We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous source terms lying in some negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.

中文翻译：

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分数阶扩散方程非齐次初边值问题的弱和强解的适定性

我们研究与具有非均匀边界和初始值的时间分数扩散方程有关的初始边界值问题的适定性。我们认为问题的解决方案既有弱也有强。对于弱解，我们引入解的定义，该解可以证明存在一些负阶Sobolev空间中具有非零初始值和边界值以及非齐次源项的初始边值问题的解的存在。对于强大的解决方案，我们引入了最佳的兼容性条件并证明了这些解决方案的存在。我们还介绍了一些尖锐的条件，这些条件可以保证在时间和空间上具有更规则的解决方案。