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

Abstract

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 $b \Bbb{R}\to\Bbb{R}$ 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 $L^1(Q)$.

Keywords

• inhomogeneous Musielak-Orlicz-Sobolev space
• parabolic problems
• Galerkin method