LAUSR

In this article, we deal with the existence result for nonlinear parabolic equations of the form $$\begin{aligned} \frac{\partial u}{\partial t}-{\text {div}}\>\mathcal {A}(x,t,u,\nabla u)-\hbox {div}\Phi (x,t,u)= f \quad \text {in }{\Omega _T=\Omega \times (0,T)}, \end{aligned}$$ with initial datum and source term which are only summable. The main term in the equation which contains the space derivatives is in divergence form. A lower order term is considered which also is in divergence form and satisfies a so-called natural growth condition written in terms of the same Musielak function $$\varphi $$ . The main aim of the paper is to establish the existence of an entropy solution to the described problem, the use of the concept of entropy solution is quite natural in this context.