On nonlinear problems of parabolic type with implicit constitutive equations involving flux

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.

Automated summary

The study investigates systems of nonlinear partial differential equations of parabolic type by introducing an additional implicit equation to relate the flux function to the spatial gradient of the unknown. By formulating four conditions concerning the form of the implicit equation, it is shown that these conditions describe a maximal monotone p-coercive graph. The study establishes the global-in-time and large-data existence of a weak solution and its uniqueness through adopting and generalizing Minty's method of monotone mappings.