| Abstract: |
| We study the nonlinear inhomogeneous Schr\odinger equation
$
i\partial_t u+\Delta u=\mu |x|^{-b}|u|^{\alpha}u
$
and prove global existence, including self-similar solutions, for small initial data in suitable Besov spaces. The results extend to equations with general potentials.
We then describe the large-time behavior: depending on the decay of the initial data, solutions are either asymptotic to self-similar solutions of the nonlinear equation or governed by the linear flow. In particular, for sufficiently decaying potentials, solutions exhibit asymptotic behavior described by self-similar solutions of the linear Schr\odinger equation. |
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