Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form

summary:We prove the existence of solutions to nonlinear parabolic problems of the following type: $$ \begin {cases} \dfrac {\partial b(u)}{\partial t}+ A(u) = f + {\rm div}(\Theta (x; t; u))& \text {in}\ Q,\\ u(x; t) = 0 & \text {on}\ \partial \Omega \times [0; T],\\ b(u)(t = 0) = b(u_0) & \text {on}\ \Omega , \end {cases} $$ where $b\colon \Bbb {R}\to \Bbb {R}$ is a strictly increasing function of class ${\mathcal C}^1$, the term $$ A(u) = -{\rm div} (a(x, t, u,\nabla u)) $$ is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, $\Theta \colon \Omega \times [0; T]\times \Bbb {R}\to \Bbb {R}$ is a Carathéodory, noncoercive function which satisfies the following condition: $\sup _{|s|\le k} |\Theta ({\cdot },{\cdot },s)| \in E_{\psi }(Q)$ for all $k > 0$, where $\psi $ is the Musielak complementary function of $\Theta $, and the second term $f$ belongs to $L^{1}(Q)$