In this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation with Caputo fractional derivative of space-dependent variable order where the coefficients are dependent on spatial and time variables. We consider a bounded Lipschitz domain and a homogeneous Dirichlet boundary condition. Variable-order fractional differential operators originate in anomalous diffusion modelling. Using the strongly positive definiteness of the governing kernel, we establish the existence of a unique weak solution in \(u\in \operatorname{L}^{\infty } ((0,T),\operatorname{H}^{1}_{0}( \Omega ) )\) to the problem if the initial data belongs to \(\operatorname{H}^{1}_{0}(\Omega )\). We show that the solution belongs to \(\operatorname{C} ([0,T],{\operatorname{H}^{1}_{0}(\Omega )}^{*} )\) in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form \(\frac{\mathrm{d}}{\mathrm{d}t} (k\ast v)(t)\) to a convolution kernel that is also space-dependent and employ this result when searching for more regular solutions. We also discuss the situation that the domain consists of separated subdomains.

在这个贡献中，我们研究了一个分数扩散方程的初始边界值问题，该方程具有空间相关变量阶数的 Caputo 分数阶导数，其中系数取决于空间和时间变量。我们考虑有界 Lipschitz 域和齐次 Dirichlet 边界条件。变阶分数微分算子起源于异常扩散建模。使用控制核的强正定性，我们在\(u\in \operatorname{L}^{\infty } ((0,T),\operatorname{H}^{1 }_{0}( \Omega ) )\)如果初始数据属于\(\operatorname{H}^{1}_{0}(\Omega )\) 问题。我们证明该解决方案属于\(\operatorname{C} ([0,T],{\operatorname{H}^{1}_{0}(\Omega )}^{*} )\)在常数的 Caputo 分数阶导数的情况下命令。我们将形式为\(\frac{\mathrm{d}}{\mathrm{d}t} (k\ast v)(t)\) 的积分微分算子的基本恒等式推广到卷积核，该卷积核也是空间相关并在搜索更规则的解决方案时使用此结果。我们还讨论了域由分离的子域组成的情况。