Some noncoercive parabolic equations with lower order terms in divergence form

Abstract

This paper deals with existence and regularity results for the problem

$$ \left\{ \begin{gathered} {u_{t}} - {\text{div}}(a(x,t,u)\nabla u) = - {\text{div}}(u{\text{E}}), in \Omega x{\text{ }}(0,T), \hfill \\ u = 0 on \partial \Omega x (0,T), \hfill \\ u(0) = {u_{0}} in \Omega , \hfill \\ \end{gathered} \right. $$

, under various assumptions on E and u ⁰. The main difficulty in studying this problem is due to the presence of the term div(uE), which makes the differential operator non coercive on the “energy space” L²(0T; H ¹₀ (Ω)).