This paper focuses on a class of variation-inequality problems involving non-divergence polytropic parabolic operators. The penalty method is employed, along with the Leray Schauder fixed point theory and limit progress, to determine the existence of solutions. The study also delves into the blow-up phenomena of the solution, revealing that under certain conditions, the solution will blow up in finite time.

In this paper, we study variational inequality systems with quasilinear degenerate parabolic operators in a bounded domain. As a series of penalty problems, the existence of the solutions in the weak sense is proved by a limit process. The uniqueness of the solution is also proved.