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2024 | OriginalPaper | Chapter

On the Loss of Regularity in a Degenerate Vibrating Beam Equation

Authors : Petar Popivanov, Borislav Yordanov

Published in: New Trends in the Applications of Differential Equations in Sciences

Publisher: Springer Nature Switzerland

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Abstract

We study the well-posedness in Sobolev spaces \(H^s(\textbf{R})\) of the Cauchy problem for \(D_t^2u=y(1+D_y^2)^2yu\), where \((t,y)\in [0,\infty )\times \textbf{R}\). Our results show that solutions u(t, y) undergo infinite losses of derivatives \(D_y\) on a subset of positive time measure \(S\subset [0,T]\) for every \(T>\pi .\) We also find explicitly a basis of eigenfunctions for the degenerate elliptic operator which do not belong to any Sobolev space with index \(s\ge 5/2.\)

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previous chapter A Didactic Approach to Study Mass—Inertial Characteristics of Bodies in Plane Motion
next chapter Mittag-Leffler Stability for Non-instantaneous Impulsive Generalized Proportional Caputo Fractional Differential Equations
Literature
1.
Alessia Ascanelli, Massimo Cicognani, Ferruccio Colombini: The global Cauchy problem for a vibrating beam equation. Journal of Differential Equations Vol 247, Issue 5, 1 September (2009).
2.
Bateman, H., Erdelyi, A.: Higher transcendental functions. vol 2, NY-Toronto-London, McGraw-Hill Book Company, (1953).
3.
Alessandro Camasta, Genni Fragnelli: Fourth-order differential operators with interior degeneracy and generalized Wentzell boundary conditions. Electronic Journal of Diff. Equations, Vol. 2022 No. 01-87 (2022).
4.
Massimo Cicognani, Ferruccio Colombini: Loss of derivatives in evolution Cauchy problems. Annali dell’Università di Ferrara 52: 271-280 (2006).
5.
Hörmander, L.: Linear partial differential operators. Berlin, Springer, (1964).
6.
Hugh L. Montgomery: The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978).
7.
Serov, V.: Fundamental solution and Fourier series in eigenfunctions of degenerate elliptic operator. J. Math. Anal. Appl. 329 (2007).
Metadata
Title
On the Loss of Regularity in a Degenerate Vibrating Beam Equation
Authors
Petar Popivanov
Borislav Yordanov
Copyright Year
2024
Book
New Trends in the Applications of Differential Equations in Sciences

Print ISBN: 978-3-031-53211-5

Electronic ISBN: 978-3-031-53212-2

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-53212-2_18

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