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| On the Well-Posedness for Stochastic Schr¨odinger Equations with Quadratic Potential |
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Citation： |
Daoyuan FANG,Linzi ZHANG,Ting ZHANG.On the Well-Posedness for Stochastic Schr¨odinger Equations with Quadratic Potential[J].Chinese Annals of Mathematics B,2011,32(5):711~728 |
| Page view： 1945
Net amount： 1129 |
Authors： |
Daoyuan FANG; Linzi ZHANG; Ting ZHANG; |
Foundation： |
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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| Abstract： |
The authors investigate the influence of a harmonic potential and
random perturbations on the nonlinear Schr\"o\-din\-ger equations.
The local and global well-posedness are proved with values in the
space $\Sigma(\BR^n)=\{f\!\in\! H^1(\BR^n), |\cdot|f\!\in\!
L^2(\BR^n)\}$. \!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 $\phi=0$, our results coincide with the
ones for the deterministic equation. |
Keywords： |
Stochastic Schr¨odinger equation, Well-posedness, Blow up |
Classification： |
35Q55, 60H15, 60H30, 60J60 |
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