| Abstract: |
| In this paper, we study the well-posedness and finite time extinction of solutions to a defocusing nonlinear Schr{o}dinger equation on smooth bounded domains, the whole space, and exterior regions. We consider a locally distributed non-Lipschitz damping term. First, we construct approximate solutions using monotone operator theory. By employing multiplier methods and a unique continuation property, we show that these approximations decay exponentially in the $L^2$-norm. Through a limit passage with weak lower semicontinuity, we then prove global existence and $L^2$-decay for solutions of the original model. Depending on spatial dimension and solution regularity, two distinct asymptotic behaviors arise: finite time extinction occurs for weak solutions if $N=1$ and for regular solutions if $N |
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