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Periodic solutions of four-order degenerate differential equations with finite delay in vector-valued function spaces

Part of: Harmonic analysis in one variable Parabolic equations and systems Functional-differential and differential-difference equations General theory of linear operators Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  14 September 2023

Shangquan Bu  and
Gang Cai
Shangquan Bu
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China (sbu@math.tsinghua.edu.cn)
Gang Cai*
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China (caigang-aaaa@163.com)
*
*Corresponding author.
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Abstract

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ($P_4$): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$, where $A$, $B$, $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$, $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$) into $X$, $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$. We completely characterize the well-posedness of ($P_4$) in the above two function spaces by using known operator-valued Fourier multiplier theorems.

Keywords

well-posednessdegenerate differential equationFourier multiplierLebesgue-Bochner spacesBesov spaces

MSC classification

Primary: 47D06: One-parameter semigroups and linear evolution equations
Secondary: 35K65: Degenerate parabolic equations 34K13: Periodic solutions 47A10: Spectrum, resolvent 42A45: Multipliers
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 18
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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