In this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain \(Q=\varOmega \times (0,\tau )\): For \(k=1,\dots , m\), find \(u_k\in K_k\) and \(\eta _k\in L^{p'_k}(Q)\) such that

where \(K_k \) is a closed and convex subset of \(L^{p_k}(0,\tau ;W_0^{1,p_k}(\varOmega ))\), \(A_k\) is a time-dependent quasilinear elliptic operator, and \(f_k:Q\times \mathbb {R}^m\rightarrow 2^{\mathbb {R}}\) is an upper semicontinuous multivalued function with respect to \(s\in {\mathbb R}^m\). We provide an existence theory for the above system under certain coercivity assumptions. In the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence and enclosure results. As an application, a multivalued parabolic obstacle system is treated. Moreover, under a lattice condition on the constraints \(K_k\), systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities.

在本文中，我们为圆​​柱域\（Q = \ varOmega \ times（0，\ tau）\）中的以下多值抛物变分不等式系统提供了一个分析框架：对于\（k = 1，\ dots，m \ ），找到\（U_K \在K_k \）和\）在L ^ {p'_k}（Q \（\ ETA _K \） ，使得

其中\（K_k \）是\（L ^ {p_k}（0，\ tau; W_0 ^ {1，p_k}（\ varOmega））\）的闭合凸集，\（A_k \）是一个时间-相依的拟线性椭圆算子和\（f_k：Q \ times \ mathbb {R} ^ m \ rightarrow 2 ^ {\ mathbb {R}} \）是关于\（s \ in {\ mathbb R} ^ m \）。我们在某些矫顽力假设下为上述系统提供了一个存在理论。在非强制性情况下，我们建立了适当的子上解方法，该方法可让我们获得存在和封闭的结果。作为一种应用，处理了多值抛物线障碍系统。而且，在晶格条件下的约束\（K_k \），演化变分-半变分不等式的系统显示为上述多值抛物变分不等式的系统的子类。