Parabolic problems with slightly superlinear convection terms

In this paper we deal with a non-linear parabolic problem which involving a convection term with super--linear growth, whose model is $$\frac{\partial u}{\partial t}-\div(\mathcal{M}(x,t)\nabla u)= -\div(u\log (e+|u|)E(x,t))+f(x,t),$$ where $\mathcal{M}$ is a bounded measurable matrix, the vector field $E$ and the function $f$ belong to suitable Lebesgue spaces. We prove the existence of a unique bounded and unbounded weak solution.