Using the theory of operator-valued Fourier multipliers, we establish characterizations for well-posedness of a large class of degenerate integro-differential equations of second order in time in Banach spaces. We are concerned with the spaces \(L^p({\mathbb {R}},X), \, 1\leqslant p<\infty \) where X is a given Banach space. When X is a UMD space and \(1<p<\infty \), we obtain concrete conditions for well-posedness based on the concept of R-boundedness (or Rademacher boundedness) for operator families. We rely on a transfer to weighted vector-valued \(L^p\)-spaces in order to prove the results. The results are applied to some concrete equations. The equation that we consider appear in several models in the applied sciences, particularly physics, rheology, material science and more generally in phenomena where memory effects play an important role.

中文翻译：

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$$ L ^ p $$ Lp-Banach空间中一类具有无限延迟的退化积分微分方程的最大正则性

使用算子值傅立叶乘法器理论，我们建立了Banach空间中一类二阶退化简积分积分微分方程的适定性的刻画。我们关心空间\（L ^ p（{\ mathbb {R}}，X），\，1 \ leqslant p <\ infty \）其中X是给定的Banach空间。当X是UMD空间且\（1 <p <\ infty \）时，我们基于算子族的R界（或Rademacher界）的概念，获得了适定性的具体条件。我们依靠转移到加权向量值\（L ^ p \）-空格以证明结果。将结果应用于一些具体方程式。我们考虑的方程式出现在应用科学的几种模型中，特别是物理，流变学，材料科学，更普遍地讲，是在记忆效应起重要作用的现象中。