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<p>In this paper, we study the long-time behavior of global solutions to the Schrödinger–Choquard equation <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_4015_Article_Equ1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="287" /> </MediaObject> <EquationSource Format="TEX">\({\rm{i}}{\partial _t}u + \Delta u = - ( {{I_\alpha } * {{\vert \cdot \vert}^b}{{\vert u \vert}^p}} ){\vert \cdot \vert^b}{\vert u \vert^{p - 2}}u.\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mi>t</mi> </msub> </mrow> <mi>u</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mrow> <mrow> <msub> <mi>I</mi> <mi>α</mi> </msub> </mrow> <mo>∗</mo> <mrow> <msup> <mrow> <mo fence="false" stretchy="false">∣</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">∣</mo> </mrow> <mi>b</mi> </msup> </mrow> <mrow> <msup> <mrow> <mo fence="false" stretchy="false">∣</mo> <mi>u</mi> <mo fence="false" stretchy="false">∣</mo> </mrow> <mi>p</mi> </msup> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow> <mo fence="false" stretchy="false">∣</mo> <mo>⋅</mo> <msup> <mo fence="false" stretchy="false">∣</mo> <mi>b</mi> </msup> </mrow> <mrow> <mo fence="false" stretchy="false">∣</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">∣</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> <mi>u</mi> <mo>.</mo> </math></EquationSource> </Equation></p><p>Inspired by Murphy who gave a simple proof of scattering for the non-radial INLS, we find that the inhomogeneous term ∣<i>x</i>∣<sup><i>b</i></sup> can replace the radial Sobolev embedding theorem, which allows us to prove scattering theory below the ground state for the intercritical case in energy space without radial assumption.</p>

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Scattering for the Non-Radial Focusing Inhomogeneous Nonlinear Schrödinger–Choquard Equation

  • 期刊论文

  • 发表日期：

    2025年7月15日

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  • Chengbin Xu

摘要

In this paper, we study the long-time behavior of global solutions to the Schrödinger–Choquard equation \({\rm{i}}{\partial _t}u + \Delta u = - ( {{I_\alpha } * {{\vert \cdot \vert}^b}{{\vert u \vert}^p}} ){\vert \cdot \vert^b}{\vert u \vert^{p - 2}}u.\) i ∂ t u + Δ u = − ( I α ∗ ∣ ⋅ ∣ b ∣ u ∣ p ) ∣ ⋅ ∣ b ∣ u ∣ p − 2 u .

Inspired by Murphy who gave a simple proof of scattering for the non-radial INLS, we find that the inhomogeneous term ∣x∣b can replace the radial Sobolev embedding theorem, which allows us to prove scattering theory below the ground state for the intercritical case in energy space without radial assumption.