Abstract In this work, we study a new type of linear partial differential equations – the variable-order subdiffusion equation. Here the Laplace operator in space acts on the Riemann-Liouville time derivative of space-dependent order. We construct a variable-order Sobolev and prove the weak solvability of the initial-boundary value problem for this equation, which confirms the well-posedness of the problem. Finally, we briefly discuss the application of the developed approach to the more general variable-order reaction-subdiffusion equation.

中文翻译：

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变阶次扩散方程的弱可解性

摘要 在这项工作中，我们研究了一种新型的线性偏微分方程——变阶次扩散方程。这里空间中的拉普拉斯算子作用于空间相关阶次的黎曼-刘维尔时间导数。我们构造了一个变阶Sobolev，并证明了该方程的初边值问题的弱可解性，证实了问题的适定性。最后，我们简要讨论了所开发的方法在更一般的可变阶反应-亚扩散方程中的应用。