On Global Existence and Blow-up for Damped Stochastic Nonlinear Schr\"odinger Equation

In this paper, we consider the well-posedness of the weakly damped stochastic nonlinear Schr\"odinger(NLS) equation driven by multiplicative noise. First, we show the global existence of the unique solution for the damped stochastic NLS equation in critical case. Meanwhile, the exponential integrability of the solution is proved, which implies the continuous dependence on the initial data. Then, we analyze the effect of the damped term and noise on the blow-up phenomenon. By modifying the associated energy, momentum and variance identity, we deduce a sharp blow-up condition for damped stochastic NLS equation in supercritical case. Moreover, we show that when the damped effect is large enough, the damped effect can prevent the blow-up of the solution with high probability.


Introduction
The nonlinear Schrödinger equation, as one of the basic models for nonlinear waves, has many physical applications to, e.g. nonlinear optics, plasma physics and quantum field theory and so on (see e.g. [3,5,10,12,17]).
In this paper, we consider the weakly damped stochastic NLS equation driven by a linear multiplicative noise in focusing mass-(super)critical range, In recent twenty years, much effort has been devoted to studying the wellposedness of stochastic NLS equation, see [1,4,10,12,14] and the references therein. In [10] and [12], the local and global existence of the mild solution of stochastic NLS equation in L 2 and in H 1 are investigated, respectively. For stochastic NLS equation on a manifold, [4] considers the existence and uniqueness of the solution based on the stochastic Strichartz estimate in UMD Banach space. [14] shows the local well-posedness in L 2 via stochastic Strichartz estimates and the global well-posedness in subcritical case. With the help of the rescaling transformation, [1] obtains that the local well-posedness in H 1 for σ < 2 (d−2) + with some decay conditions on the noise, and that the global existence when λ = −1, σ < 2 (d−2) + or λ = 1, σ < 2 d . The global existence of the solution in H 2 is presented in [8] for one dimensional stochastic NLS equation driven by linear multiplicative noises. For the global existence of the solution of the NLS equation in critical case, there exist much more results in the deterministic case than those in stochastic case. For instance, in the deterministic case, [19] finds a threshold R by the optimal constant of Gagliardo-Nirenberg's inequality, and proves the global existence of the solution when σ = 2 d and u 0 < R . After adding the noise and damped effect, we wonder whether these effect will influence the existence of a threshold in mass critical case for the damped stochastic NLS equation, which is one of the main interests in this paper.
The another main interest is to study the damped effect and the noise effect on the blow-up in the focusing mass-(super)critical case. It's well known that the solution of deterministic NLS equation with a = 0 in the focusing mass-(super)critical case will blow up in some finite time when u 0 possesses some negative Hamiltonian, see [3,5,17] and references therein. When a > 0, the damped term has the effect to delay the blow-up, see [16,17,18] and references therein. For instance, for σ > 2 d , the blow-up may occur for small values of a (see e.g. [18]) and large enough values of a can ensure the global existence of the solution for σ ≥ 2 d (see e.g. [16]). In stochastic case, the noise also has an impact on blow-up solutions. [13] shows that the noise effect can accelerate the formation of singularity, and that the solution of Eq. (1) with a = 0 in focusing supercritical case will blow up in a finite time with a positive probability when the variance of the initial datum is finite. The blow-up solution for the stochastic NLS equation driven by additive noises is considered in [11]. When the noise of stochastic NLS equation is non-conservative, [2] shows that adding a large multiplicative Gaussian noise can prevent the blow-up in any finite time with high probability.
Throughout this paper, we assume that the local-wellposedness of the solution of Eq. (1) holds. The local solution u(·) is defined on a random interval [0, τ * (u 0 , ω)), where τ * (u 0 , ω) is a stopping time such that τ * (u 0 , ω) = +∞, or lim First, the evolution of charge and energy of the local solution are introduced. By using the optimal constant of Gagliardo-Nirenberg's inequality, we show the a priori estimation in H 1 -norm, and prove that the threshold R is unchanged when a ≥ 0, σ = d 2 and initial datum is deterministic. Moreover, based on the proved exponential integrability of the solution u, i.e., with α depending on u 0 , a and Q, we obtain the strong continuous dependence on the initial data in one dimensional case, which is not a trivial property for stochastic partial differential equation with non-global coefficients, see [6,8] and references therein. We would like to mention that this exponential integrability is useful for studying the continuous dependence on noises, exponential tail estimate of the solution, strong and weak convergence rates of numerical approximations, see [6,7,8,9,15] and references therein. Next we consider the influence of damped term and noise on the blow-up. For the damped stochastic NLS equation, that is, a > 0, the method used in [13] to get the blow-up condition is not available since the variance identity of Eq. (1) do not have a polynomial expansion. To overcome this difficulty, we modify the energy, momentum and variance identity which is similar to [18], and deduce a sharp blow-up condition. Indeed, we show that under some mild assumptions on u 0 and Q, if there exist z ≥ 4aσ σd−2 andt such that This implies that no matter how large the damped effect is, the blow-up phenomenon will not disappear. We remark that the above blow-up condition can be degenerated to the blow-up condition in conservative stochastic case and in the deterministic case. On the other hand, if the noise satisfies more conditions, using the rescaling transform idea in [1], we prove that when a → ∞ and σ ≥ 2 d , for any fixed time T, the blow-up of the solution does not happen with probability 1.
This paper is organized as follows. In Section 2, we study the evolution of charge and energy, and show the global existence of the unique solution. In Section 3, the modified variance identity is given. Based on it, we obtain a sharp blow-up condition. Furthermore, we prove that when the value of the damped coefficient a becomes large enough, the solution does not blow up at any finite time with high probability. At last, We give a short conclusion in Section 4.

Global existence of solutions for critical stochastic NLS equations
In this section, we focus on the global existence and some properties of the solution for Eq. (1). Throughout this paper, we assume that the local well-posedness for Eq. (1) holds. For the local well-posedness for Eq. (1), we refer to [1,12,14] and references therein. When consider the focusing mass-(super)critical case, [11] proves that the solution of Eq. (1) blows up with any initial data for the additive case. For the stochastic NLS driven by the multiplicative noise, similar situation happens with any initial datum in the super-critical case (see e.g. [2,13]). This phenomenon is different from the deterministic case, where the solution will blow up in some finite time when u 0 possesses some negative Hamiltonian in the focusing mass-(super)critical case (see e.g. [3,5,17]). However, it is still not clear on whether or not the solution of Eq. (1) equation globally exists in critical case. Notice that when the noise is independent of space and a = 0, i.e., the global existence and blow-up results become more clear. In this case, one can use the infinite dimensional Doss-Sussman type transformation u(t) = exp(iβ(t))y(t) to get the well-posedness and blow-up results, where y(t) satisfies We first study the global exsitence of the solution of Eq. (1) in the focusing critical case. This suggests that the critical nonlinearity in multiplicative cases is different from the supercritical nonlinearity, and that the critical nonlinearity combined with the dispersion term dominates the behavior of the solution. For convenience, we assume that u 0 ∈ H 1 is a deterministic function and that To get a priori estimate of u, we first study the evolution of charge M (u(t)) := u(t) 2 and energy H(u) := 1 2 ∇u 2 − λ 2σ+2 u 2σ+2 L 2σ+2 in the following lemma. and Proof For purpose of obtaining the charge and energy evolution of u, the truncated argument in [12] is applied. In detail, let N ∈ N + and K > 0 and define the operators Θ N , N ∈ N by where F is the Fourier transform and θ ∈ C ∞ c is a real-valued and nonnegative function satisfying θ(x) = 1 for |x| ≤ 1, θ(x) = 0 for |x| > 2. Using the above notation, we have the truncated approximation, for m = (m 1 , m 2 ) ∈ N 2 , where Combining with Itô formula in [0, τ ] and taking limits as m → ∞, the evolution of the charge (2) is obtained by choosing a large enough K. Similarly, using the above arguments, the energy evolution law (3) can be proved.
Remark 2.1. The truncated argument is also available for stochastic NLS equation with the homogenous Dirichlet boundary condition. In this case, replacing Θ N by the projection operator P N , then the truncated Galerkin approximated equation becomes where K > 0, N ∈ N + . The inverse inequality, for s ≥ 1, implies the coefficients of Eq. (5) are globally Lipschitz. Therefore, by the arguments in [12], the result of Lemma 2.1 holds.
In order to illustrate the global well-posedness result, we introduce the optimal constant for Gagliardo-Nirenberg inequality and its corresponding ground state solution (see e.g. [19]).
Based on Lemma 2.1 and Lemma 2.2, we are in position to show the global existence of u. For the sake of simplicity, the procedure about truncated arguments and taking limits is omitted in the rest of this paper.
Proof Since the local well-posedness of Eq. (1) is shown by [12,4,1], we only need to get the uniform boundedness of u H 1 to ensure the global existence of the solution. By the charge conservation law, Gagliardo-Nirenberg inequality and σd = 2, where C σ,d = σ+1 R 2σ and R is the ground state solution of ∆R − R + R 2σ+1 = 0. Then the energy evolution of u implies that for any T 0 > 0, any stopping time τ < inf(T 0 , τ * (u 0 )) and any time t ≤ τ , Then Hölder inequality and Sobolev embedding theorem yield that Gronwall inequality implies that sup t≤τ E ∇u(t) 2 ≤ C(u 0 , R, a, Q, T 0 ).

ON GLOBAL EXISTENCE AND BLOW-UP FOR DAMPED STOCHASTIC NLS EQUATION 7
Moreover, using Burkholder-Davis-Gundy inequality and Young inequality, where ǫ < 1 2 . The last term of the above inequality is bounded by Hölder inequality and the charge evolution law (2), which in turns implies that The condition u 0 < R in Theorem 2.1 is a sufficient condition for the global existence of the solution. Based on it, we get a upper bound of the probability of the blow-up with a random initial datum at any finite time.
Corollary 2.1. Assume that u 0 is random initial datum. Under the condition of Theorem 2.1, we have Moreover, we can get the following exponential integrability of u, which is useful for studying the continuous dependence on initial data and noise, exponential tail estimate of the solution, strong and weak convergence rates of numerical approximations (see e.g. [7,8,9]).
Again by the Gagliardo-Nirenberg inequality (6) and the Sobolev embedding theorem, we obtain Then Lemma 3.1 in [8] implies that Applying the above exponential integrability of exact solution, we deduce the following strongly continuous dependence on the initial data.
Proof Applying the truncated arguments, Itô fromula and taking limits yield that Since for a, b ∈ C, |a| 4 a−|b| 4 b = (|a| 4 +|b| 4 )(a−b)+ab(|a| 2 +|b| 2 )(ā−b)+|a| 2 |b| 2 (a− b), combining with Young inequality and Gagliardo-Nirenberg inequality, we get Gronwall inequality and Gagliardo-Nirenberg inequality lead that After taking expectation, by Young inequality, we obtain It is obvious that if these two exponential moments in the above inequality are bounded, the theorem is proved. For simplicity, we take E exp Then we take α = −2a+ 2aσ c(σ+1) u 0 2σ C σ,d + 4 c 2 u 0 2 f Q L ∞ such that 16 2a−α u 0 2 ≤ 1 and α < 2a. Indeed, the assumption on u 0 and v 0 , together with Gagliardo-Nirenberg inequality Eq. (6), implies which ensure that 16 2a−α u 0 2 ≤ 1 and α < 2a. Proposition 2.1, combining the above estimations, implies the uniform boundedness of the exponential moments of Eq. (7), which completes the proof. with R 2 = π √ 3, the above strong continuous dependence result on initial data holds with max( u 0 , v 0 ) < 4 3π 2 2 when a becomes large enough. For u 0 < R , a ≥ 0, by the above arguments, we can get the continuous dependence on initial data in pathwise sense,

Blow-up of solutions in focusing mass-(super)critical case
As shown in Section 2, the result about well-posedness for Eq. (1) in the critical case is similar to that in deterministic case. Notice that this phenomenon is different from the additive case with σ ≥ d 2 and the multiplicative case with σ > d 2 (see e.g. [11,13]), where the singularity happens in any finite time with a positive probability for any initial datum. In fact, the authors in [13] show that for σ ≥ 2 d , if u 0 ∈ L 2 (Ω; Σ)∩L 2σ+2 (Ω; L 2σ+2 (R d )), f Q = k∈N + |∇Q 1 2 e k | 2 and for somet > 0, then P(τ * (u 0 ) ≤t) > 0. The above result implies that if the energy of u 0 is a.s. negative, then P(τ * (u 0 ) ≤ t) > 0 for some t > 0 provided the noise is not too strong, i.e., f Q L ∞ is small enough. The natural question is whether the damped effect can prevent the blow-up phenomenon or not in stochastic case.
To study the blow-up phenomenon, we introduce the finite variance space With the help of a smoothing procedure and truncated arguments (see e.g. [13]), we can prove rigorously the evolution laws of V and G for the damped stochastic NLS equation.
Proposition 3.1. Assume that u 0 ∈ Σ. Under the conditions of Lemma 2.1, for any stopping time τ < τ * (u 0 ) a.s., we have Proof Applying Itô formula to V and G, integration by parts and taking the imaginary part of the integration, we obtain By the definition of H and σd = 2, we get For the damped stochastic NLS equation, the method in [13] is not available since the damped effect will lead that the expansion of V produces many addition terms which can not be estimated directly. We introduce the modified energy, invariance and momentum as in [18] and study the evolution of these modified quantities to investigate the blow-up condition for supercritical case σd > 2, a > 0. Proof The proof is similar to the proof of Lemma 2.1 and Proposition 3.1 by using smoothing procedures, truncated arguments, integration by parts and Itô formula. More details, we refer to [13].
Based on Lemma 3.1, we prove a preliminary result on the blow-up condition for Eq. (1) in the supercritical case.
then for somet, we have Proof We prove the assertion by contradiction. Assume that the solution u exists globally. Then for any t > 0, τ * (u 0 ) > t a.s. Then we take τ = t. The evolution law of modified energy Eq. (8) , charge evolution law Eq. (2) and taking expectation leads that Next, we aim to show a priori estimate on e bt G(u(t)). For simplicity, we denotẽ H(u) := ∇u 2 − σd 2σ+2 u 2σ+2 2σ+2 . Applying the evolution of modified momentum Eq. (9) and taking expectation, we obtain t 0 E e bs G(u(s)) ds (12) To control the second term E H (u(s)) uniformly, we take b ≤ a[2 − 4σ σd−2 ] such that =H(u(s)).
Then the fact thatH(u) ≤ 2H(u) leads that Using Gronwall inequality, we have The above estimation and Eq. (12) yield that t 0 e bs G(u(s))ds Again by Gronwall inequality, E t 0 e bs G(u(s))ds The above inequality, Eq. (10) and the non-negativity of V yield that Since the assumption means that By the above inequality and the positivity of e bt , there exists somet such that lim t→t E V (u(t)) = 0. By the uncertainty principle u 2 ≤ 2 d ∇u |x|u , we get .
The above estimation yields that E ∇u(t) 2 goes into ∞ when t tends tot, which leads to a contradiction and finishes the proof.
Remark 3.1. The above proposition implies that when the solution of Eq. (1) will blow up in a finite time with a positive probability. Clearly, if the energy of u 0 is negative a.s., damped effect is not strong and the noise is small enough, the blow up phenomenon of the solution always happens.
The blow-up condition Eq. (11) presents the effect of the damped term. The following result gives a sharp time-dependent blow-up condition.
Theorem 3.1. Let σd > 2, a ≥ 0, u 0 ∈ L 2 (Ω; Σ) ∩ L 2σ+2 (Ω; L 2σ+2 ), and Assume also that for some z ≥ 4aσ σd−2 andt such that then we have Proof The proof is similar to the proof of Proposition 3.2. Using the evolutions of the modified energy Eq. (8), the new energyH in Proposition 3.2 leads that for z ≥ 4aσ σd−2 , t 0 E e bsH (u(s)) ds It seems that when a becomes larger, the blow-up time becomes longer and that when a goes to ∞, the blow-up condition is not satisfied. Indeed, we can show that when the damped effect is large enough, the damped effect can prevent the blowup of the solution with high probability. The key of the proof is using the infinite dimensional Doss-Sussman type transformation in [1,2]. In the following theorem, we assume that the noise satisfies k∈N + Q    where η(x) = 1 + |x| 2 if d = 2, and η(x) = (1 + |x| 2 )(ln(2 + |x| 2 )) 2 if d = 2. Under these assumptions, the local well-posedness is obtained in [1]. We also remark when {e k } k∈N + is an orthonormal basis of H, the decay condition natural holds. In this case, e k can be chosen as the k-th Hermite function in H, and meanwhile as the k-th eigenvector of the operator Q Proof We apply the rescaling transformation v(t) = e at−iW (t) u(t) to Eq. (1) and get the following random partial differential equation dv = i exp at − iW (t) ∆u(t)dt + i exp at − iW (t) |u| 2σ udt (14) = i ∆ + 2i∇(W (t)) · ∇ + |∇W (t)| 2 + i∆W (t) v + i exp − 2aσt |v| 2σ vdt := A(t)vdt + i exp − 2aσt |v| 2σ vdt.
Considering the solution map G of Eq. (14), by the random Strichartz estimate and similar arguments in [1], we obtain where v, w ∈ X τ R , C S = (2σ + 1)D 2σ , D is the Sobolev embedding coefficient form L 2σ+2 to H 1 , C τ is the random Strichartz estimate coefficient, q > 1 and 1 q = 1 − 2 r > 0. Now we take R = 4C τ u 0 H 1 and τ (a) = inf t > 0 2C S C t aσ R 2σ > 1 such that the mapping G : X τ R −→ X τ R has a fixed point in the Banach space X τ R , · C(0,τ ;H) + · L r (0,τ ;L 2σ+2 ) , which implies the local well-posedness of Eq. (14). Now, we aim to show that lim a→∞ P(τ * (u 0 , a) > T ) = 1. Based on the result in [1, Lemma 2.7] that C t , t > 0 is F t measurable, increasing and continuous, the definition of τ * (u 0 , a) and the a.s. boundedness of C t yield that lim a→∞ P(τ * (u 0 , a) > T ) ≥ lim a→∞ P(τ (a) > T ) Remark 3.4. If the noise is space-independent or disappears, applying the arguments in Theorem 2.1 and Theorem 3.2, one can get global existence and blow-up results of the solutions for the damped stochastic NLS equation.

Conclusions
In this paper, we consider the influence of both damped term and noise on the stochastic nonlinear Schrödinger equation driven by multiplicative noise. We first show the global existence of the unique solution for damped stochastic NLS equation in critical case and study the exponential integrability and the continuous dependence on the initial data of the solution. Then based on the modified variance identity, we deduce a sharp blow-up condition for damped stochastic NLS equation in supercritical case. Moreover, we prove that the large damped effect can prevent the blow-up with high probability.