Well-posedness results for superlinear Fokker-Planck equations

TL;DR

This paper establishes existence and qualitative properties of solutions for a class of nonlinear superlinear Fokker-Planck equations with complex growth and structure.

Contribution

It provides new existence results and qualitative analysis for superlinear Fokker-Planck equations with non-standard growth conditions.

Findings

Existence of distributional solutions in $C([0,T),L^1)$

Qualitative properties of solutions analyzed

Applicable to equations with superlinear growth in $u$

Abstract

In this manuscript we deal with a class of nonlinear Fokker-Planck equations with the following structure \[ \partial_t u - \div\big(M\nabla u+ E h(u)\big)=0, \] 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.