Asymptotic behavior result for obstacle parabolic problems with measure data

The paper deals with the asymptotic behavior, as n tends to $$+\infty $$ , of distributional solutions for a class of variational parabolic inequalities of the following form: $$\begin{aligned}{(\mathcal {P})\left\{ \begin{aligned}&\int _{0}^{T}\langle (u_{n})_{t}, u_{n}-w\rangle +\langle -\text {div }a(t,x,\nabla u_{n}) ,u_{n}-w\rangle _{W^{-1,p'}(\varOmega ), W^{1,p}_{0}(\varOmega )} \mathrm{d}t\\&\quad \le \langle \mu ,u_{n}-w\rangle _{\mathcal {M}_{b}(Q),C^{0}(\overline{Q})},\quad \forall u_{n},w\in \mathcal {K}_{n},\\&\text { with }\mathcal {K}_{n}=\Big \lbrace z\in L^{p}(0,T;W^{1,p}_{0}(\varOmega ))\text { such that } z_{t}\in L^{p'}(0,T;W^{-1,p'}(\varOmega )),\ z(0,x)=0,\\&\qquad \qquad \qquad \quad \text { and }\Vert z\Vert _{L^{\infty }(0,T;W^{1,\infty }_{0}(\varOmega ))}:=\Vert \nabla z\Vert _{L^{\infty }(Q)}=\underset{t\in [0,T]}{\text {sup}}\Vert \nabla z\Vert _{L^{\infty }(\varOmega )}\le n\Big \rbrace \end{aligned}\right. }\end{aligned}$$ where $$\varOmega $$ is a bounded open set in $$\mathbb {R}^{N}$$ ( $$N\ge 2$$ ), $$T>0$$ , $$\mu $$ is a nonhomogeneous Radon measure and $$u\mapsto -\text {div }a(t,x,\nabla u_{n})$$ is a defined operator satisfying Leray–Lions assumptions. We provide a proof of the convergence of $$u_{n}$$ (solution of this problem) to the distributional solution u of the corresponding homogeneous parabolic equation $$u_{t}-\text {div }a(t,x,\nabla u)=\mu $$ in Q with $$u(0,x)=0$$ and the same datum $$\mu $$ .