In this article, our purpose is to study the existence and uniqueness of a solution to a damped variable order fractional subdiffusion equation with time delay. Under weak assumptions on the data, we prove the uniqueness of a weak solution to the problem under consideration. The method of semi-discretization is extended to this kind of time fractional parabolic problem with delay in the case that the time delay parameter $s>0$ satisfies $s⩽T$, where $T$ denotes the final time. As a consequence, two a priori estimates are predicted based on a discrete variational formulation of the problem. The existence of the problem’s weak solution on the time frame $\left[0,\lfloor \frac{T}{s}\rfloor s\right]$ is established by the aid of these derived a priori estimates. The paper is closed by introducing a fully discrete scheme based on Galerkin Legendre spectral approximation for the spatial operator and the backward Euler difference approximation for the temporal variable order operator. Accordingly, the accuracy and efficiency of the proposed scheme are justified by giving some numerical experiments for the sake of clearness.

在本文中，我们的目的是研究具有时滞的阻尼变阶分数次扩散方程解的存在性和唯一性。在对数据的弱假设下，我们证明了所考虑问题的弱解的唯一性。在时延参数为$s>0$满足$s⩽吨$， 在哪里$吨$表示最后的时间。因此，基于问题的离散变分公式预测了两个先验估计。时间框架上问题弱解的存在$\left[0,\lfloor \frac{吨}{s}\rfloor s\right]$是通过这些导出的先验估计建立的。本文通过引入基于空间算子的 Galerkin Legendre 谱近似和时间变阶算子的后向欧拉差分近似的完全离散方案来结束。因此，为了清楚起见，通过给出一些数值实验来证明所提出方案的准确性和效率是合理的。