Asymptotic behavior of solutions to a dissipative nonlinear Schrödinger equation with time dependent harmonic potentials

We consider the Cauchy problem of a dissipative nonlinear Schrödinger equation with a time dependent harmonic potential. We find a critical situation that the $L^2$-norm of dissipative solutions decays or not and which is decided by a nonlinear power and time decay order of harmonic potential.

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