Quantitative and asymptotic estimates for parabolic obstacle problems with a drift term

We deal with the obstacle problem related to an operator with a drift-type lower-order term that in the linear case represents the one related to the Fokker–Plank equation, whose (normalized) solution describes the evolution of the probability density for a stochastic process. As the simplest possible model, we can consider the operator

The main novelty is the presence in the coefficient of the lower-order term of a singularity in the spatial variable. More precisely, we assume that the coefficient of the drift term lies in the Marcinkiewicz class weak-for a.e. time and satisfies the minimal time integrability assumption. The obstacle function is assumed to be time-continuous. Despite the lack of coercivity, we prove the well-posedness of a global solution to the obstacle problem and we describe the asymptotic behavior of such a solution. Moreover, we give quantitative asymptotic, stability estimates for the solutions to different problems. More precisely, we measure the distance in time of a solution to a parabolic obstacle problem from a solution to a stationary one. Fundamental tools in proving our results are a regularizing-in-time procedure and a suitable application of a Gronwall’s type lemma. A bound on the distance of from bounded functions is needed. However, this restriction holds whenever is in the Lebesgue space .