Abstract
We investigate the well-posedness of the third-order integro-differential equations \({{(P_{3})} : {\alpha u'''(t)+u''(t)} {=\beta Au(t)+\beta\int_{-\infty}^t a(t-s)Au(s){\rm d}s+\gamma Bu'(t)+f(t),(t \in {\mathbb{T}}:=[0,2\pi])}}\) with periodic boundary conditions \({u(0)=u(2\pi),\ u'(0)=u'(2\pi),\ u''(0)=u''(2\pi)}\), in periodic Lebesgue–Bochner spaces \({L^p({\mathbb{T}}; X)}\), periodic Besov spaces \({B_{p,q}^{s}({\mathbb{T}}; X)}\) and periodic Triebel–Lizorkin spaces \({F_{p,q}^{s}({\mathbb{T}}; X)}\), where A and B are closed linear operators on a Banach space X satisfying \({D(A)\cap D(B) \neq \{0\}, a\in L^{1}({\mathbb{R}_{+}})}\) and \({\alpha,\beta,\gamma}\) are positive constants. We completely characterize the well-posedness of (P 3) in the above three function spaces by using operator-valued Fourier multiplier theorems.
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This work was supported by the NSF of China (No. 11571194,11401063), the Specialized Research Fund for the Doctoral Program of Higher Education (20120002110044), the Natural Science Foundation of Chongqing (cstc2014jcyjA00016) and Science and Technology Project of Chongqing Education Committee (Grant No. KJ1500314).
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Cai, G., Bu, S. Periodic solutions of third-order integro-differential equations in vector-valued functional spaces. J. Evol. Equ. 17, 749–780 (2017). https://doi.org/10.1007/s00028-016-0335-5
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DOI: https://doi.org/10.1007/s00028-016-0335-5