Well-posedness results for superlinear Fokker-Planck equations

In this manuscript we deal with a class of nonlinear Fokker-Planck equations with the following structure $${\mathrm{\partial }}_{t}u-÷(M\mathrm{\nabla }u+Eh(u))=0,$$ with $M$ a bounded elliptic matrix, $E$ a vector field in a suitable Lebesgue space, and $h(u)$ featuring a superlinear growth for $u$ large. We provide existence results of $C([0,T),L^1)$ distributional solutions to initial-boundary value problems related to the equation above together with some qualitative properties of solutions.