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<p>In this paper, we investigate the Cauchy problem for the degenerate and nondegenerate Kirchhoff-type wave equations involving the fractional Laplacian and a damping. With the help of potential well theory and convexity method, we get some new sufficient conditions on the existence of collapse solutions. Moreover, by constructing invariant sets, we find the exact sharp threshold of global existence and collapse, which corresponds to the exact sharp threshold of wave’s collapse to the elastic strings. Here, the threshold is explicitly expressed by the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm of the ground state solutions to the nonlinear fractional Schrödinger equation.</p>

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On the Collapse to the Degenerate and non-Degenerate Kirchhoff-Type Equations Involving the Fractional Laplacian

  • 期刊论文

  • 发表日期：

    2025年10月9日

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  • Guoyi Fu,
  • Shanshan Fu,
  • Yiyin Yuan,
  • Shihui Zhu

摘要

In this paper, we investigate the Cauchy problem for the degenerate and nondegenerate Kirchhoff-type wave equations involving the fractional Laplacian and a damping. With the help of potential well theory and convexity method, we get some new sufficient conditions on the existence of collapse solutions. Moreover, by constructing invariant sets, we find the exact sharp threshold of global existence and collapse, which corresponds to the exact sharp threshold of wave’s collapse to the elastic strings. Here, the threshold is explicitly expressed by the \(L^2\) L 2 -norm of the ground state solutions to the nonlinear fractional Schrödinger equation.