Some noncoercive parabolic equations with lower order termsin divergence form

This paper deals with existence and regularity results for the problem$ cases{u_t-mathrm{div}(a(x,t,u )nabla u)=-mathrm{div}(u,E)qquad in Omegatimes (0,T),cr u=0 qquad on partial Omegatimes (0,T), cr u (0)= u_0 qquad in Omega ,cr} $under various assumptions on E and $ u_0 $. The main difculty in studying this problem is due to the presence of theterm div(uE), which makes the differential operator non coercive on the "energy space" $ L^2 (0, T; H_0^1 (Omega)) $.AMS Subject Classification: 35K10, 35K15, 35K65.