We investigate the defocusing inhomogeneous nonlinear Schrödinger equation
$ i \partial_tu + \Delta u = |x|^{-b} \left({\rm e}^{\alpha|u|^2} - 1- \alpha |u|^2 \right) u, \quad u(0) = u_0, \quad x \in \mathbb{R}^2, $
with $ 0<b<1 $ and $ \alpha = 2\pi(2-b) $. First we show the decay of global solutions by assuming that the initial data $ u_0 $ belongs to the weighted space $ \Sigma(\mathbb{R}^2) = \{\,u\in H^1(\mathbb{R}^2) \ : \ |x|u\in L^2(\mathbb{R}^2)\,\} $. Then we combine the local theory with the decay estimate to obtain scattering in $ \Sigma $ when the Hamiltonian is below the value $ \frac{2}{(1+b)(2-b)} $.
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