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In this paper, we consider a class of the defocusing inhomogeneous nonlinear Schrödinger equation $$\begin{aligned} i\partial _t u + \varDelta u - |x|^{-b} |u|^\alpha u = 0, \quad u(0)=u_0 \in H^1, \end{aligned}$$ i ∂ t u + Δ u - | x | - b | u | α u = 0 , u ( 0 ) = u 0 ∈ H 1 , with $$b, \alpha >0$$ b , α > 0 . We first study the decaying property of global solutions for the equation when $$0<\alpha <\alpha ^\star $$ 0 < α < α ⋆ where $$\alpha ^\star = \frac{4-2b}{d-2}$$ α ⋆ = 4 - 2 b d - 2 for $$d\ge 3$$ d ≥ 3 . The proof makes use of an argument of Visciglia (Math Res Lett 16(5):919–926, 2009). We next use this decay to show the energy scattering for the equation in the case $$\alpha _\star<\alpha <\alpha ^\star $$ α ⋆ < α < α ⋆ , where $$\alpha _\star = \frac{4-2b}{d}$$ α ⋆ = 4 - 2 b d .
Journal of Evolution Equations – Springer Journals
Published: Jan 19, 2019
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