Regularity results and asymptotic behavior for a noncoercive parabolic problem

In this paper we study the regularity and the behavior in time of the solutions to a quasilinear class of noncoercive problems whose prototype is $\begin{array}{c}\left(\begin{array}{cc}{u}_t-\mathrm{div}\left(a(x,t,u)\nabla u\right)=-\mathrm{div}\left(uE(x,t)\right)& \text{in}\Omega \times (0,T),\\ u(x,t)=0& \text{on}\partial \Omega \times (0,T),\\ u(x,0)=u_0\left(x\right)& \text{in}\Omega .\end{array}\right)\end{array}$ In particular we show that under suitable conditions on the vector field E, even if the problem is noncoercive and although the initial datum $u_0$ is only an $L^1\left(\Omega \right)$ function, there exist solutions that immediately improve their regularity and belong to every Lebesgue space. We also prove that solutions may become immediately bounded. Finally, we study the behavior in time of such regular solutions and we prove estimates that allow to describe their blow-up for t near zero.