Long-time behavior of solutions to nonlinear parabolic problems with irregular obstacles

In this paper we study weak solutions to general non-homogeneous parabolic obstacle problems, whose obstacle function is time-dependent irregular. We prove the well-posedness of a global solution to the obstacle problem and we describe the asymptotic behavior of such a solution. In particular, we measure the distance in time of the solutions to two parabolic problems with different initial data and forcing terms. Moreover, in the autonomous case, we prove that, as the time approaches infinity, the global solution of our obstacle problem converges to the solution of the corresponding elliptic obstacle problem. Finally, we compare the solution to a quasi-harmonic evolution problem from the solution to the stationary case.