In this paper, we consider the global existence, uniqueness and $L^{\infty }$ estimates of weak solutions to quasilinear parabolic equation of $m$-Laplacian type $u_t-{\mathrm{div}\left(\right|\nabla u|}^{m-2}{\nabla u)=u|u|}^{\beta -1}{\int }_{\Omega }{\left|u\right|}^{\alpha }\mathrm{d}x$ in $\Omega \times (0,\infty )$ with zero Dirichlet boundary condition in $\partial \Omega$. Further, we obtain the $L^{\infty }$ estimate of the solution $u\left(t\right)$ and $\nabla u\left(t\right)$ for $t>0$ with the initial data $u_0\in L^q\left(\Omega \right)$$(q>...$

This paper proves the local well-posedness of strong solutions to a two-phase model with magnetic field and vacuum in a bounded domain $\Omega \subset {ℝ}^3$ without the standard compatibility conditions.