Obstacle problem for semilinear parabolic equations with measure data

TL;DR

This paper establishes existence and uniqueness of solutions for a semilinear parabolic obstacle problem with irregular barriers and measure data, using probabilistic and approximation methods.

Contribution

It introduces a novel approach combining probabilistic techniques and penalization to solve obstacle problems with irregular barriers and measure data.

Findings

Proved existence and uniqueness of renormalized solutions.

Developed approximation schemes via penalization.

Applied probabilistic methods to analyze solutions.

Abstract

We consider the obstacle problem with two irregular reflecting barriers for the Cauchy-Dirichlet problem for semilinear parabolic equations with measure data. We prove the existence and uniqueness of renormalized solutions of the problem and well as results on approximation of the solutions by the penaliztion method. In the proofs we use probabilistic methods of the theory of Markov processes and the theory of backward stochastic differential equations.