Parabolic obstacle problems with a drift term: Existence, uniqueness and asymptotic behavior

Abstract

In this paper we are concerned 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. The main novelty is the presence in the coefficient of the lower order term of a singularity in the spatial variable and minimal-in-time integrability assumption. 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, in the autonomous case, we prove that the global solution of our obstacle problem converges to the solution of the corresponding elliptic obstacle.