Degenerate parabolic equations with singular lower order terms

Abstract

In this paper, we give existence and regularity results for nonlinear parabolic problems with degenerate coercivity and singular lower order terms, whose simplest example is \begin{eqnarray*} \begin{cases} u_t-\Delta_p u= {\frac{f(x,t)}{u^\gamma}} & \mbox{in}\;\Omega\times (0,T)\\ u(x,t)=0 & \mbox{on}\;\partial\Omega\times(0,T)\\ u(x,0)=u_0 (x) & \mbox{in}\;\Omega\; \end{cases} \end{eqnarray*} with $\gamma>0$, $p\geq 2$, $\Omega$ a bounded open set of $\mathbb{R}^{\mathrm{N}}$ ($N\geq 2$), $0 < T < +\infty$, $f\geq 0$, $f\in L^m(Q_T)$, $m\geq 1$ and $u_0\in L^\infty(\Omega)$ such that $$ \forall \, \omega\subset\subset\Omega\; \exists\;d_{\omega} > 0\,:\,u_{0}\geq d_{\omega}\;\mbox{in}\,\;\omega\,. $$ The aim of the paper is to extend the existence and regularity results recently obtained for the associated singular stationary problem. One of the main difficulties that arises in the parabolic case is the proof of the strict positivity of the solution in the interior of the parabolic cylinder, in order to give sense to the weak formulation of the problem. The proof of this property uses Harnack's inequality.