In this paper, we study a pseudo-parabolic equation with the Caputo fractional derivative. By applying the properties of Mittag–Leffler functions and the method of eigenvalue expansion, under a suitable definition of mild solution of our problem, we obtain the existence result and \(L^p\) regularity of the mild solution by using some Sobolev embeddings. Finally, we also give some examples to illustrate the proposed method.

中文翻译：

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分数阶伪抛物方程的后向问题的适定性和不适性结果

在本文中，我们使用Caputo分数导数研究了伪抛物线方程。通过应用Mittag–Leffler函数的性质和特征值展开的方法，在我们问题的温和解的适当定义下，我们使用一些Sobolev嵌入获得了温和解的存在结果和\（L ^ p \）正则性。最后，我们还给出一些例子来说明所提出的方法。