Existence of global weak solutions to a parabolic $p$-Laplacian problem with convective term

For a given bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$, with $C^2$ boundary, and a given instant of time $T>0$, we prove the existence of a global weak solution on $(0,T)$, which satisfies a maximum principle, to a parabolic $p$-Laplacian system with convective term without divergence constraint for any $p\in (1,2)$.