On qualitative properties of solutions of equations and inequalities with KPZ-nonlinearities

For quasilinear partial differential equations and inequalities containing nonlinearities of the Kardar–Parisi–Zhang type (i.e., the scalar square of the gradient and its generalizations), we present a summary of (old and recent) results regarding the stabilization of solutions (for the parabolic and elliptic cases), the blow-up of solutions, and specific phenomena (e.g., the extinction of solutions). Descriptive examples demonstrating the Bitsadze approach (the technique of monotone maps) applied in this research area are provided.