The results of this work deal with the existence and blow up of solutions for the following damped extensible beam with degenerate nonlocal damping and source term <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11401_2025_10_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="370" /> </InlineMediaObject> <EquationSource Format="TEX">$u_{tt}+\Delta^{2}u-M(\Vert\nabla u \Vert^{2})\Delta u+\Vert \Delta u\Vert^{2\alpha}\vert u_{t}\vert^{\gamma}u_{t}=\vert u\vert^{\rho}u$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>u</mi> <mrow> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>−</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mo>∥</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <msup> <mo>∥</mo> <mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mo>∥</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <msup> <mo>∥</mo> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </msup> <mo fence="false" stretchy="false">∣</mo> <msub> <mi>u</mi> <mrow> <mi>t</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">∣</mo> <mrow> <mi>γ</mi> </mrow> </msup> <msub> <mi>u</mi> <mrow> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">∣</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">∣</mo> <mrow> <mi>ρ</mi> </mrow> </msup> <mi>u</mi> </math></EquationSource> </InlineEquation>. It is regarded as the second part of the paper by Narciso et al. (in 2023), where global existence, uniqueness and asymptotic stability of strong solutions were obtained for regular initial data in the case ∣u∣ρu ≡ 0.

A Blow-up Result for an Extensible Beam with Degenerate Nonlocal Nonlinear Damping

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发表日期：

2025年3月1日

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Vando Narciso,
Fatma Ekinci,
Erhan Pişkin

摘要

The results of this work deal with the existence and blow up of solutions for the following damped extensible beam with degenerate nonlocal damping and source term $u_{tt}+\Delta^{2}u-M(\Vert\nabla u \Vert^{2})\Delta u+\Vert \Delta u\Vert^{2\alpha}\vert u_{t}\vert^{\gamma}u_{t}=\vert u\vert^{\rho}u$ $u_{t t} + Δ^{2} u - M (∥ \nabla u ∥^{2}) Δ u + ∥ Δ u ∥^{2 α} ∣ u_{t} ∣^{γ} u_{t} = ∣ u ∣^{ρ} u$ . It is regarded as the second part of the paper by Narciso et al. (in 2023), where global existence, uniqueness and asymptotic stability of strong solutions were obtained for regular initial data in the case ∣u∣^ρu ≡ 0.