In this paper, we study a diffusion equation of the Kirchhoff type with a conformable fractional derivative. The global existence and uniqueness of mild solutions are established. Some regularity results for the mild solution are also derived. The main tools for analysis in this paper are the Banach fixed point theory and Sobolev embeddings. In addition, to investigate the regularity, we also further study the nonwell-posed and give the regularized methods to get the correct approximate solution. With reasonable and appropriate input conditions, we can prove that the error between the regularized solution and the search solution is towards zero when tends to zero.

中文翻译：

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带分数拉普拉斯算子和基尔霍夫项的抛物线方程的终值问题

在本文中，我们研究了具有可合分数阶导数的基尔霍夫型扩散方程。建立了温和解的全局存在性和唯一性。还导出了温和解的一些规律性结果。本文分析的主要工具是 Banach 不动点理论和 Sobolev 嵌入。此外，为了研究正则性，我们还进一步研究了非适定并给出了正则化方法以获得正确的近似解。在合理适当的输入条件下，我们可以证明正则化解与搜索解之间的误差在趋于零时也趋于零。