We study the well-posedness of the third order degenerate differential equations with infinite delay$$(P_3): (Mu)'''(t) + (Lu)''(t) + (Bu)'(t)= Au(t) + \int _{-\infty }^t a(t-s)Au(s)ds + f(t){\text{ on }}[0, 2\pi ]$$in 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 satisfying $$D(A)\subset D(B)\cap D(L)\cap D(M)$$ and $$ a\in L^1(\mathbb{R}_+)$$. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for $$(P_3)$$ to be $$L^p$$-well-posed(or $$B_{p,q}^s$$-well-posed). Concrete examples are also given to support our main abstract results.

中文翻译：

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Banach空间中无限时滞三阶退化方程的解

我们研究了具有无限延迟的三阶退化微分方程的适定性$$(P_3): (Mu)'''(t) + (Lu)''(t) + (Bu)'(t)= Au (t) + \int _{-\infty }^ta(ts)Au(s)ds + f(t){\text{ on }}[0, 2\pi ]$$in Lebesgue–Bochner 空间 $$ L^p(\mathbb{T};\; X)$$ 和周期 Besov 空间 $$B_{p,\,q}^s(\mathbb{T};\; X)$$，其中 A, B , L 和 M 是 Banach 空间 X 上的闭线性算子，满足 $$D(A)\subset D(B)\cap D(L)\cap D(M)$$ 和 $$ a\in L^1( \mathbb{R}_+)$$。使用已知的运算符值傅立叶乘子定理，我们给出了 $$(P_3)$$ 为 $$L^p$$-well-posed(或 $$B_{p,q}^s$$ - 姿势很好）。还给出了具体的例子来支持我们的主要抽象结果。