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

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