Long-time behavior for evolution processes associated with non-autonomous nonlinear Schrödinger equation. (English) Zbl 1532.35422
Summary: We consider non-autonomous nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in a bounded smooth domain and time-dependent forcing that models the motion of waves in a quantum-mechanical system. We address the problem of the local and global well posedness and using rescaling of time we prove the existence of a compact pullback attractor for the associated evolution process. Moreover, we prove that the pullback attractor has finite fractal dimension. To the best of our knowledge, our approach has not been used in the literature to treat the non-autonomous Schrödinger equation.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B41 | Attractors |
| 35B45 | A priori estimates in context of PDEs |
| 28A80 | Fractals |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
non-autonomous Schrödinger equation; evolution process; global well-posedness; compact pullback attractor; fractal dimensionReferences:
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