We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone p-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations.

中文翻译：

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含通量隐式本构方程的抛物型非线性问题

我们研究抛物型非线性偏微分方程系统，其中椭圆算子被作用于通量函数的一阶散度算子所取代，该通量函数通过附加的隐式方程与未知物的空间梯度相关。这种设置在应用方面足够广泛，显着扩展了非线性抛物线问题的范式。制定关于隐式方程形式的四个条件，我们首先证明这些条件描述了一个最大单调p-强制图。然后，我们确定（弱）解决方案的全局时间和大数据存在及其唯一性。为此，我们采用并显着推广了 Minty 的单调映射方法。从数值近似的角度来看，以一种易于处理的方式开发了一个包含几个新工具的统一理论。