } such that t k → T and (5.3).
The following variational characterization of the ground state Q plays an important role in the proof of Theorem 10.
Lemma 12. Let p = 1 + 4/n, and Q be the ground state of (5.1). Then we have For the proof of Lemma 12, see Weinstein [29] and also [18].