Parabolic problems with slightly superlinear convection terms

TL;DR

This paper investigates a nonlinear parabolic PDE with a superlinear convection term, establishing the existence and uniqueness of both bounded and unbounded weak solutions under certain conditions.

Contribution

It introduces a novel analysis of a parabolic problem with superlinear convection, proving existence and uniqueness of solutions in a general setting.

Findings

Existence of weak solutions under specified conditions

Uniqueness of solutions in the considered framework

Solutions can be both bounded and unbounded

Abstract

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.