We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities. Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.

我们研究障碍物类型的抛物线准变分不等式（QVI）。在障碍物映射的适当假设下，我们通过两种方法证明了此类QVI的解的存在：一种是通过椭圆QVI进行时间离散化，另一种是通过抛物线变分不等式进行迭代。使用这些结果，我们展示了解图的方向可微性（在某种意义上），该解将图的抛物线QVI的源项纳入解集，并将此结果与上述图的或有导数相关联。我们以一个由抛物线微分算子的逆给出障碍物映射的示例结束。