Entropy Unilateral Solution for Some Noncoercive Nonlinear Parabolic Problems via a Sequence of Penalized Equations

Abstract

We give an existence result of the obstacle parabolic equations $$\frac{\partial b(x,u)}{\partial t} -div(a(x,t,u,\nabla u))+div(\phi(x,t,u)) =f\quad \text{in}\ \ Q_T,$$ where $b(x,u)$ is bounded function of $u$, the term $- {\rm div}(a(x,t,u,\nabla u))$ is a Leray-Lions type operator and the function $\phi$ is a nonlinear lower order and satisfy only the growth condition. The second term $f$  belongs to $L^{1}(Q_T)$. The proof of an existence solution is based on the penalization methods.