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

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(ℝ ⁿ) = {f ∈ H ₁(ℝ ⁿ), |·|f ∈ L ²(ℝ 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 ϕ = 0, our results coincide with the ones for the deterministic equation.

Keywords

Stochastic Schrödinger equation / Well-posedness / Blow up

References

[1] de Bouard A., Debussche A.. The stochastic nonlinear Schrödinger equation in H ₁. Stochastic Anal. Appl., 2003, 21(1): 97-126

[2] de Bouard A., Debussche A.. On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation. Probab. Theory Related Fields, 2002, 123: 76-96

[3] de Bouard A., Debussche A.. Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise. Ann. Probab., 2005, 33(3): 1078-1110

[4] de Bouard A., Debussche A.. A stochastic nonlinear Schrödinger equation with multiplicative noise. Comm. Math. Phys., 1999, 205(1): 161-181

[5] Carles R.. Remarks on nonlinear Schrödinger equations with harmonic potential. Ann. Henri Poincaré, 2002, 3(4): 757-772

[6] Carles R.. Nonlinear Schrödinger equations with repulsive harmonic potential and applications. SIAM J. Math. Anal., 2003, 35(4): 823-843

[7] Carles R.. Critical nonlinear Schrödinger equations with and without harmonic potential. Math. Models Methods Appl. Sci., 2002, 12(10): 1513-1523

[8] Cazenave T.. Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, 2003, Providence, RI: Courant Institute of Mathematical Sciences, A. M. S.

[9] Da Prato G., Zabczyk J.. Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications, 1992, Cambridge: Cambridge University Press