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<p>We deal with an abstract nonlocal differential equation involving delays in separable Hilbert spaces, where the considered equation is a general model for some equations that arose in fluid dynamics. Some sufficient conditions are analyzed to ensure the global existence, regularity, and stability for a class of nonlocal in time semilinear differential equations with time-varying delays. Our approach is based on the theory of completely positive functions, local estimates, as well as the fixed point argument. We prove the Hölder regularity in the case of nonuniqueness and the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2452_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{1}-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>C</mi> <mn>1</mn> </msup> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation> regularity in the case of uniqueness by using the assumption of regularity of the initial datum to overcome the difficulty caused by delay factors. Furthermore, some results on the long-time behavior of mild solutions are established by employing a new Halanay-type inequality. The obtained results can be applied to some concrete time-delayed partial differential equations, such as the classical reaction diffusion, the Rayleigh–Stokes equation and the diffusion equation with memory.</p>

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Regularity and stability for a class of time-delayed nonlocal semilinear equations

  • 期刊论文

  • 发表日期：

    2025年2月26日

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  • Tran Van Tuan,
  • Nguyen Van Dac

摘要

We deal with an abstract nonlocal differential equation involving delays in separable Hilbert spaces, where the considered equation is a general model for some equations that arose in fluid dynamics. Some sufficient conditions are analyzed to ensure the global existence, regularity, and stability for a class of nonlocal in time semilinear differential equations with time-varying delays. Our approach is based on the theory of completely positive functions, local estimates, as well as the fixed point argument. We prove the Hölder regularity in the case of nonuniqueness and the \(C^{1}-\) C 1 - regularity in the case of uniqueness by using the assumption of regularity of the initial datum to overcome the difficulty caused by delay factors. Furthermore, some results on the long-time behavior of mild solutions are established by employing a new Halanay-type inequality. The obtained results can be applied to some concrete time-delayed partial differential equations, such as the classical reaction diffusion, the Rayleigh–Stokes equation and the diffusion equation with memory.