<p>The main goal of this paper is to study the asymptotic behavior of a weakly damped forced fractional nonlinear Klein–Gordon–Schrödinger system in one dimensional unbounded domain. We prove the existence of a global attractor <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_313_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}_{\alpha }\; \left( \alpha \in (\frac{1}{2},1)\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">A</mi> <mi>α</mi> </msub> <mspace width="0.277778em" /> <mfenced close=")" open="("> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mfenced> </mrow> </math></EquationSource> </InlineEquation> of the systems of the fractional nonlinear Klein–Gordon–Schrödinger (FNLKGS) equations in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_313_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="176" /> </InlineMediaObject> <EquationSource Format="TEX">$H^{\alpha }(\mathbb {R})\times H^{\alpha }(\mathbb {R})\times L^2(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msup> <mi>H</mi> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and more particularly that this attractor is in fact a compact set of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41808_2025_313_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="193" /> </InlineMediaObject> <EquationSource Format="TEX">$H^{2\alpha }(\mathbb {R})\times H^{2\alpha }(\mathbb {R})\times H^{\alpha }(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msup> <mi>H</mi> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msup> <mi>H</mi> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the regularity of the attractor for a new class of fractional nonlinear Klein–Gordon–Schrödinger systems

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发表日期：

2025年1月19日

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Salah Missaoui

摘要

The main goal of this paper is to study the asymptotic behavior of a weakly damped forced fractional nonlinear Klein–Gordon–Schrödinger system in one dimensional unbounded domain. We prove the existence of a global attractor $\mathcal {A}_{\alpha }\; \left( \alpha \in (\frac{1}{2},1)\right) $ $A_{α} (α \in (\frac{1}{2}, 1))$ of the systems of the fractional nonlinear Klein–Gordon–Schrödinger (FNLKGS) equations in $H^{\alpha }(\mathbb {R})\times H^{\alpha }(\mathbb {R})\times L^2(\mathbb {R})$ $H^{α} (R) \times H^{α} (R) \times L^{2} (R)$ and more particularly that this attractor is in fact a compact set of $H^{2\alpha }(\mathbb {R})\times H^{2\alpha }(\mathbb {R})\times H^{\alpha }(\mathbb {R})$ $H^{2 α} (R) \times H^{2 α} (R) \times H^{α} (R)$ .