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On the well-posedness for stochastic Schrödinger equations with quadratic potential

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Abstract

The authors investigate the influence of a harmonic potential and random perturbations on the nonlinear Schrödinger equations. The local and global well-posedness are proved with values in the space Σ(ℝn) = {f ∈ H 1(ℝn), |·|f ∈ L 2(ℝn)}. When the nonlinearity is focusing and L 2-supercritical, the authors give sufficient conditions for the solutions to blow up in finite time for both confining and repulsive potential. Especially for the repulsive case, the solution to the deterministic equation with the initial data satisfying the stochastic blow-up condition will also blow up in finite time. Thus, compared with the deterministic equation for the repulsive case, the blow-up condition is stronger on average, and depends on the regularity of the noise. If ϕ = 0, our results coincide with the ones for the deterministic equation.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Daoyuan Fang, Linzi Zhang & Ting Zhang

Authors
  1. Daoyuan Fang
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  2. Linzi Zhang
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  3. Ting Zhang
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Corresponding author

Correspondence to Ting Zhang.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 10871175, 10931007, 10901137), the Zhejiang Provincial Natural Science Foundation of China (No. Z6100217) and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20090101120005).

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Fang, D., Zhang, L. & Zhang, T. On the well-posedness for stochastic Schrödinger equations with quadratic potential. Chin. Ann. Math. Ser. B 32, 711–728 (2011). https://doi.org/10.1007/s11401-011-0670-3

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  • DOI: https://doi.org/10.1007/s11401-011-0670-3

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