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Abstract
In this paper, we consider the well-posedness of the weakly damped stochastic nonlinear Schrödinger(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.
Highlights
IntroductionThe nonlinear Schrodinger 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])
The nonlinear Schrodinger 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.In this paper, we consider the weakly damped stochastic NLS equation driven by a linear multiplicative noise in focusing mass-(super)critical range, (1)du = i(∆u + λ|u|2σu)dt − audt + iu ◦ dW (t), u(0) = u0, where 2 d ≤ σ < 2 (d−2)+, λ =
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
Summary
The nonlinear Schrodinger 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 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 2 d and u0. 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. 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 u0 possesses some negative Hamiltonian, see [3, 5, 17] and references therein. Based on the proved exponential integrability of the solution u, i.e., sup E exp t∈[0,∞)
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