Non-stability result of entropy solutions for nonlinear parabolic problems with singular measures

Abstract

In this paper, we study the nonlinear parabolic equation given by $$\begin{aligned} u_{t}-\text {div}(a(t,x,\nabla u))+|u|^{q-1}u=f+\lambda ,\quad \text { in }(0,T)\times \Omega , \end{aligned}$$ where \(1 1\), \(f\in L^{1}(Q)\), \(\lambda \) is a measure concentrated on a set of zero parabolic r-capacity and \( u\mapsto -\text {div}(a(t,x,\nabla u))\) is a pseudo-monotone operator. We also consider the corresponding bilateral obstacle problem with measure data concentrated on a set of zero parabolic p-capacity whose model is $$\begin{aligned} \langle u_{t}-\text {div}(a(t,x,\nabla u))-\lambda , v-u\rangle \ge 0, \end{aligned}$$ with \(u\in K=\lbrace w\in L^{p}(0,T;W^{1,p}_{0}(\Omega )): |w|\le 1\rbrace \) for every \(v\in K\). We define a notion of entropy solutions, we give convergence properties essential to our proofs and we establish a non-stability result.